Paper Title: Revisiting the Classical McKay Correspondence, Derived Equivalences and the Spectrum of Kleinian Surface Singularities: A Look Through the Mirror.




[2020]14-02,14E16,14F08, 14J33,14B05
	
	MIST: A Simple and Scalable End-To-End 3D Medical Imaging Segmentation Framework
    
Adrian Celaya1,2 Evan Lim1 Rachel Glenn1 Brayden Mi1 Alex Balsells1,2 Tucker Netherton1 Caroline Chung1 Beatrice Riviere2 David Fuentes1

    August 5, 2024
==============================================================================================================================================


To Kaxjuu, on the occasion of her first nine months in this world.

	
	§ ABSTRACT  In this article, we revisit the classical McKay correspondence via  homological mirror symmetry. Specifically, we demonstrate how this correspondence can be articulated as a derived equivalence between the category of vanishing cycles associated with a Kleinian surface singularity and the category of perfect complexes on the corresponding quotient orbifold. We further illustrate how this equivalence allows for the interpretation of the spectrum of a Kleinian surface singularity solely in terms of the representation-theoretic data of the associated binary polyhedral group.
	


	

§ INTRODUCTORY OVERVIEW

	
	

 §.§ Kleinian Surface Singularities

	
	In his seminal 1884 work “Lectures on the Icosahedron and Equations of the Fifth Degree” <cit.>, F. Klein accomplished, among many other things, the classification of all finite subgroups of (2, ℂ). Klein demonstrated that every finite subgroup G of (2, ℂ) must be isomorphic to one of three types: a cyclic group, a binary dihedral group, or a binary polyhedral group.
	
	
	
	
	We can arrive at this classification as follows. Choose a G-invariant Hermitian metric on ℂ^2. Consequently, any such G becomes conjugate to a subgroup of the corresponding special unitary group (2) ⊂(2, ℂ). On the other hand, we have the group extension:
	

		
			1 [r]     ℤ/2ℤ [r]     (2) [r, "π"]     (3) [r]     1
		
	

	
	This extension can be elegantly realized through Hamilton's algebra of quaternions:
	
    ℍ≅{ q = [  z  w; -w  z ] : z, w ∈ℂ}⊂ M_2(ℂ)

	
	Note that one can identify the special unitary group (2) with the group of unit quaternions ℍ^* := { q ∈ℍ : q^2 := (q) = 1 }≅(2), which acts on the Euclidean space ℝ^3 by conjugation after identifying the latter with the space of purely imaginary quaternions:
	
    ℝ^3 ≅{[  r√(-1)       z;      -z -r√(-1) ] : r ∈ℝ, z ∈ℂ}⊂ℍ.

	
	Thus, the assignment of taking a unit quaternion q ∈ℍ^* to the corresponding conjugation operator v ∈ℝ^3 ↦ q · v · q^-1 produces the two-fold covering of Lie groups π: (2) →(3). Now, there are two possibilities for a finite subgroup G ⊂(2). Either we have an induced group extension:
	
    1 →ℤ/2ℤ→ G →π(G) → 1,

	This occurs if and only if G is of even order, or we have an isomorphism G ≅π(G), if and only if G is a cyclic group of odd order. Therefore, the classification of finite subgroups of (2, ℂ) can be reduced to the classification of finite subgroups of (3). The problem of classifying finite rotation groups in three-dimensional Euclidean space dates back to the ancient Greeks. Indeed, a finite subgroup of (3) is either isomorphic to a cyclic group, a dihedral group, or the group of symmetries of a Platonic solid (see, for instance, Chapter 8 of <cit.> for a proof).
	
	
	
	Furthermore, Klein completely solved the invariant theory for each finite subgroup G ⊂(2, ℂ). Specifically, G acts on the polynomial algebra ℂ[u, v] by means of linear changes of variables. As Klein observed, there exists a triple of invariant polynomials x, y, z ∈ℂ[u, v]^G subject to a single algebraic relation f(x, y, z) = 0 for some quasi-homogeneous polynomial f ∈ℂ[x, y, z] such that ℂ[u, v]^G = ℂ[x, y, z]. This implies that the algebra of invariant polynomials ℂ[u, v]^G is finitely generated. Therefore, the corresponding quotient variety X = 𝔸^2_ℂ/G := (ℂ[u, v]^G) can be realized geometrically by means of the hypersurface X ≅ f^-1(0) defined by the regular map f: 𝔸^3_ℂ→𝔸^1_ℂ. The polynomials f identified by Klein are explicitly given in Table <ref>.
	
	

 §.§ Du Val's Resolution of Kleinian Singularities

	
	The simply laced Dynkin diagrams (see Fig. <ref>) are at the core of the so-called ADE pattern, which governs several distinct classification schemes in mathematics. For instance, simple Lie algebras, Coxeter reflection groups, quivers of finite type, and the Platonic solids are all primary examples of this mysterious universal pattern (see <cit.> for an exposition on this fascinating subject).
	
	
	
	In the 1920s, P. Du Val <cit.> discovered a remarkable relationship between Kleinian surface singularities and the simply laced Dynkin diagrams, unveiling another instance of the ADE pattern. The idea is as follows: for any finite subgroup G ⊂(2, ℂ), the quotient variety X := 𝔸^2_ℂ / G has a single isolated singularity that can be resolved by a finite sequence of blow-ups at certain points. In fact, we can produce a minimal resolution π: Y → X := 𝔸^2_ℂ / G in the sense of birational geometry. Each irreducible component of the corresponding exceptional divisor E ⊂ Y is a rational (-2)-curve, which means that it is isomorphic to the projective line ℙ^1_ℂ with self-intersection -2 within the smooth surface Y. Interestingly, the dual graph of the exceptional divisor E, which has a vertex for each of its irreducible components with an edge connecting them whenever they intersect, coincides with the corresponding simply laced Dynkin diagram. The type of this diagram is the same as that corresponding to the group G according to the ADE classification in Table <ref>.
	
	
	
	
	Moreover, Du Val also provided a characterization of Kleinian surface singularities as those isolated surface singularities that do not affect the conditions of adjunction (see <cit.> for a discussion of the various ways to characterize Kleinian surface singularities).
	
	

 §.§ The Classical McKay Correspondence

	
	As the 20th century progressed, several distinct aspects of Kleinian surface singularities and their role in the ADE classification became better understood. Notably, Brieskorn's solution <cit.> in 1970 of a conjecture by A. Grothendieck demonstrated the realization of Kleinian surface singularities in terms of the nilpotent variety of the simply laced semi-simple Lie algebra of the corresponding type. However, a straightforward relationship between the simply laced Dynkin diagrams and binary polyhedral groups was discovered by J. McKay <cit.> in 1980 within the realm of representation theory. As McKay observed, for any finite subgroup G of (2, ℂ), we can associate a certain quiver Q_G in the following manner. Let V denote the standard representation of G given by the inclusion G ⊂(2, ℂ). The vertices of the quiver Q_G correspond to the irreducible representations of G, and there is an edge connecting a pair of irreducible representations V_α and V_β whenever there is an inclusion V_α⊂ V ⊗ V_β. The resulting quiver Q_G is often called the extended McKay graph of G. On the other hand, the sub-quiver Q_G ⊂Q_G, obtained by deleting the vertex corresponding to the trivial representation of G, is referred to as the McKay quiver of G.
	
	
	
	
	In these terms, the McKay correspondence can be stated as an isomorphism of quivers:
	
    Q_G ≅ E^∨,

	where E^∨ denotes the quiver given by the dual graph of the exceptional divisor E of the minimal resolution π: Y → X := 𝔸^2_ℂ / G. This implies a one-to-one correspondence between the non-trivial representations of G and the irreducible components of the exceptional divisor E. A more geometric perspective on the above isomorphism is due to González-Sprinberg and Verdier <cit.>, who associated to each irreducible representation V_α of G a vector bundle ρ_α over Y such that the Chern classes c_1(ρ_α) provide a basis for the Picard group 𝖯𝗂𝖼(Y). This construction can be extended to an isomorphism in K-theory:
	
    (G) ≅ K_G(𝔸^2_ℂ) ≅ K(Y).

	Moreover, a stronger version of McKay's correspondence states that there is a triangulated equivalence of derived categories (see <cit.>):
	
    D^b_G(𝔸^2_ℂ) ≃ D^b(Y).

	This equivalence can be interpreted as the ability to read off the geometry of the minimal resolution Y directly from the representation theory of the finite group G, and vice versa.
	
	

 §.§ The Spectrum of an Isolated Hypersurface Singularity

	
	The notion of the spectrum of isolated hypersurface singularities was introduced in 1976 by J. Steenbrink <cit.> based on previous work by V. Arnold <cit.> (we also refer the reader to <cit.> for an introduction to this subject). Let us briefly recall some relevant notions. Consider a polynomial f ∈ℂ[x_0, …, x_n] with f(0) = 0 such that the central fiber f^-1(0) ⊂𝔸^n_ℂ has an isolated singularity at the origin. A classical result by J. Milnor <cit.> states that there exists a ball X := B_δ⊂𝔸^n+1_ℂ as well as a disc S := Δ_ϵ⊂𝔸^1_ℂ such that the restriction f: X^* → S^* is a C^∞-smooth fibration, where S^* := Δ_ϵ - {0} and X^* := X - X_0 for X_0 := f^-1(0) ∩ B_δ. In particular, the fibers X_t := f^-1(t) ⊂ X for any value of the parameter t ∈ S^* are all diffeomorphic. Moreover, each X_t has the same homotopy type as a bouquet of n-spheres, as many as the dimension of the Jacobian algebra of f:
	
    μ := _ℂℂ[x_0, …, x_n] / (∂_0 f, …, ∂_n f).

	Thus, by circling the origin 0 ∈ S along a path starting and ending at any fixed value t, we get the corresponding monodromy operator:
	
    T: H^n(X_t) → H^n(X_t).

	
	
	
	
	The celebrated Brieskorn's monodromy theorem <cit.> states that the above-defined operator T is quasi-unipotent. More precisely, we have (T^q - 1)^n+1 = 0 for some positive integer q. Hence, the eigenvalues of T are all roots of unity. On the other hand, a result by Steenbrink <cit.> states that there is a Mixed Hodge structure on the cohomology H^n(X_t). With the aid of this mixed Hodge structure, we can choose specific logarithms λ of the eigenvalues e^2πλ√(-1) of T together with some multiplicity m_λ. These rational numbers λ are called the spectral numbers of the singularity and can be conveniently organized into the so-called spectral polynomial:
	
    Sp(f) = ∑_λ∈ℚ m_λ t^λ∈ℤ[t^λ, λ∈ℚ].

	The spectrum of an isolated hypersurface singularity is a powerful invariant due to its remarkable properties. For instance, the semi-continuity of the spectrum under deformations led to a proof by A. Varchenko <cit.> of a conjecture by V. Arnold regarding an upper bound for the number of ordinary double points in a projective hypersurface.
	
	
	For instance, the spectrum of the Kleinian surface singularities in Table <ref> is depicted in Table <ref>. Naturally, a question arises: Is there a purely representation-theoretic interpretation for the spectral numbers associated with a finite subgroup of (2, ℂ)?
	
	

 §.§ A Look Through the Mirror

	
	Since the spectacular achievement by Candelas, de la Ossa, Green, and Parkes <cit.> regarding the counting of rational curves on a quintic threefold, the phenomenon of mirror symmetry observed in string theory has evolved into a rich theory at the interface between physics and mathematics. From a mathematical perspective, a possible rigorous formulation of this phenomenon is given by the celebrated homological mirror symmetry conjecture proposed by M. Kontsevich <cit.> at the International Congress of Mathematicians in 1994. In broad terms, certain types of geometric entities in a wide sense (ranging from varieties, stacks, singularities, or noncommutative spaces, etc.) can be associated with two types of models: the A-model (related to symplectic geometry) and the B-model (related to complex geometry), encoded by suitably defined categories of A-branes and B-branes, respectively. Homological mirror symmetry predicts that for any such geometric entity 𝒳, there exists a mirror dual 𝒳^∨ such that the corresponding A and B models are interchanged by means of categorical equivalences (we refer the reader to <cit.> for an accessible introduction to homological mirror symmetry):
	
    A()≃ B(^∨) and B()≃ A(^∨).

	
	In the context of the present work, homological mirror symmetry reveals itself in the following manner. For any finite subgroup G of (2, ℂ), let us consider the corresponding quotient orbifold 𝒳 := 𝔸^2_ℂ G. It turns out that the mirror dual of 𝒳 is given by the polynomial map f: 𝔸^2_ℂ→𝔸^1_ℂ associated with G as depicted in Table <ref>. In physical jargon, this map is the superpotential defining a Landau-Ginzburg model. As we will explain in Section <ref>, the A-model attached to the polynomial f: 𝔸^2_ℂ→𝔸^1_ℂ can be realized using the category of twisted complexes of the so-called directed Fukaya category ^→(f), which can be thought of as a categorification of a distinguished basis of vanishing cycles of the Milnor fibration f: X^* → S^*, that is, those cycles in H^n(X_t) that collapse to zero as t → 0. 
	
	On the other hand, as we will explain in Section <ref>, the B-model attached to 𝒳 is based on the category of perfect complexes (𝒳) ≅_G(𝔸^2_ℂ), that is, bounded complexes of G-equivariant vector bundles over the complex affine plane 𝔸^2_ℂ. Hence, as we argue in Section <ref>, the classical McKay correspondence can be stated as a triangulated equivalence:
	
    (^→(f)) ≃_G(𝔸^2_ℂ).

	
	Of course, the above derived equivalence provides only half of the whole homological mirror symmetry scheme taking place here. Let us mention that the B-model associated with the map f: 𝔸^2_ℂ→𝔸^1_ℂ can be related to the so-called category of matrix factorizations (f), which turns out to be equivalent to the category of singularities (X_0) over the central fiber X_0 := f^-1(0) (see <cit.>). On the other hand, the A-model associated with the orbifold 𝒳 should be given by a suitably defined equivariant Fukaya category _G(𝔸^2_ℂ). Thus, the remaining half of the homological mirror symmetry scheme would be determined by a corresponding triangulated equivalence:
	
    (f) ≃(_G(𝔸^2_ℂ)).

	
	Even though it would be quite interesting to have a complete picture of the above equivalence, as well as an interpretation of its meaning, we do not attempt to pursue these problems in the present work.
	
	
	

 §.§ The Stringy Spectrum of Orbifolds

	
	As the reader probably guessed, the directed Fukaya category ^→(f) encodes a great deal of the singularity invariants of f in some way. In fact, the spectrum of f can be extracted from the category of twisted complexes (^→(f)), as we will explain in Section <ref>. On the other hand, due to the triangulated equivalence (^→(f)) ≃_G(𝔸^2_ℂ) given by the McKay correspondence, it is reasonable to ask how one can compute the spectrum of f in terms of _G(𝔸^2_ℂ). Moreover, it turns out that the latter category is closely related to the representation theory of G, as we will explain in Section <ref>. Hence, one should be able to define the spectrum of f in terms of the representation-theoretic data coming from G. Indeed, as we will explain in Section <ref>, one can define the stringy spectrum of the orbifold 𝒳 = 𝔸^2_ℂ G as an invariant of the category _G(𝔸^2_ℂ), and this coincides with the classical spectrum of f, answering the question posed at the end of Section 1.4.
	
	

 §.§ Final Considerations

	
	We would like to wrap up this introduction by pointing out that there are multiple formulations of the mirror duality for ADE singularities in the existing literature, with various interpretations and levels of generality. For example, but not limited to, you can refer to <cit.>. As far as we are aware, the specific formulation of the classical McKay correspondence as a mirror duality that we present here is novel. However, it's important to note that as this is a survey article, we haven't included many details, as those will be included in future works. Additionally, the connection we've described in the last section of this article, regarding the representation of the spectrum in terms of orbifold data, is a new perspective. It provides the initial evidence that motivated us to develop a much more comprehensive theory of the stringy spectrum of orbifolds. This work serves as an initial announcement of the results, which will be further detailed in <cit.>.
	
	

§ THE A-SIDE: THE CATEGORY OF VANISHING CYCLES

	
	In this section, we will briefly describe the A-model associated with a Kleinian surface singularity. Fix a polynomial map f:^2_→^1_ as given in Table <ref> from now on. The last map can be thought as the superpotential defining a Landau-Ginzburg model. In this setting, one can construct the so-called directed Fukaya category ^→(f), an A_∞-category that enhances the collection of vanishing cycles of f. In this sense, the category ^→(f) acts as a receptacle for the singularity invariants attached to the surface defined by the zero loci V(f):=f^-1(0)⊂^2_. Moreover, the corresponding category of complex twisted (^→(f)) with its triangulated structure provides a suitable model for the A-branes of the Landau-Ginzburg model defined by f (we refer the reader to Chapter 3 of <cit.> for a readable discussion on this notion).    
	

 §.§ The Milnor fibration
 Fortunately, the polynomials of Table <ref> belong to a class of polynomials for which the corresponding singularity invariants can be described in rather explicit manner; namely, the class of weighted homogeneous polynomials. Consider a polynomial 
    f=∑_n=(n_x,n_y,n_z)∈^3c_n x^n_xy^n_yz^n_z
 in [x,y,z]. The support of f is defined as the set {n∈^3|c_n≠0} of ^3. We say that f in [x,y,z] is weighted homogeneous with weights (w_x,w_y,w_z)∈_>0 if its support is contained in the hyperplane of with equation 
    w_xn_x+w_yn_y+w_zn_z=1.
 Thus, if d the smallest common denominator of w_x,w_y,w_z and we define the integer numbers a=dw_x, b=dx_y and c=dw_z, we have 
    f(λ^ax,λ^by,λ^cz)=λ^df(x,y,z)
 for λ∈^*. In such a case, the restriction f:^3_-f^-1(0)→^1_-{0} turns out to be a C^∞-smooth fibration, called the global affine Milnor fibration of f. By moving around the unit circle in ^1_-{0}, we obtain the corresponding monodormy homeomorphism h:F→ F where F:=f^-1(1)⊂^3_ is the global affine Milnor fiber and 
    h(x,y,z):=e^2π√(-1)/d·(x,y,z).
 Moreover, if f has just an isolated singularity at the origin 0∈^3_, then the associate Milnor number is given by 
    μ:=(f)=d-a/a·d-b/b·d-c/c
 where 
    (f):=[x,y,z]/(∂_xf,∂_yf,∂_zf)
 is the Jacobian algebra of f. On the other hand, it is well-known that the global affine Milnor fiber F has the homotopy type of a bouquet of spheres ⋁_i=1^μ S^2, so that the cohomology of F is given by: 
    H^n(F,)=   if n=0
    ^μ   if n=2
    
    		0    otherwise
 In fact, we have the following (cf. Theorem 9.2.8 of <cit.>):
	
	 There are identifications H^2(F,)≅ H^2_(F)⊗≅Ω_f, where 
    Ω_f:=Ω^3_^3_/df∧Ω^2_^3_≅(f)· dx∧ dy∧ dz
 is the so-called Milnor module of f, induced by the residue map 
    ω∈Ω^3_^3_↦_F(ω)∈Ω^2_F
 where η:=_F(ω) satisfies ω=η∧df/f+ω' for some regular 3-form ω' in ^3_ 
	
	
	

 §.§ Vanishing cycles
 In fact, we can do better. Let A⊂^3_≥ 0 be a finite sub-set such that the monomials x^α_xy^α_yz^α_z for α=(α_x,α_y,α_z)∈ A for a basis of the Jacobian algebra (f). For α in A, define 
    n(α):=ω_x(α_x+1)+ω_y(α_y+1)+ω_z(α_z+1)
 and 
    ω_α:=x^α_xy^α_yz^α_z/(f-1)^[n(α)]dx∧ dy∧ dz.
 Thus, we have the following (which is a particular case of Theorem 9.2.8 of <cit.>):
	
	 The collection {[η_α]∈ H^2(F,)}_α∈ A where η_α:=_F(ω_α) form a basis of H^2(F,).
	
	Moreover, one can also prove a geometric version of the above cohomological result:
	
	 For each α in A, let V_α⊂ F be the Poincaré dual of η_α. Then, the collection {V_α}_α∈ A is distiguished basis of vanishing cycles for f.
	
	

 §.§ The directed Fukaya category
 We define ^→(f) as the A_∞-category whose objects are given by a distinguished basis of vanishing cycles {V_α}_α∈ A as in the last paragraph and whose morphism spaces are given by Lagrangian intersection Floer complexes. Roughly speaking, the corresponding triangulated category of twisted complexes (^→(f)) has for objects the pairs (C,d) where C belongs to the additive completion of ^→(F) and d is a cohomological differential operator on C satisfying the Maurer-Cartan equation (see Chapter 3 of <cit.> for details) 
	
	

§ THE B-SIDE: THE CATEGORY OF PERFECT COMPLEXES

	
	The goal of this section is to provide a brief description of the B-model associated to a Kleinian quotient singularity. For any finite sub-group G of (2,), the corresponding quotient variety X:=^2_/G is affine X≅([x,y]^G) with a single isolated singularity. On the other hand, the natural G-action on the canonical bundle ω_^2_ is trivial and, therefore, it descends into the canonical sheaf ω_X, which is trivial ω_X≅_X. In other words, the variety X is a singular and non-compact Calabi-Yau surface. In fact, we can do better by considering instead the quotient stack :=^2_ G, which is a Calabi-Yau orbifold. As is explain in Section <ref>, the orbifold  turns out to be the mirror dual of the Landau-Ginzburg model given by the corresponding Kleinian polynomial f_G:^2_→^1_ as in Table <ref>. In order to see this, we will define the B-model attached to  as the triangulated category of perfect complexes B():=(), as we presently describe in more detail.     
	
	

 §.§ The category of perfect complexes
 Let us start by recalling that there is a linear equivalence ()≅_G(^2_) between the category of coherent sheaves over  and the category of G-equivariant coherent sheaves over ^2_. Thus, we also have a triangulated equivalence between derived categories D^b()≅ D^b_G(^2_). Let us denote by ()⊂ D^b() the subcategory of perfect complexes, that is, those bounded complexes of coherent sheaves over  that are locally quasi-isomorphic to a bounded complex of vector bundles in the étale topology of . However, since the orbifold =^2_ G is a global quotient, it has the resolution property (which means that every coherent sheaf over  is the quotient of a vector bundle), and this implies in turn that every perfect complex over  is in fact quasi-isomorphic to a bounded complex of global vector bundles over  (see <cit.>). Let us denote by ()⊂() the exact sub-category of vector bundles over  (so that we have a corresponding equivalence ()≅_G(^2_)). Thus, we have the following:
	
	 There is a triangulated equivalence ()≃ D^b(()).
	
	

 §.§ The skew group algebra
 There is rather nice algebraic description of the category of perfect complexes () which runs as follows. Note that the abelian category () has a projective generator. Indeed, if [G] denotes the regular representation of G, the trivial vector bundle [G]×^2_→^2_ can be endowed with a G-equivariant structure by means of the diagonal G-action on the total space [G]×^2_. We will denote by _[G] the corresponding stacky vector bundle over =^2_ G. Hence, it is easy to see that any coherent sheaf  over  admits an epimorphism _[G]^n→→ 0 for some positive integer n. Thus, _[G] provides a projective generator of (), as have been claimed. Note that, for any coherent sheaf  over , the linear space (_[G],) acquires the structure of a module over the endomorphism algebra (_[G]) by composition of morphisms. The last immediately implies that:
	
	 The assignment ↦(_[G],) induces a linear equivalence of categories ()≃((_[G]))
	
	In fact, it is well-known that there is a much more concrete description of the endomorphism algebra (_[G]). Note that the group G acts on the polynomial algebra [x,y]. In fact, a matrix g=[ a b; c d ] in G⊂(2,) acts on a given polynomial f in [x,y] by changing variables f^g(x,y)=f(ax+by,cx+dy). We can define the skew group algebra [x,y]⋊ G as the -algebra generated by the pairs (f,g) where f∈[x,y] and g∈ G, subject to the relations 
    (f_1,g_1)· (f_2,g_2)=(f_1f_2^g_1,g_1g_2).
 It turns out that the quotient orbifold =^2_ G is affine and the skew group algebra [x,y]⋊ G plays the role of its algebra of regular functions (which is not commutative in general). More precisely, we have the following: 
	
	 There is an isomorphism of algebras (_[G])≅[x,y]⋊ G. In particular, there is an equivalence ()≃([x,y]⋊ G) between the category of vector bundles over  and the category of finitely generated projective modules over [x,y]⋊ G, and this induces a triangulated equivalence: 
    ()≃ D^b(([x,y]⋊ G))

	
	

 §.§ Orbifold cohomology
 The triangulated category () can be thought as a sort of universal invariant of the orbifold . In principle, any additive invariant of , like the orbifold Euler number χ^() or the orbifold K-theory K^(), can be extracted out of () by means of a definite procedure. In this work, we are mainly interested in the so-called orbifold cohomology (sometimes also referred as the Chen-Ruan cohomology) introduced in <cit.>. Let us recall that the orbifold cohomology of  with complex coefficients is defined by 
    H_^∙(,):=H^∙(Λ,)
 where Λ is the inertia orbifold of  and the right hand side in the above equality is the usual cohomology of Λ. On the other hand, due to the localization principlie for orbifold theories, we can write down the orbifold cohomology of  as 
    H_^∙(,)≅⊕_α∈π_0(Λ)H^∙-(α)(_α,)
 where _α⊂ is the connected component corresponding to α and (α) denotes its age (see <cit.>). In our particular case where =^2_ G is a global quotient orbifold, each _α is contractible and we get an identification 
    H_^∙()≅(G)⊗_
 where (G) denotes the representation ring of G. However, note that we have an additional grading by age in H_^∙(,), which will become relevant later on. Finally, let us just mention that the orbifold cohomology of  can be recovered out of () as the periodic cyclic cohomology of this last (see <cit.>):  
	
	 There is an identification H^∙_(,)≅^∙((X)) between the orbifold cohomology of  and the periodic cyclic cohomology of ().
	
	

§ THE CLASSICAL MCKAY CORRESPONDENCE AS A MIRROR DUALITY

	
	This section will formulate the classical Mckay correspondence as a mirror duality relating the A-side of a Kleinian surface singularity and the B-side of the quotient orbifold of the associated binary polyhedral group. To this end, we will show that each side can be realized by means of the derived category of the Dynkin quiver of the corresponding type.
	
	

 §.§ The simply laced Dynkin diagrams
 By a quiver we just mean a finite directed graph. A representation V of a quiver Q consists of finite-dimensional vector space V_q for each vertex q of Q and a linear map T_l:V_q→ V_p for each directed edge l:q→ p. It turns out that there is an equivalence 
    (Q)≅(P_Q)
 between the category of representations of Q and the category of finitely generated modules over the path algebra P_Q. We will say that Q is of finite type iff there is a finite collection of indecomposable representations of Q. As another instance of the  pattern, a celebrated result of P. Gabriel <cit.> states that (we also refer the reader to Chapter 6 of <cit.> for a proof of this fact):
	
	 The quivers of finite type are given exactly by the simply laced Dynkin diagrams of  type as those in Figure <ref>.
	
	

 §.§ The A-side
 For any polynomial f as those in Table <ref>, the distinguished basis of vanishing cycles {V_α}_α∈ A given in Theorem <ref> provides a strong exceptional collection in the triangulated category (^→(f)). Moreover, the corresponding semi-orthogonal decomposition of the last category can be encoded by a Dynkin quiver Q_f, called the quiver of vanishing cycles of f, having the same  type as the polynomial f. Thus, we have the following:
	
	 There is a triangulated equivalence (^→(f))≃ D^b(Q_f).
	
	

 §.§ The B-side
 On the other hand, for any finite sub-group G of (2,) we have the corresponding quotient orbifold :=^2_ G. Hence, it turns out that the connected components of the inertia orbifold Λ, which are sometimes called the twisted sector of , give rise to a strong exceptional collection in the triangulated category (). Similarly as in paragraph (4.2), this induces a semi-orthogonal decomposition of () which can be encoded by the Mckay quiver Q_G. In this manner, we have the following:  
	
	 There is a triangulated equivalence _G(^2_)≃ D^b(Q_G).
	
	

 §.§ The classical Mckay correspondence
 Now, suppose that G is a finite sub-group of (2,) and f:^3_→^1_ the corresponding polynomial map as in the classification of Table <ref>. Thus, we have the Mckay quiver Q_G and the quiver of vanishing cycles Q_f, respectively, as in paragraphs (4.2) and (4.3). Hence, the classical Mckay correspondence can be stated as follows:
	
	 There is an isomorphism of quivers Q_f≅ Q_G.
	
	Finally, ass a collective consequence of Theorems <ref>,<ref> and <ref>, we get the following homological form of the classical Mckay correspondence:
	
	There is a triangulated equivalence:
    (^→(f))≃_G(^2_)

	
	

§ THE SPECTRUM OF KLEINIAN SURFACE SINGULARITIES

	
	To conclude this article, we will explain how one can extract the spectrum of a Kleinian surface singularity out of the representation theoretic data coming from the corresponding binary polyhedral group, answering the question stated at the end of paragraph (1.4).
	
	

 §.§ The Picard-Lefschetz formula
 Let {V_α}_α∈ A be a distinguished basis of vanishing cycles as in paragraph (2.2). We denote by δ_α the reflection along the normal hyperplane to the class [η_α]∈ H^2(F,) with respect to the intersection form of H^2(F,). Thus, the cohomological monodromy operator 
    T:=h^*:H^2(F,)→ H^2(F,)
 is given by the Picard-Lefschetz formula:
    T=∏_α∈ Aδ_α
 Note that, in this case, the monodromy operator T is of finite order. In fact, we have T^d=1.
	
	

 §.§ The Milnor module
 Note for a weighted homogeneous polynomial f with weights (w_x,w_y,w_z)∈_>0^3, the Milnor module 
    Ω_f:=Ω^3_^3_/df∧Ω^2_^3_
 introduced in Paragraph 2.1 has a -grading defined in the following manner. We define the degree of a monomial x^n_xy^n_yz^n_z dx∧ dy∧ dz as:
    (x^n_xy^n_yz^n_z dx∧ dy∧ dz):=a(n_x+1)+b(n_y+1)+c(n_z+1)
 Thus, for any rational number λ we define Ω_f,λ as the sub-module of Ω_f whose elements are given by those polynomials whose constituent monomials have degree equal to λ. In this manner, we get a decomposition:
    Ω_f:=⊕_λ∈Ω_f,λ
 It turns out that one can write down the spectral polynomial of f explicitly as the graded dimension of Ω_f, namely:
	
	 The identity (f)=∑_λ∈(Ω_f,λ)t^λ holds true.
	
	On the other hand, we can recover the Milnor module Ω_f out of the triangulated category (^→(f)) as follows:
	
	 There is a canonical identification Ω_f≅((^→(f))) between the Milnor module of f and the periodic cyclic cohomology of (^→(f)).
	
	Additionally, one can recover the Hodge filtration F^∙ of Ω_f by taking the graded piece F^pΩ_f as the subspaces generated by the elements η_α in the distinguished basis having n(α)≤ n-p+1.
	
	

 §.§ The stringy spectrum
 Of course, due to the triangulated equivalence 
    _G(^2_)≃(^→(f))
 
	given by the homological Mckay correspondence as in Theorem <ref>, we have the following consequence of Theorem <ref>:
	
	 There is an identification Ω_f≅ H_^∙(,) where :=^2 G. 
	
	Moreover, it turns out that, under the above identification 
    Ω_f≅ H_^∙(,),
 the Picard-Lefschetz formula regarding the monodromy operator on Ω_f can be translated in terms of the Frobenius algebra structure of H_^∙(,), which is given using the so-called Chen-Ruan product (see <cit.>). One can also recover the -grading of Ω_f from a suitably defined -grading on H_^∙(,) which is related to the grading by age. Therefore, due to Theorem <ref>, it should be possible to recover the spectral polynomial of f from the representation ring 
    (G)≅ H_^∙(,).
 However, we must conclude here and refer the reader to <cit.> for a complete account of this story.
	
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 Paper Title: Enumerating Finite Braid Group Orbits on $SL_2(\C)$-Character Varieties.







 









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	Enumerating Finite Braid Group Orbits on SL_2()-Character Varieties
    Amal Vayalinkal
    August 5, 2024
===================================================================

§ ABSTRACT 
		We analyze finite orbits of the natural braid group action on the character variety of the n times punctured sphere. Building on recent results relating middle convolution and finite complex reflection groups, our work implements Katz's middle convolution to explicitly classify finite orbits in the SL_2()-character variety of the punctured sphere. We provide theoretical results on the existence of finite orbits arising from the imprimitive finite complex reflection groups and formulas for constructing such examples when they exist. In the primitive finite complex reflection groups, we perform an exhaustive search and provide computational results. Our contributions also include Magma computer code for middle convolution and for computing the orbit under this action when it is known to be finite. 
	

	

§ INTRODUCTION

	Dynamics of the natural braid group action on character varieties has been studied for over a century. The growing interest in classifying the finite orbits of this action has led to a wealth of new techniques and examples. Inspired by Boalch <cit.> who explored finite complex reflection groups as a source of finite orbits, Lam-Landesman-Litt <cit.> showed, up to mild conditions, all finite orbits come from finite complex reflection groups via Katz's middle convolution operation <cit.>. With the goal of completing their analysis, we search the finite complex reflection groups to enumerate all possibilities and study them. We classify all finite orbits arising via middle convolution from the finite complex reflection groups. 
	
	Let Σ_0, T+1 denote the T+1-punctured sphere. Given T and n, a rank n representation of the fundamental group of Σ_0, T+1 can be identified with a tuple of matrices (A_1, … A_T+1) ∈ GL_n()^T+1 satisfying 
    ∏_i = 1^T+1 A_i = Id

	
	The mapping class group of Σ_0, T+1, denoted  _0, T+1 has a well-defined action on such representations. It is well known that _0, T+1 is isomorphic to the spherical braid group on T+1 strands, denoted B_T+1.  This group has T generators σ_1,  …, σ_T and acts on T+1-tuples in the following way: 
	
    σ_i(A_1, A_2, …, A_T, A_T+1) = (A_1, …, A_i-1, A_iA_i+1A_i^-1, A_i, A_i+2, …, A_T+1)
	
	If we further consider such representations up to simultaneous conjugation, this action descends to an action on the character variety i.e. on the isomorphism classes of representations.
	
		Conjugacy classes of tuples [A_1, ⋯, A_T+1] with finite orbit under this action are called MCG-finite and we say a tuple is irreducible if the associated representation is irreducible.
	

		Recently, Lam-Landesman-Litt <cit.> classified finite orbits of rank two tuples with at least one matrix of infinite order.  They proved that such Zariski dense representations come from pullbacks of curves or arise via middle convolution from finite complex reflection groups.
	
	The first kind, “pullback" representations were classified by Diarra <cit.>. 
	
	Finite irreducible complex reflection groups, which can be viewed as generalizations of Coxeter groups, were classified by Shephard and Todd in 1954 as belonging to an infinite family G(m, p, n) for each rank n or being one of 34 exceptional cases denoted G_3 to G_37 [9]. In view of this classification, Lam-Landesman-Litt <cit.> prove that for T+1 ≥ 7, MCG-finite representations do not come from groups of rank five or greater. Their paper leaves open for analysis the infinite family of rank three and four and most of the exceptional groups of rank three and four, as well as the case when T + 1 < 7, which we investigate.
	
	Katz <cit.> originally introduced middle convolution in the language of perverse sheaves and one of our contributions is implementing computer code to explicitly compute the middle convolution given a tuple of matrices, following the algebraic reformulation of Dettweiler-Reiter <cit.>. We also provide code for computing the braid group orbit for MCG-finite 4-tuples and 5-tuples in the  SL_2()-character variety of the punctured sphere. This computer code is publicly accessible online at the following link:  <amalvayalin.github.io/research.html> and we hope it encourages further applications of this operation.
	
	
		A 𝐧𝐢𝐜𝐞 T-tuple is a tuple (up to simultaneous conjugation) from a finite complex reflection group such that its middle convolution produces a rank 2 MCG-finite T+1-tuple. 
	
	For the precise definition of when a tuple is nice, see Definition <ref>. An explanation for how we get MCG-finite T+1-tuples from a T-tuple is provided in Remark <ref>. Since middle convolution depends on a parameter λ (see Section <ref>), the same nice T-tuple can give rise to distinct conjugacy classes of rank 2 MCG-finite T+1-tuples, as discussed in Remark <ref>.
	
	With this definition in mind, our problem of studying rank 2 MCG-finite tuples changes to studying nice tuples in the finite complex reflection groups. 
	Using properties of complex reflection groups, as well as prior work on this topic, it remains to study nice n and n+1-tuples in the rank n = 3, 4 primitive groups, nice n+1-tuples in the imprimitive groups of rank 2 <n ≤ 5 and  nice n-tuples in imprimitive groups of rank n ≥ 3. More details on why these are the only remaining cases is provided in Remark <ref>, Remark <ref> and <cit.>.
	
	
	In the imprimitive groups, we either prove no such nice tuples exist, or provide a system of equations for an infinite family of solutions. We found nice tuples in all the primitive reflection groups of rank three. Among the rank four primitive reflection groups, all but G_29 and G_31 contain nice tuples. For each nice tuple in the primitive groups, we provide representatives (up to simultaneous conjugation and braid group orbit equivalence) before and after middle convolution. We also list the size of the subgroup generated by the rank 2 matrices, and in most cases we provide an approximate `size' for the braid group orbit. 
	 
		There are no nice n-tuples in G(m, p, n) for n ≥ 5. 
	
	
		In the rank 3 imprimitive groups, an infinite family of nice 3-tuples exist in G(m, 1, 3) and G(m, m, 3) and in each case we provide conditions on the eigenvalues of the product of the matrices in the tuple. Nice 4-tuples exist in G(m, p, 3) for m >1 if and only if one of the following holds: 
		
			
  * p = m and m ≠ 3. 
			
  * p = 1 and (m, 3) = 1.
			
  * p ≠ 1, m and p = 3. 
		
		In all three cases the parameter for middle convolution is a root of unity of order m.
	
	 
		In the rank 4 imprimitive groups, nice 4-tuples and nice 5-tuples exist in G(m, p, 4) if and only if p = m and m =2 or m = 4. 
		
	The proof of these theorems, as well as further conditions on the eigenvalues of the product of the matrices and equations for how to construct such examples are provided in Section <ref>.
	
		In our computations for the primitive groups, nice tuples within each group are partitioned into different 𝐭𝐲𝐩𝐞𝐬 based on eigenvalues (see Definition <ref> and Remark <ref>). 
	 
		For each primitive reflection group of rank three, we summarize the distinct rank 2 MCG-finite tuples (up to simultaneous conjugation and braid group equivalence) produced by taking middle convolution of the different nice tuples (up to type equivalence). We also note how many of the rank 2 tuples generate an infinite subgroup. 
		
			
  * In G_23, there are three nice 3-tuples, resulting in six distinct MCG-finite 4-tuples after middle convolution. Three of the six generate infinite subgroups. This group also has one nice 4-tuple, which after middle convolution produces one MCG-finite 5-tuple that generates an infinite subgroup. 
			
  * In G_24,  there is only one nice 3-tuple, which produced three distinct MCG-finite 4-tuples after middle convolution, all generating infinite subgroups. There does not exist any nice 4-tuples in this group. 
			
  * In G_25, there are four nice 3-tuples, resulting in eight distinct MCG-finite 4-tuples after middle convolution. Five of these generate infinite subgroups. This group also has three nice 4-tuples, which produced two distinct MCG-finite 4-tuples after middle convolution. Only one of these 4-tuples generates an infinite subgroup. 
			
  * In G_26, there are three nice 3-tuples, resulting in seven distinct MCG-finite 4-tuples after middle convolution. Five of these generate an infinite subgroup. In this group there are no nice 4-tuples. 
			
  * In G_27, there are three nice 3-tuples, resulting in nine distinct MCG-finite 4-tuples after middle convolution. Six of these generate an infinite subgroup.
			This group also has one nice 4-tuple, which produced one MCG-finite 5-tuple after middle convolution. This 5-tuple does not generate an infinite subgroup.
		
	
	
	 
		For each primitive reflection group of rank four, we summarize the distinct rank 2 MCG-finite tuples (up to simultaneous conjugation and braid group equivalence) produced by taking middle convolution of the different nice tuples (up to type equivalence). Only nice tuples in G_32 produces an MCG-finite tuple that generates an infinite subgroup after middle convolution. 
		
			
  * In G_28, there is only one nice 4-tuple, which produces one MCG-finite 5-tuple after middle convolution. 
			
  * In G_29 there are no nice 4-tuples nor any nice 5-tuples. 
			
  * In G_30, there are three nice 4-tuples, each producing one distinct MCG-finite 5-tuple after middle convolution. For each of these 5-tuples, after tensoring with a character so the matrices are in SL_2( ) we end up with one matrix being identity, which means these tuples reduce to MCG-finite 4-tuples instead. There are no nice 5-tuples in this group. 
			
  * In G_31 there are no nice 4-tuples nor any nice 5-tuples.
			
  * In G_32 there are three nice 4-tuples, resulting in four distinct MCG-finite 5-tuples after middle convolution. One of these 5-tuples generates an infinite subgroup. After tensoring with a character so the matrices are in SL_2(), for two of the remaining three 5-tuples, we get the identity matrix, so these reduce to MCG-finite 4-tuples instead. This group also contains at least two nice 5-tuples, producing at least one example of an MCG-finite 6-tuple.  The 6-tuple did not generate an infinite subgroup. The search for nice 5-tuples in this group was too large computationally, so this list may be incomplete. 
					
		
	

 §.§ Prior Work

	MCG-finite representations correspond to algebraic solutions to nonlinear differential equatons, such as the Painlevé VI equation and more generally the  Schlesinger equations <cit.>.
	Regarded as a nonlinear analogue of the hypergeometric equation, the Painlevé VI (PVI) equation posed over 100 years ago, has inspired much work on the study of finite orbits. Algebraic solutions to the PVI equation correspond to irreducible MCG-finite representations when n = 2 and T+1 = 4 and motivates progress towards a nonlinear Schwartz list <cit.>. Among the many advances made in this area, notable contributions on finite orbits come from Hitchin <cit.>, Dubrovin-Mazzocco <cit.>, Boalch <cit.>, Kitaev <cit.> and an eventual computer-aided classification by Lisovy-Tykhyy <cit.>. 
	
	The work of Dubrovin-Mazzocco originally studied the PVI equation in connection with reflection groups <cit.>. Expanding on this idea,  solutions constructed from finite reflection groups through special cases of middle convolution, like the “Okamoto transform,"  were explored by others in <cit.>, <cit.>, <cit.>. More generally, a computer-aided classification was provided by Tykhyy in <cit.>, with explicit orbits listed when T+1 = 4, 5 and conjectural descriptions of the different possibilities for T+1 ≥ 6.  We found exact matches for orbits given in this list, except in two cases where the closest match differs in just one choice of parameter. As both these examples almost match with the same orbit they provided, we suspect this difference might be due to a typo. Details on these two tuples are provided in Section <ref> and Section <ref> and the differing coordinates are marked with an asterisk in the tables.
	
	The connection between finite complex reflection groups and middle convolutions was further explored more recently in the work of Lam-Landesman-Litt <cit.> who proved that all MCG-finite tuples with at least one matrix of infinite order are either pullback representations (classified in <cit.>) or come from finite complex reflection groups via middle convolution. 
	
	
	Our work investigates the finite complex reflection groups for MCG-finite tuples and examines their middle convolution in relation to the braid group orbit. In the applicable cases, we also explore how the examples we find relate to the classifications provided by Lisovy-Tykhyy <cit.> and Tykhyy <cit.>. Note that the analysis in these two papers exclude the tuples where all the matrices share a common eigenvector; these correspond to non-irreducible cases and were classified by Cousin-Moussard <cit.>.
	
	

§ BACKGROUND

	

 §.§ Middle Convolution
 
	Defined by Katz <cit.>, middle convolution  is an invertible functor in the category of perverse sheaves depending on some parameter λ∈^× and is denoted MC_λ. Middle convolution can also be viewed as a functor in the category of local systems on the punctured sphere to itself, and with this interpretation, Katz proved that all irreducible rigid local systems on the punctured sphere can be constructed from rank 1 local systems via scaling or middle convolution.  
	
	Letting k be a field and F_T denote the free group on T generators, middle convolution can equivalently be understood as a functor on the category of finite dimensional k[F_T] modules to itself as described by Dettweiler-Reiter <cit.>. The relation to local systems on the T+1-punctured sphere is by taking k = and viewing a tuple of T matrices as defining the action of [F_T] on the module i.e. a representation of [F_T]. In this way irreducible tuples correspond to irreducible representations.
	While middle convolution preserves irreducibility <cit.>, it typically changes the dimension of the module (rank of the local system on T+1-punctured sphere) and does not preserve basic properties e.g. finiteness of the subgroup generated by the matrices (i.e. monodromy group). Important to the study of finite orbits, one desirable property of middle convolution is that it is equivariant with the action of the braid group B_T <cit.>. One way to construct nontrivial finite orbits is to take irreducible tuples generating a finite subgroup, hence finite braid group orbit, and taking the middle convolution to (hopefully) get a tuple generating an infinite group which will necessarily be irreducible and have finite braid group orbit. 
	
	The purely algebraic analogue by Dettweiler-Reiter <cit.> gives an explicit description of middle convolution in terms of tuples of matrices and relates some properties of middle convolution to the matrices in the tuple.
	
		Given a tuple of T matrices A = [A_1, A_2, …, A_T] with A_i ∈ GL_n(), the 𝐩𝐫𝐨𝐝𝐮𝐜𝐭 of such a tuple is the product of the matrices in the tuple, in the ordering given by the tuple. So the product of the tuple 𝐀 is A_1A_2⋯ A_T. The 𝐢𝐧𝐯𝐞𝐫𝐬𝐞 of the tuple A is the tuple [A_T^-1, …, A_1^-1], denoted 𝐀^-1. The 𝐢𝐧𝐯𝐞𝐫𝐬𝐞 𝐩𝐫𝐨𝐝𝐮𝐜𝐭 of a tuple is the matrix product of the inverse tuple.
	
	
		Given a tuple of T matrices A = [A_1, A_2, …, A_T] with A_i ∈ GL_n(), we denote the middle convolution of 𝐀 with parameter λ∈^× as MC_λ(A) = [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃]. Note that MC_λ(A) will again be a tuple of T matrices, well-defined up to simultaneous conjugation, with Ã_̃ĩ∈ GL_n'() for some n' not necessarily equal to n. 
	
	
	
		Middle convolution does not preserve the product of the matrices as it is only well-defined up to simultaneous conjugation. 
	
	
	Letting V denote the underlying vector space associated to the [F_T] module defined by a tuple 𝐀. Then the relation between n and n' is as follows. 	
	
		Let V denote the [F_T]-module defined by the tuple A = [A_1, A_2, …, A_T] and let λ∈^×. 
		
			
  * If λ≠ 1 then 
    (MC_λ(V)) = ∑_i = 1^T(A_i -I) - ((V) - (λ· A_1A_2 ⋯ A_T -I))

		
	 
	
		Proofs of this statement, as well as further details and properties of the middle convolution can be found in <cit.> and <cit.>. 
	
	To get an MCG-finite T+1-tuple of rank 2, we want to find a T-tuple of reflections (hence (A_i -I) = 1) such that their product has a nontrivial eigenvalue of multiplicity T-2. This is why we search for nice tuples, Definition <ref> in Section <ref>.

	Another property of middle convolution that will greatly simplify our computations is the following. 
	 (Dettweiler and Reiter, 2000)
		If 𝐀 is an irreduble tuple of rank greater than 1, then the modules MC_λ^-1(𝐀^-1) and MC_λ(𝐀)^-1 are isomorphic. 
	
	
		Proved in <cit.>.
	
	
	Our code for middle convolution uses the formulation given by Dettweiler-Reiter in <cit.> as opposed to their formulation in <cit.> which only differs by changing the order of blocks in the matrices. The code requires a matrix group W containing the matrices in the tuple 𝐀 and the base ring of W should contain the parameter λ used for middle convolution. Note that the final function Compute_MC will automatically check if our intermediate constructions act invariantly on the subspaces K and L, as defined in the notation of <cit.>. The full computer code can be found online at  <amalvayalin.github.io/research.html>.
	
	

 §.§ Finite Complex Reflection Groups
 
	A real reflection can be identified as a matrix with one eigenvalue -1 and all other eigenvalues are 1, and pseudo-reflections refers to matrices with one eigenvalue a root of unity and all other eigenvalues 1.  A finite complex reflection group (FCRG) is a finite group generated by pseudo-reflections. 
	As we are only working with complex groups, the distinction between real or complex eigenvalues is not relevant and henceforth reflections can refer to real reflections or pseudo-reflections.
	
	All irreducible finite complex reflection groups  were classified by Shephard-Todd <cit.>, as belonging to an infinite family of imprimitive reflection groups or being one of 34 exceptional primitive cases. 
	 
		The rank of a reflection group is the smallest dimension of the vector space on which the group acts irreducibly.
		Every finite complex reflection group of rank n has a minimal generating set consisting of either n or n+1 reflections. The group is said to be well-generated if it can be generated by n reflections. 
	
	As we want irreducible MCG-finite T+1-tuples, we know that our nice T-tuple (see Defintion <ref>) must at least be of length n to generate an irreducible FCRG, so in each group we only need to search for T-tuples with T > n. Among the rank three and four primitive irreducible groups, only G_31 is not well-generated (see Cartan matrices in <cit.>. An imprimitive group G(m, p, n) is well-generated if p = m or p = 1. 
	
	Associated to each FCRG of rank n is a set of n invariants called the degrees of the reflection group <cit.>, <cit.>. An important property of these invariants relevant for our purposes is their relation to the eigenvalues of the group elements. Note that we already know the eigenvalues for any element of the group must be roots of unity since the group is finite. 
	
		Let W be a FCRG of rank n and let {d_i}_i = 1, …, n denote the degrees of W. Suppose λ is some primitive d-th root of unity that is an eigenvalue for some element of W. Then, d | d_i for some degree d_i. Moreover, the maximum dimension of the eigenspace corresponding to λ for any element of the group is the number of degrees d_i that is divisible by d, and for some element in W the eigenspace corresponding to λ achieves this maximal dimension. 
	
	
		Proved in <cit.>.
	
	The theorem above will be helpful in our computations for the primitive groups as we know how large our field needs to be to ensure the code finds all eigenvalues. 
	
	Further properties specific to imprimitive reflection groups are described in Section <ref>.
	
	

 §.§ Representations and Character Varieties

	
	Given a finitely generated group Γ, a representations of Γ is a homomorphism ρ: Γ⟶ G where G < GL_n() is some algebraic group. Considering the vector space V the matrices ρ(γ) act on, ρ is said to be irreducible if there is no nonzero subspace U ⊊ V that is left invariant by the matrix ρ(γ) for all γ∈Γ. Two representations ρ and ρ' are isomorphic if there exists some g ∈ G such that ρ(γ) = g^-1ρ'(γ) g for all γ∈Γ. 	The character of a representation  ρ is the map χ_ρ: Γ⟶ given by χ_ρ(γ) = ρ(γ). Isomorphic representations have the same character since trace is fixed under conjugation. The character variety 𝔛(Γ, G) parametrizes the isomorphism classes of representations ρ: Γ⟶ G in terms of suitable traces.  
	
	In the case of the punctured sphere Σ_0, T+1, the fundamental group is π_1(Σ_0, T+1) = γ_1, …, γ_T+1|γ_1*γ_2*⋯ *γ_T+1 = 1. Any representation of Γ = π_1(Σ_0, T+1) is identified by ρ(γ_i) = M_i ∈ G such that M_1 · M_2 ⋯ M_T+1 = I is the identity matrix. Two such representations  ρ and ρ' are equivalent if the tuple [ρ(γ_1), …, ρ(γ_T+1)] is related to [ρ'(γ_1), …, ρ'(γ_T+1)] by simultaneous conjugation.
	
	Taking G = SL_2(), we can consider the SL_2()-character variety of π_1(Σ_0, T+1). 
	
	Given a tuple of of matrices [M_1, M_2, …, M_T+1], let 
    t_i = (M_i)      t_ij = (M_iM_j)      t_ijk = (M_iM_jM_k) *for 1≤ i < j< k ≤ T+1

	Since π_1(Σ_0, T+1) = F_T we have a natural inclusion 𝔛(π_1(Σ_0, T+1), SL_2()) ⊆𝔛(F_T, SL_2()). This means for our purposes, M_T+1 will always be (M_1M_2⋯ M_T)^-1 so t_T+1 = (M_1M_2⋯ M_T) as all the matrices M_i will be in SL_2() where matrices have the same trace as their inverse.
	
	
		Let [M_1, M_2, …, M_T+1] ∈ SL_2()^T+1 such that Π M_i = I. Then up to simultaneous conjugation, any such (T+1)-tuple can be identified by the following trace coordinates: 
		
			
  * For T = 3: (t_1, t_2, t_3, t_12, t_13, t_23, t_4) 
			
  * For T = 4: (t_1, t_2, t_3, t_4, t_12, t_13, t_14, t_23, t_24, t_34, t_123, t_124, t_134, t_234, t_5)
		
	
	
		For proof and further discussion of character varieties of the punctured sphere see <cit.> and <cit.>. 
	
	Such trace coordinates are also sometimes referred to as the 𝐬𝐢𝐠𝐧𝐚𝐭𝐮𝐫𝐞 of the tuple. 
	
	

§ METHODS
 
	For the rest of this written work, fix ζ_d = e^2π i/d as a primitive d-th root of unity. 
	
	
	Since the product of [A_1, ⋯, A_T+1] must be the identity, we know any MCG-finite tuple is determined by the first T matrices, up to simultaneous conjugation. As middle convolution does not preserve products (see Remark <ref>), our idea is to take middle convolution of the first T matrices to get 2× 2 matrices and then induce a representation of π_1(Σ_0, T+1) by taking the T+1-th matrix to be the inverse product. The condition for middle convolution being rank 2, (stated in Theorem <ref>) depends on the dimension of the subspace fixed by, or equivalently the multiplicity of the eigenvalue 1 of each matrix in the tuple and the eigenvalues of their product. For much better control on the former, we consider reflections since they only have one nontrivial eigenvalue, of multiplicity 1. 
	
	Thus, for each irreducible finite complex reflection group W of rank n, we want to find all irreducible T-tuples of reflections A = [A_1, A_2, …, A_T] in W such that for a suitable choice of parameter λ, MC_λ(A) has rank 2. Given T, the formula given in Theorem <ref> tells us that the product Π_i = 1^T A_i of reflections in A should have a nontrivial eigenvalue λ^-1 of multiplicity T-2 so that MC_λ(A) = [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃] is a T-tuple of rank 2. We give a name to tuples satisfying these conditions.
	
		A T-tuple A = [A_1, A_2, …, A_T] of matrices from a finite complex reflection group W is called 𝐧𝐢𝐜𝐞 if A satisfies the following: 
		
			
  * All A_i are reflections in W
			
  * W = A_1, A_2, …, A_T
			
  * (A_1A_2 ⋯ A_T)^-1 has an eigenvalue λ≠ 1 of multiplicity T-2. 
		
	
	
	The condition of generating the whole group is to ensure that we are not “over-counting" as some reflection groups contain other reflection groups as subgroups, and we only want to study irreducible MCG-finite tuples. 
	
	
		Given a nice T-tuple 𝐀 = [A_1, A_2, …, A_T], we know there exists some λ≠ 1 such that MC_λ(𝐀) = [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃] is a T-tuple of rank 2. taking the T+1-th matrix to be 
    A_T+1 = (Π_i = 1^T Ã_̃ĩ)^-1
 we get a T+1-tuple [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃, A_T+1] whose product is the identity. As 𝐀 comes from a finite group we know it has finite braid group orbit, and since middle convolution is equivariant with the braid group action (see Section <ref>), we know that [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃, A_T+1] must be MCG-finite. Moreover, if {A_1, A_2, …, A_T} generates an irreducible group, then as middle convolution preserves irreducibility, we know our T+1-tuple is irreducible as well. Thus, we have used a nice T-tuple to produce an irreducible MCG-finite T+1-tuple. 
	
	 
		For some nice T-tuples, it is possible that multiple choices of λ have multiplicity T-2 and hence the different choices of parameter λ gives rise to possibly different rank 2 MCG-finite T+1-tuples. 
	
	
	 
		The eigenvalue condition of multiplicity T-2 tells us that in any complex reflection group of rank n, for T > n+2 the middle convolution will not be rank 2 so there are no nice T-tuples.  By Definition <ref>, since we need at least n reflections to generate the whole group, we know there are no nice tuples when T < n. For T = n+2, the product of the T matrices would be a scalar matrix (as there must be an eigenvalue of multiplicity n) which we can take to be the identity, reducing to the T= n+1 case. 	So for each n ∈{3, 4}, we only search the primitive reflection groups for nice tuples of length T = n or T = n+1. 
	
	
		

 §.§ Computing The Braid Group Orbit

	The braid group can be presented as B_T+1 = σ_1, …, σ_T |σ_iσ_i+1σ_i = σ_i+1σ_iσ_i+1, σ_iσ_j = σ_jσ_i where the first relation holds for 1 ≤ i≤ T-3 and the second for i-j ≥ 2. A generator σ_i acts on T+1-tuples in the following way: 
	
    σ_i(M_1, M_2, …, M_T, M_T+1) = (M_1, …, M_i-1, M_iM_i+1M_i^-1, M_i, M_i+2, …, M_T+1)
	
	
	Let F_+ denote the set of all words in positive powers of these generators. Further, let Orb(B_T+1, x) denote the orbit of some tuple x under the action of B_T+1 and Orb(F_+, x) denote the set of images of x under the action of words in F_+. 
	
	
		For a tuple x of matrices, Orb(B_T+1, x) is finite if and only if Orb(F_+, x) is finite. 
	
	
		Since F_+ ⊂ B_T+1, one direction is obvious. For the other direction first note that for any fixed g ∈ B_T+1, if the set {g^i x : i ≥ 0} is finite, then we can rewrite g^-1x = g^n x for some n ≥ 0, so we get that {g^i x: i ∈} must be finite. Further, since the elements of B_T+1 act one at a time, it is enough to study how words in positive powers of the generators act on x. Thus, if Orb(F_+, x) is finite, then the braid group orbit must be finite as well. 
	
	
	To help with computing the braid group orbit, we need a way to capture the equivalence relation of tuples being related by simultaneous conjugation. Using character varieties, we can write down trace coordinates for our tuple, which we know how to do from Theorem <ref> when the matrices are in SL_2().
	
	
		Given a nice tuple 𝐀 = [A_1, A_2, …, A_T], let  MC_λ(𝐀) = [Ã_̃1̃, Ã_̃2̃, …, Ã_̃T̃] denote the middle convolution. If (a_1, a_2, …, a_T+1) ∈ (^×)^T+1 is such that a_i Ã_̃ĩ∈ SL_2() for i = 1, … T and a_T+1 ((Ã_̃1̃Ã_̃2̃⋯Ã_̃T̃)^-1) ∈ SL_2() with Π^T+1 a_i = 1, the tuple 
		
    Â = [a_1 Ã_̃1̃, a_2 Ã_̃2̃, …, a_T+1(Ã_̃1̃Ã_̃2̃⋯Ã_̃T̃)^-1]

		is said to be 𝐢𝐧𝐝𝐮𝐜𝐞𝐝 by MC_λ(𝐀) or by  𝐀 if the parameter λ is clear. Â is a tuple in (SL_2())^T+1 which multiplies to the identity.
	
	When we say trace coordinate of a tuple MC_λ(𝐀) we will always mean the trace coordinates of the induced tuple Â unless otherwise specified.
	
	We want to compute the orbit of induced tuples Â up to simultaneous conjugation. This is equivalent to computing the orbit of the point associated to Â in the character variety 𝔛(π_1(Σ_0, T+1), SL_2()). Note that since all of our tuples come from FCRG and middle convolution is equivariant with the action of the braid group we already know all the orbits are finite. 
	
	The code for the algorithm described below can be found online at <https://amalvayalin.github.io/research.html>.
	
	Given a tuple 𝐌 = [M_1, …, M_T+1] of SL_2()^T+1, such that their product is the identity, the braid group B_T+1 orbit is computed as follows. 
	
	We first compute the signature of 𝐌, and then apply each of the T generators to the tuple 𝐌. If σ_i(𝐌) has a new signature, then we store the tuple in To_Test and the signature in Signatures and add 𝐌 to Tested.  We then apply σ_i again to the (atmost T) new tuples in To_Test we got from the action on 𝐌, once again storing only tuples which have new signatures. We repeat this process until the sets To_Test and Tested are the same. The function returns the set Tested, which should contain only one representative per distinct signature, and can optionally return the set Signatures as well. 
	Note that here our definition of signature is the trace coordinates, which  differs from the definition used in Tykhyy's classification in terms of residues <cit.>.
	
	

 §.§ Finding Nice Tuples
 
	
	
	Nice T-tuples correspond to  MCG-finite T+1-tuples in the following way: given a nice tuple [A_1, …, A_T], we get a corresponding representation  [A_1, …, A_T, A_T+1] of the T+1-punctured sphere if we take A_T+1 = (A_1A_2 ⋯ A_T)^-1. 
	Thus, it is enough to search each complex reflection group for nice tuples.
	
	By Theorem <ref>, an eigenvalue of multiplicity T-2 for an element in W must be a d-th root of unity for d dividing at least T-2 degrees of W. 
	Any reflection group can be constructed in Magma by providing the generating reflections; the irreducible finite complex reflection groups can be easily constructed in Magma by providing the Shephard-Todd number or the corresponding parameters for G(m, p, r). In most cases Magma generates the group W via the reflection representation over the trace field of W <cit.>. 
	
	
	One issue is that not all characteristic polynomials will split over the trace field, as Magma computes eigenvalues for matrices over the field of entries. Since our search depends on the eigenvalues of the product of matrices in the tuple(s), we must change the base field of W to the smallest cyclotomic field containing all the possible eigenvalues. The largest this extension will be (ζ_d), where d is the least common multiple of all the degrees of W, but in certain cases smaller fields might suffice. For large d computation time is typically slower, but we have expanded the field to the LCM of the degrees unless otherwise stated. For example, in the case of W = G_32, the degrees are (W) = {12,18,24,30 } and so ((W)) = 360 which slows down computation time considerably. For this group, we prove in Section <ref> that it is enough to search for 4-tuples over (ζ_12) by showing any eigenvalue of multiplicity 2 must be a 12-th root of unity. 
	
	Before outlining the steps of our search, we define some terms. 
	
	 
		Given a tuple of matrices 𝐀 = [A_1, A_2, …, A_T], the 𝐭𝐲𝐩𝐞 of 𝐀 is determined by the eigenvalues of (A_1A_2 ⋯ A_T)^-1 counted with multiplicity. Tuples whose inverse product has the same eigenvalues with multiplicity are said to be of the same type.  An 𝐞𝐱𝐞𝐦𝐩𝐥𝐚𝐫 of a type is a choice of nice tuple of that type. Given a tuple of some type A, the type of the inverse tuple is called the 𝐢𝐧𝐯𝐞𝐫𝐬𝐞 𝐭𝐲𝐩𝐞 of A and the two types together make an 𝐢𝐧𝐯𝐞𝐫𝐬𝐞 𝐩𝐚𝐢𝐫. 
	

		The use of types allows us to partition a set of nice tuples based on the eigenvalues of the inverse product. Since there are many nice tuples of the same type, we choose exemplars for each type to perform the in-depth computations on. Note that the choice of exemplar for any type is arbitrary and it may be possible that different exemplars result in different orbits. To ensure this does not happen, we go back through all the tuples for each type and make sure that it belongs to the braid group orbit we compute for the middle convolution of the exemplar. More details on the braid group computations can be found in Section <ref>.  
	
	In some groups we get types that are inverse pairs. In these cases we only provide the computations for one exemplar from each pair. To ensure the inverse type would not produce a new orbit, we check that the inverse tuple of the middle convolution lies in the orbit of the middle convolution of the exemplar; this is enough by the relationship between inverse tuples and inverse parameters given in Theorem <ref>. 
	The code for how we checked each type is available at <https://amalvayalin.github.io/research.html>.
	 
		Aside from a few  exceptional cases that were computationally too large, we verified that for each choice of the parameter λ, the rank 2 MCG-finite tuples coming from nice tuples of the same type are in the same braid group orbit, as are exemplars from the inverse type. In this sense, types can be viewed as `almost' equivalence classes for nice tuples up to inverse pairs.
	
	
	After coercing the matrix group into a larger field if necessary, the general process for searching for T-tuples is as follows.
	
		
  * Fix a representative Q_i for each conjugacy class of reflections in W and fix an ordering [Q_1, ⋯, Q_n].
		
  * Given T, construct the list M of distinct sets of T reflections in W, using the ordering of Q to eliminate some tuples equivalent up to simultaneous conjugacy. 
		
  * Search M for all sets such the matrices generate W (means the tuple is irreducible) and the product of the matrices, for some order of multiplying, has an eigenvalue of multiplicity T-2. Each such set is then organized into a tuple based on the order of multiplication and stored in a list X of irreducible tuples. 
		
  * For each nice tuple x ∈ X, compute the eigenvalues of the inverse product of x and partition X into different types corresponding to the different sets of eigenvalues of the inverse product. 
		
  * Choose exemplars 𝐀, 𝐁... for each type and compute the middle convolution MC_λ_1(𝐀), MC_λ_2(𝐀), MC_λ(𝐁)... for suitable choice(s) of λ.
		
  * Compute the subgroup of GL_2() generated by MC_λ(𝐀), MC_λ(𝐁), …
		
  * For each middle convolution of the exemplars, find a suitable character [a_1, …, a_T+1] and construct the induced tuple Â
		
  * Compute the braid group orbit of Â and identify the SL_2() subgroup generated by the matrices of Â
		
  * Check the other exemplars and inverse tuple do not produce new orbits
		
  * Use the tuples in the braid group orbit to compute the parameter and residue signatures of the full equivalence class as defined in the previous classifications to try and match our examples to theirs. 
	
	We recorded all the details for our exemplars and the middle convolution computations for each of the primitive reflection groups of rank 3 and 4. The middle convolution tables are listed in the Appendix. The types and exemplars are listed in the section for each group, as is the comparison tables with the other classifications.  
	More details on what we use to compare with the existing classifications of <cit.> and <cit.> are given in Sections <ref> and <ref> .
	
		

§ IMPRIMITIVE REFLECTION GROUPS

	Imprimitive reflection groups G(m, p, n) are parameterized by m, p , n ≥ 1 with p | m. These groups can be viewed as n × n generalized permutation matrices, which each row and column containing only one non-zero entry which is either 1 or an m/p-th root of unity.  For m > 1 and (m, p, n) ≠ (2, 2, 2) these groups are always irreducible, meaning they act irreducibly on ^n (see <cit.>). 
	
	An element A  of G(m, p, n) can be written as A = [a_1, a_2, …, a_n |σ] for σ∈ S_n and p | a_1 + a_2 … + a_n. This element can be identified with the matrix where the (i, σ(i))-th entry of A is ζ_m^a_i and all the other entries are 0 <cit.>. Note that all these a_i should be read modulo m since we are working with powers of an m-th root of unity. 
	Additionally, there is a natural surjective homomorphism : G(m, p, n) ⟶ S_n given by sending A ↦σ. 
	
	One observation that follows from the definition is that for any p,  G(m, m, n) ≤ G(m, p, n) and G(m, p, n) ≤ G(m, 1, n) as subgroups.  
	
	Just as in the primitive case, the degrees of imprimitive groups tell us about the possible eigenvalues.  Writing q = m/p, we have: 
	
    (G(m, p, n))    = {m, 2m, …, (n-1)m, qn}
    
    		|ZG(m, p, n)|    = q (p, n)

	where ZG(m, p, n) is the center of G(m, p, n) and a cyclic subgroup.  
	
	
	Reflections in the group G(m, p, n) will be of one of the following two types:
	
		
  * Type 1: These are elements such that σ = (ij) is a transposition with a_k ≡ 0  m for k≠ i, j and a_i + a_j ≡ 0  m. All reflections of Type 1 are of order 2 and lie in the subgroup G(m, m, n) of G(m, p, n). Using the notation from <cit.> we write a Type 1 reflection as s(i, j; a_i) with the understanding that this is the same as s(j, i; -a_i).
		
  * Type 2: These are elements such that σ = e and for some i, a_i ≢0   m and a_k ≡ 0  m for all k ≠ i. These matrices are diagonal matrices of order m/(m, a_i) and will be denoted s(i; a_i). Type 2 matrices can only exist when p < m. 
	

	
		Let S be a set of reflections with minimal possible cardinality such that S generates G(m,p,n). Then S satisfies the following conditions: 
		
			
  * When p = 1, S contains n-1 reflections of Type 1 and one reflection of Type 2 of order m
			
  * When p = m, S contains n reflections of Type 1
			
  * When p ≠ 1, m, S contains n reflections of Type 1 and one reflection of Type 2 of order m/p
		
	
	
		Stated in <cit.> with proofs in <cit.> and <cit.>.
	
	
	
		For any reflection r ∈ G(m, p, n), the homomorphism : G(m, p, n) ⟶ S_n, one of the following must be true: 
		
			
  * (r) is a transposition and r has order 2
			
  * (r) is the identity.
		 
	
	
		This is true by the definition of reflections and the definition of the homomorphism. For a more thorough examination of related properties, we refer to <cit.>.
	
	
	Given any set of reflections generating an irreducible imprimitive group the permutations associated to the  reflections in the set must generate a transitive subgroup of S_n, or else the group cannot be irreducible. Note that in the case where the associated permutations are transpositions (i.e. the reflections are Type 1), the only transitive subgroup of S_n generated by transpositions is S_n itself <cit.>. 
	
	A subgroup of a reflection group generated by reflections is called a reflection subgroup. We modify some notation from <cit.> to state relevant theorems for identifying reflection subgroups in the imprimitive groups. Let X = {s(a_h, a_h+1; b_k)} be a set of reflections in G(m, m, n). We can associate to X the graph Γ_X with nodes {a_i} and edges (a_h, a_h+1) for s(a_h, a_h+1; b_k) in X. Letting X denote the reflection group generated by the reflections in X, we see that this group is irreducible if and only if the graph Γ_X is connected. 
	Suppose Γ_X is such that the graph contains exactly one cycle, given by r edges corresponding to {s(a_h, a_h+1; b_h)} for 1 ≤ h ≤ r for some 2 ≤ r ≤ n. Then we define the parameter δ(X) to be the absolute value of ∑_1^r b_h. 
	If p ≠ m, we say X is a nice reflection set of G(m, p, n) if X contains exactly one reflection of Type 2.
	
	
		Note that s(a_h, a_h+1; b_h) = s(a_h+1, a_h; -b_h), so if X is such that the graph contains only one cycle of length r, then the cycle corresponds to some permutation (a_1a_2 ⋯ a_r) (in cycle notation) and the signs of b_h must be chosen accordingly when computing δ(X). In particular, if the cycle is of length two, i.e. given as a double edge between two nodes corresponding to reflections s(a_1, a_2; b_1) and s(a_1, a_2; b_2) then δ(X) = |b_1 -b_2|.
	

	
		With the notation as defined above, we have the following relations for the reflection subgroup generated by X.  
		
			
  * Let X be reflection set of G(m, m, n) such that the graph Γ_X is connected and contains exactly one cycle. Then X = G(m, m, n) if and only if the integer δ(X) is coprime to m.
			
  * Let X be a nice reflection set of G(m, p, n) with s(a_1; b) the one reflection of Type 2, such that Γ_X is connected with n_1 nodes and exactly one cycle. Then X = G(m/m_1, (b, m)/m_1, n_1) where m_1 = (b, m, δ(X)).
			
  * Let X be a nice reflection set of G(m, p, n) with s(a_1; b) the unique reflection of Type 2 such that the graph Γ_X is a connected tree with n_1 nodes. Then X = G(m/(b, m), 1, n_1)
		
	
	
		The first is proven in <cit.> and the second is proven in <cit.>. Note that in both these papers they refer to cycles as circles. The third statement is proven in <cit.>, where they define rooted trees to capture the presence of Type 2 reflections. We refer to these paper for more details on the graph theoretic interpretation of generating sets and their connection to reflection subgroups of the imprimitive reflection groups.
	

		
		Among the imprimitive groups, we have to check for n-tuples and n+1-tuples in G(m, p, n). By Theorem <ref>, the only possible groups that can contain nice n-tuples are the well-generated groups G(m, 1, n) and G(m, m, n). We prove there are no nice n-tuples in either of these groups when n ≥ 5 in Section <ref>. By results in <cit.>, we know that n+1-tuples can only exist in G(m, p, n) for n < 5. This work will fully analyze the existence of nice n+1-tuples for 2< n <5.
	
	
		

 §.§ Nice 3-Tuples in G(m, p, 3)

	In this case we are seeking nice tuples [r_1, r_2, r_3] from G(m, p, 3) such that their product has a nontrivial eigenvalue of multiplicity 1. Note that since r_1, r_2, r_3 = G(m, p, 3) we know that p = 1 or p = m are the only possibilities as these are the only well-generated imprimitive groups (see Definition <ref>)
	
		In G = G(m, 1, 3)  for a nice tuple [r_1, r_2, r_3] with r_1, r_2 of Type 1 and r_3 of Type 2 with nontrivial eigenvalue ζ_m' of order m, letting A = r_1r_2r_3, the eigenvalues of A must be the distinct cube roots of ζ_m'. 
	
	
		Say we have three reflections 
    r_1r_2r_3 = A
 for some A ∈ G with Eigv(A) = {λ_1, λ_2, λ_3} with some λ_i ≠ 1. Note that since the r_i generate G, exactly two of them are Type 1 corresponding to distinct transpositions and the third is Type 2 with order m, say with nontrivial eigenvalue ζ_m'. Using the homomorphism : G ⟶ S_3, we know that (A) is even and since it cannot be the identity it must be a 3-cycle.
		Since A is a generalized permutation matrix, we know (A) = 0 and we get the following equalities must be true: 
		
    ζ_m'    =  (r_1r_2r_3) = (A) = λ_1λ_2λ_3 
    
    			0    = (A) = λ_1 + λ_2 + λ_3

		Since λ_i are all roots of unity, we know their sum being zero means the argument differs by 2 π/3.
		Further writing A = [a_1, a_2, a_3 | (123)], without loss of generality, gives us the matrix: 
		
    A = [       0 ζ_m^a_1       0;       0       0 ζ_m^a_2; ζ_m^a_3       0       0 ]

		Thus we can compute the characteristic polynomial as p_A(x) = -(x^3 - ζ_m^a_1 + a_2 + a_3) and since the constant term of the characteristic polynomial is the determinant of A, we get that ζ_m' = ζ_m^a_1 + a_2 + a_3. Thus λ_i must be the distinct cube roots of ζ_m'.
	
	
		There are nice 3-tuples in G(m, 1, 3) for all m >1. 
	
	
		The reflections in the tuple [s(2, 3; 1), s(1, 2; 1), s(3; 1)] will generate G(m, 1, 3) by Theorem <ref>. By the preceding lemma, the product will have eigenvalues corresponding to the three distinct cube roots of ζ_m, all of which cannot be 1.  
	

	
		In G = G(m, m , 3), if we have r_1r_2r_3 = A for reflections r_i generating G, then the eigenvalues of A are {λ^-2, λ, -λ}
	
	
		Since all the r_i are of Type 1, using : G ⟶ S_3 we see that (A) must be a transposition, say (ij) ∈ S_3. Writing A = [a_1, a_2, a_3 | (ij)] we see that (A) = ζ_m^a_k where k ≠ i, j and (A) = -1. Letting {λ_1, λ_2, λ_3} denote the eigenvalues of A, we get that 
		
    ζ_m^a_k   = λ_1 + λ_2 + λ_3 
    
    			-1    = λ_1 λ_2λ_3

		The characteristic polynomial of A is p_A(x) = -(x^2 -ζ_m^a_i + a_j)(x- ζ_m^a_k). Let λ_k = ζ_m^a_k so then we get that λ_i = -λ_j since these are the roots of x^2 - ζ_m^a_i + a_j. Using the determinant equation we get 1 = λ_i^2 λ_k hence λ_i^2 = λ_k^-1 = ζ_m^-a_k = ζ_m^a_i + a_j so both λ_i and λ_j square to the inverse of λ_k.
	
	
	
		Nice 3-tuples exist in G(m, m, 3) for all m >1. 
	
	
		In this case we can consider reflections [s(2, 3; b_1), s(1, 2; b_2), s(1, 2; b_3)] with b_2 and b_3 chosen so that |b_2 -b_3| is coprime to m. Taking b_1 = 1, b_2 = 1 and b_3 = 0, we get δ(X) = |b_2 -b_3| is coprime to m and so by Theorem <ref>, we know these reflections will generate G(m, m, 3). The characteristic polynomial of the product r_1r_2r_3 = A will in this case be p_A(x) = -(x-ζ_m)(x^2 -ζ_m^-1). Since the roots of the quadratic will be distinct, we know we will have at least two distinct eigenvalues, so the only worry is if the eigenvalues might be {ζ_m, ζ_m, 1} as then there is no nontrivial eigenvalue of multiplicity 1. If 1 is a root of the quadratic, then we know ζ_m^-1 = 1 and ζ_m = 1 which is not possible. Hence, such nice tuples always exist. 
	
	
	

 §.§ Nice 4-Tuples in G(m, p, 3)

	We are seeking nice tuples of four reflections from G(m, p, 3) such that their product has an nontrivial eigenvalue λ of multiplicity 2. This can be written as the identity 
	
    r_1r_2r_3r_4 = λ r_5^-1

	where the r_i generate G(m, p, 3) and r_5 is some reflection, not necessarily in G(m, p, 3). 
	
	
		In the notation of equation <ref> above, for any such nice tuple, it must be that r_5^-1 is in G(m, 1, 3) and λ is an m-th root of unity. Moreover, in any G(m, p,3 ), (including p = 1 and p= m) it must be that exactly four of these reflections are Type 1 and only one is of Type 2. 
	
	
		Note that since λ is an eigenvalue of multiplicity 2 for some matrix in G(m, 1, 3) it must be that the order of λ divides at least 2 degrees in G(m, 1, 3) = {m, 2m , 3m}. Thus the order of λ divides m and so λ I is in the center of G(m, 1, 3). This tells us that r_5 is always in G(m, 1, 3). To generate any of G(m, p, 3) we know the set {r_i}_i ≤ 5 must contain at least 2 reflections of Type 1 by Theorem <ref>.  For any p, rearranging the equation and using the homomorphism : G(m, 1, 3) ⟶ S_3 we get that (r_1r_2r_3r_4r_5) = (λ I) = e. Since e is even we know there must be exactly 2 or 4 reflections of Type 1. The product of two transpositions can only be the identity if they are the same transposition and we need at least two distinct transpositions to generate a transitive subgroup of S_3, hence there are exactly 4 reflections of Type 1 and one reflection of Type 2 for any G(m, p, 3).  
	
	Note that this does not prove that λ is primitive m-th root, however in the next three theorems we will conclude this is indeed always the case. 
	Up to rearranging the equation <ref>, we may assume r_5^-1 is the unique reflection of Type 2 and label its nontrivial eigenvalue ζ which we know must be some m-th root of unity, not necessarily primitive. 
	
	 
		To study the tuples in the cases where they might exist, we first study the number of ways to factorize the identity in S_3 into a transitive product of 4 transpositions. In S_3 any two distinct transpositions will generate a transitive subgroup which must be all of S_3 so fixing the first transposition as (12) we only need to find, up to simultaneous conjugacy, tuples of 3 transpositions that multiply to (12). Using computer search to find such conjugacy classes of 3-tuples, then adjoining with (12) on the left yields four options: 
		
			
  * [(12), (12), (23), (23)]
			
  * [(12), (23), (12), (13)]
			
  * [(12), (23), (23), (12)] 
			
  * [(12), (13), (23), (13)]
		
		Note that these tuples are multiplied starting at the leftmost transposition, although that does not matter in this case. 
		Converting these tuples to Type 1 reflections we can compute the product for each option. 
		
			
  * s(1, 2; b_1)s(1, 2; b_2)s(2, 3; b_3)s(2, 3; b_4) = [           ζ_m^b_1 -b_2                      0                      0;                      0 ζ_m^-b_1 +b_2 +b_3-b_4                      0;                      0                      0         ζ_m^-b_3 + b_4 ]
			
  * s(1, 2; b_1)s(2, 3; b_2)s(1, 2; b_3)s(1, 3; b_4) = [  ζ_m^b_1 +b_2 -b_4                  0                  0;                  0      ζ_m^-b_1 +b_3                  0;                  0                  0 ζ_m^-b_2 -b_3+ b_4 ]
			
  * s(1, 2; b_1)s(2, 3; b_2)s(2, 3; b_3)s(1, 2; b_4) = [ ζ_m^b_1 +b_2 -b_3-b_4                     0                     0;                     0         ζ_m^-b_1 +b_4                     0;                     0                     0        ζ_m^-b_2 + b_3 ]
			
  * s(1, 2; b_1)s(1, 3; b_2)s(2, 3; b_3)s(1, 3; b_4) = [  ζ_m^b_1 +b_3 -b_4                  0                  0;                  0 ζ_m^-b_1 +b_2 -b_3                  0;                  0                  0     ζ_m^-b_2 + b_4 ]
		
		Equations for {b_i} can be found by setting each of these matrices equal to λ r_5^-1:
		
    [  λ  0  0;  0  λ  0;  0  0 λζ ]

		for some root ζ with order dividing m. The choice of putting ζ in the third row is arbitrary and ζ could have been placed in any diagonal entry.
	
	 
		For any nice 4-tuple in G(m, p, 3), p≠ 1 it must be that either λ or λζ in the notation of Remark <ref> has the same order as ζ_m^δ(X), where X is any generating reflection set given by the nice 4-tuple.
	
	
		By Lemma <ref>, given any nice 4-tuple in G(m, p, 3) satisfying equation <ref>, we can rearrange all the Type 1 reflections to be one of the factorizations given in Remark <ref> up to simultaneous conjugation. As any reflection set X generating G(m, p, 3) p ≠ 1 will only need at least three Type 1 reflections, we see that for each of the four factorizations above, there are multiple ways to choose X so that the associated graph only contains one cycle. For example, the first factorization gives us the possibilities: X = {s(1, 2; b_1), s(1, 2; b_2), s(2, 3; b_i)} for i = 3, 4 and X = {s(1, 2; b_i), s(2, 3; b_3), s(2, 3; b_4)} for i = 1, 2. So we know δ(X) must be of the form
		
    δ(X) = 
    			|b_1 -b_2| 
    
    			|b_3 -b_4|

		By the equations each factorization must satisfy, we know ζ_m^b_1 -b_2 = λ and ζ_m^-b_3 +b_4 = λζ so one of these must have the same order as ζ_m^δ(X). The argument is similar for the third factorization since the formula for δ(X) only differs in the indices. The possible δ(X) values from the second and fourth factorizations only differ in the indices, so we only justify the result for the second factorization. In this case the options for δ(X) are 
		
    δ(X) = 
    			|b_1 +b_2 -b_4| 
    
    			|b_1 -b_3| 
    
    			|b_2 +b_3 -b_4|

		Up to taking absolute value, the first two options must equal λ and the third must be λζ so one of these has the same order as ζ_m^δ(X).
	

	
		Nice 4-tuples exist in G(m, m, 3) if and only if m ≠ 3. For m ≠ 3, we have: 
		
			
  * If (m, 3) = 1 then order(λ) = order(ζ) = m
			
  * If (m ,3) = 3 then order(λ) = m = 3 order(ζ) with order(ζ) ≠ 1. 
		 
	
	
		To generate G(m, m, 3), we know that our nice 4-tuple must contain a generating reflection set X such that δ(X) is coprime to m, which means either λ or λζ has order m, by Lemma <ref>. If λ does not have order m, then λζ has order m, but taking the determinant of equation <ref>, we know that 1 = λ^3 ζ so λ^2 has order m which means λ has order m. Thus, λ is always a primitive m-th root. Moreover, we know m ≠ 3 since the determinant tells us λ^3 = ζ^-1 and ζ≠ 1 as it is the nontrivial eigenvalue of the Type 2 reflection r_5^-1.
		
		If (m, 3) = 1 then we know λ^3 = ζ^-1 is also a primitive m-th root and hence so is ζ. Using factorization 1, we can take the tuple [s(1, 2; 1), s(1, 2; 0), s(2, 3; 2), s(2, 3; 0)], so λ = ζ_m and ζ = ζ_m^-3. This tuple generates G(m, m, 3)  by Theorem <ref> since taking X = {s(1, 2; 1), s(1, 2; 0), s(2, 3; 2)} gives δ(X) = 1 which is coprime to m.  
		
		If (m, 3) = 3 then λ^3 has order m/3 and thus so too will ζ. Again taking the tuple as above will work, since δ(X) = 1 will still be coprime to m and the product will give λ = ζ_m and ζ = ζ_m^-3 which now is not a primitive m-th root.  
	
	
		
		Nice 4-tuples exist in G(m, 1, 3) if and only if (m, 3) = 1. In this case the order of λ is m.
	
	
		Let [r_2, r_3, r_4, r_5] be a nice 4-tuple in G(m, 1, 3). Let r_5 be the unique reflection of Type 2 with nontrivial eigenvalue ζ^-1 of order m, so we can write ζ^-1 = ζ_m^-b with (-b, m) = 1. Let d = order(λ). We know d| m since λ I is in the center ZG(m, 1, 3). Taking determinants we get the equation
		
    λ^3 = ζ^-1

		so m | d since (ζ^-1)^d = 1, hence d = m. Since ζ^-1 is a primitive m-th root, so too is λ^3 and therefore we know (m, 3) = 1.
		
		Let m be such that (m, 3) = 1, so we know m-3 is coprime to m. Taking b = m-3, we can consider the tuple [s(1, 2; 0), s(2, 3; 1), s(2, 3; -1), s(3; m-3)]. Taking X = {s(1, 2; b_2), s(2, 3; b_3), s(3; -b)} we see by Theorem <ref> that X = G(m, 1, 3) since (m-3, m) = 1.
		Taking the product, we get r_2r_3r_4r_5 = λ r_1^-1 for some Type 1 reflection r_1 = s(1, 2; -1) with λ = ζ_m.
	

	
		Suppose p ≠ 1 and p≠ m. Then there exists a nice 4-tuple in G(m, p, 3) if and only if p = 3. In this case λ must be of order m.
	
	
		Let [r_2, r_3, r_4, r_5] be a nice 4-tuple in G(m, p, 3). Letting r_5 denote the unique reflection of Type 2, we know it must have order q = p^-1m so we can denote its nontrivial eigenvalue as ζ = ζ_q^b = ζ_m^pb with (b, m) = 1. 
		Further, we have the product r_2r_3r_4r_5 = λ r_1^-1 with r_1 ∈ G(m, m, 3) ≤ G(m, p, 3), hence λ I is in the center ZG(m,p , 3). Letting d = order(λ) we see that since |ZG(m, p, 3)| = q (p, 3), we must have that d | q (p, 3). Taking determinants, we get λ^3 = ζ^-1 and hence q | d and d | 3q.  If d ∤ q then since d ≠ 3 (becasue ζ^-1≠ 1), and d | q (p, 3) we must have that (p, 3) = 3.
		This gives us two cases: 
		
			
  * d = q
			
  * d = 3q and (p, 3) = 3
		
		Recall that by Theorem <ref>, for a reflection set X to generate G(m, p, 3), we need (pb, m) = p and (pb, δ(X), m) = 1.  
		
		In the first case d  = q so we can write λ = ζ_q^a = ζ_m^pa and ζ = ζ_q^b = ζ_m^pb with (b, q)= 1 = (a, q). By Lemma <ref>, we see that 
    (δ(X), m) = (pa, m) 
    ((a+b)p, m)

		Since neither of these satisfy (pb, δ(X), m) = 1, it is not possible for a nice 4-tuple in G(m, p, 3) to have d = q. 
		
		Suppose now that p = 3k so m = 3kq. Again we can write ζ = ζ_q^b = ζ_m^pb with (b, q) = 1, and this time since d= 3q we write λ = ζ_m^ka with (a, m) = 1. In this case the possible options for δ(X) gives us
		
    (δ(X), m) = (ka, m) 
    (ka + pb, m)

		In either case we get (pb, δ(X), m) = k so we conclude k = 1. This also gives us that λ must be of order m. 
		Thus, we conclude nice 4-tuples can only exist in G(m, 3, 3). In this case, we can write λ = ζ_m^a with (a, m) = 1 and we still have ζ = ζ_m^3b with (b, m) = 1 for q = p^-1m. We can take our 4-tuple to be [s(1, 2; 0), s(2, 3; 1), s(2, 3; -1), s(3; m-3)], noting that since 3 | m we also have that (m-3, m) =3. Here δ(X) = 2 so (m-3, 2, m) = 1 and we conclude this tuple is indeed nice.  
	
	
	
		Note that in all 3 cases described above, we will have to take middle convolution with parameter λ^-1 since our equations use the tuple product and not the inverse product.
	

	

 §.§ Nice 4-Tuples in G(m, p, 4)
 
	
	We are seeking nice tuples satisfying 
    r_1r_2r_3r_4 = A
 with r_i a reflection in G(m, p,4) and A has a nontrivial eigenvalue of multiplicity 2. 		Again the only possibilities are p = 1 and p = m since the group must be generated by 4 reflections.
	
		In G = G(m, 1, 4) there are no nice tuples of 4 reflections.
	
	
		Suppose we had r_1r_2r_3r_4 = A such that r_i generate G and A has eigenvalue λ≠ 1 of multiplicity 2. Since {r_i} generate G, we know exactly three of them are of Type 1 and exactly one is of Type 2 with nontrivial eigenvalue an m-th root of unity, say ζ_m'. Note that since the three Type 1 reflections should correspond to transpositions that generate all of S_4, they must be three distinct transpositions with each of {1, 2, 3, 4} showing up at least once. This means their product cannot be a transposition. Using : G ⟶ S_4 we get that (A) = σ for some 4-cycle σ since the permutation must be odd and cannot be a transposition. 
		Writing A = [a_1, a_2, a_3, a_4 |σ] we can compute that the character polynomial of A p_A(x) must be 
    p_A(x) = x^4 - ζ_m^a_1+ a_2 + a_3 + a_4

		This gives us that all the eigenvalues of A are the distinct 4th roots of ζ_m^a_1 + a_2 + a_3 + a_4 which means A cannot have an eigenvalue of multiplicity 2. 
	
	
	In the case of G(m, m, 4), we know for any nice 4-tuple, we must have that all the reflections are of Type 1. This means they all map to transpositions in S_4 and hence the permutation associated to A must be even. In S_4 the only options are e, the disjoint product of transpositions and a three cycle. We would need at least 6 transpositions to write e as the transitive product of transpositions, so we only have two choices. Without loss of generality, if A corresponds to disjoint transpositions, we take it to be (12)(34) and if A corresponds to a 3-cycle we take it to be (132).
	
	 
		
		The number of transitive ordered factorizations of (12)(34) into four transpositions in S_4 is 96 <cit.>. Using a computer, we find that there are 12 factorizations up to simultaneous conjugation. We can reduce these 12 even further by eliminating factorizations that are related by the action of one of the generators of B_4, since this means they lie in the same braid group orbit. 
		These preliminary computations give us three equivalence classes of factorizations, (which may still lie in the same braid group orbit) which we write with representatives: 
		
			
  * [(12), (34), (13), (13)]
			
  * [(13), (13), (12), (34)]
			
  * [(13), (24), (23), (14)]
		
		Choosing general reflections in G(m, m, 4) associated to these factorization and computing the products we get:
		
			
  * s(1,2; b_1) · s(3, 4; b_2) · s(1, 3; b_3) · s(1, 3; b_4) = [                0          ζ_m^b_1                0                0; ζ_m^-b_1+b_3-b_4                0                0                0;                0                0                0          ζ_m^b_2;                0                0 ζ_m^-b_2-b_3+b_4                0 ]
			
  * s(1,3; b_1) · s(1,3; b_2) · s(1, 2; b_3) · s(3, 4; b_4) = [                0  ζ_m^b_1-b_2+b_3                0                0;         ζ_m^-b_3                0                0                0;                0                0                0 ζ_m^-b_1+b_2+b_4;                0                0         ζ_m^-b_4                0 ]
			
  * s(1, 3; b_1) · s(2,4; b_2) · s(2, 3; b_3) · s(1, 4; b_4) = [             0   ζ_m^b_1-b_3             0             0;   ζ_m^b_2-b_4             0             0             0;             0             0             0  ζ_m^-b_1+b_4;             0             0 ζ_m^-b_2 +b_3             0 ]
		
	
	
	 
		Similarly, we can consider transitive ordered factorizations of (132) into four transpositions in S_4. Up to simultaneous conjugation and the action of a generator of B_4, we again get three equivalence classes of factorizations (which may not be distinct): 
		
			
  * [(14), (14), (12), (23)]
			
  * [(14), (14), (13), (12)]
			
  * [(14), (14), (23), (13)]
		
		These factorizations give us the following matrices: 
		
			
  * s(1, 4; b_1) · s(1, 4; b_2) · s(1, 2; b_3) · s(2, 3; b_4) = [                   0                   0 ζ_m^b_1-b_2+b_3+b_4                   0;            ζ_m^-b_3                   0                   0                   0;                   0            ζ_m^-b_4                   0                   0;                   0                   0                   0       ζ_m^-b_1 +b_2 ] 
			
  * s(1, 4; b_1) · s(1, 4; b_2) · s(1, 3; b_3) · s(1,2; b_4) = [                0                0 ζ_m^b_1 -b_2+b_3                0;         ζ_m^-b_4                0                0                0;                0    ζ_m^-b_3 +b_4                0                0;                0                0                0     ζ_m^-b_1+b_2 ]
			
  * s(1, 4; b_1) · s(1, 4; b_2) · s(2, 3; b_3) · s(1, 3; b_4) = [               0               0 ζ_m^b_1-b_2+b_4               0;    ζ_m^b_3 -b_4               0               0               0;               0        ζ_m^-b_3               0               0;               0               0               0   ζ_m^-b_1 +b_2 ]
		
	
		 
			In G = G(m, m, 4) if we had r_1r_2r_3r_4 = A with r_i reflections generating G and A has a nontrivial eigenvalue λ of multiplicity 2 then the eigenvalues of A are one of the following: 
			
				
  * {λ, λ, -λ, -λ} for λ a 4-th root of unity
				
  * {λ, λ, λ_1, λ_2} for λ a 4-th root of unity and λ_i ≠λ are the two distinct cube roots of λ^-1 other than λ.  
			
		
		
			We know all the r_i are of Type 1, so using : G ⟶ S_4 we know that (A) must be an even permutation, so it is either a double transposition or a 3-cycle. Let the eigenvalues of A be {λ, λ, λ_1, λ_2}. 
			If (A) is a double transposition then we see that 
			
    0    = (A) = 2λ + λ_1 + λ_2 
    
    				1    = (A) = λ^2 λ_1 λ_2

			Rearranging the equation for trace gives us that 
    1 = |λ| = | λ_1 + λ_2|/2 = |λ_1| + |λ_2|/2 = 1

			By properties of the triangle inequality, we get that λ_1 and λ_2 are colinear, hence λ_1 = ±λ_2. However, if λ_1 = -λ_2 then the trace equation means λ = 0 which is not possible, hence λ_1 = λ_2. Now the trace equation tells us that 2(λ + λ_1) = 0, hence λ_1 = -λ. 
			Next, we can use the determinant equation to see that 1 = λ^4 and so λ must be a 4th root of unity. This gives us that the eigenvalues of A are as described in case 1). 
			
			Further, when (A) is a double transposition, say A = [a_1, a_2, a_3, a_4 | (12)(34)], the characteristic polynomial is p_A(x) = (x^2 -ζ_m^a_1 + a_2)(x^2 - ζ_m^a_3 + a_4). However, since we know the roots of these polynomials agree, we see that ζ_m^a_1 + a_2 = ζ_m^a_3 + a_4, but we already knew that ζ_m^a_1 + a_2 = ζ_m^-a_3 - a_4 since m | a_1 + a_2 + a_3 + a_4. Thus we conclude that ζ_m^a_1 + a_2 = ± 1 and so ζ_m^a_1 = ±ζ_m^-a_2 and similarly, ζ_m^a_3 = ±ζ_m^-a_4 where the signs must agree for both pairs. 
			If (A) is a 3-cycle, we write A = [a_1, a_2, a_3, a_4 | (132)], since taking a different 3-cycle just changes the indices. The determinant condition stays the same as above, however our equation for trace is now: 
			
    ζ_m^a_4 = (A) = 2λ + λ_1 + λ_2

			Computing the characteristic polynomial we get p_A(x) = (x-ζ_m^a_4)(x^3 - ζ_m^a_1+a_2+a_3) and since the cube roots of ζ_m^a_1 + a_2 + a_3 are all distinct, we know that for λ to have multiplicity two, it must be one of the cube roots and λ = ζ_m^a_4. Since ζ_m^a_1 + a_2 + a_3 = ζ_m^-a_4 = λ^-1, we see that λ, λ_1 and λ_2 must be the three distinct cube roots of λ^-1. Now we have that λ^3 = λ^-1 and hence λ^4 = 1, proving that the eigenvalues of A are as described in case 2).
		
	
	 
		Nice 4-tuples exist in G(m, m, 4) iff  m = 2 or m = 4. In either case, λ is of order m. 
	
	
		Say we had a nice tuple in G(m, m, 4) such that r_1r_2r_3r_4 = A. 
		Suppose first that A corresponds to a three-cycle, then our nice 4-tuple is equivalent to one of the factorizations in Remark <ref>.  For all three factorizations, by the previous Lemma, we know that ζ_m^-b_1 +b_2 = λ and by Theorem <ref>, we know that δ(X) = |b_1 - b_2| must be coprime to m. This means λ is a primitive m-th root and by Lemma <ref> we know m = 4 or m = 2 in this case. 
		
		If A corresponds to a double transposition, then our nice 4-tuple is equivalent to one of the factorizations in Remark <ref> and we may write A = [a_1, a_2, a_3, a_4 | (12)(34)]. From the first factorization and the proof of Lemma <ref>, we see that ± 1 = ζ_m^a_1 + a_2 = ζ_m^b_3 -b_4 = λ^2. In this case δ(X) = |b_3 -b_4| which we know must be coprime to m by Theorem <ref>, we get that λ^2 is a primitive m-th root so m = 2. The same argument will also give that m must be 2 if our nice tuple is equivalent to the second factorization in Remark <ref>. For the third possible factorization, we see that the cycle has four edges and so δ(X) = |b_1 -b_3 +b_2 -b_4 since the cycle corresponds to the permutation (1324). But again we have that ± 1 = ζ_m^a_1 + a_2 = ζ_m^b_1 + -b_3 + b_2 -b_4 so λ^2 is a primitive m-th root so m = 2. 
			If m = 4, then we can take r_1 = s(1, 4; -1), r_2 = s(1, 4; 0), r_3 = s(1, 2; -1), r_4 = s(2, 3; 0). Then δ(X) = 1 so these reflections generate G(4, 4, 4). Further, we have 
    A = [  0  0 -1  0;  i  0  0  0;  0  1  0  0;  0  0  0  i ]
 which has eigenvalue i with multiplicity 2.	
		If m = 2, then we can take r_1 = s(1, 4; 1), r_2 = s(1, 4; 0), r_3 = s(1, 2; 0), r_4 = s(2, 3; 0) so again δ(X) = 1 and generates G(2, 2, 4). In this case 
    A = [  0  0 -1  0;  1  0  0  0;  0  1  0  0;  0  0  0 -1 ]
 which has eigenvalue -1 with multiplicity 2.
	
	Note that when taking the middle convolution for such tuples, the parameter should be λ^-1 since these equations are for the tuple product and not the inverse product.
	
		

 §.§ Nice 5-Tuples in G(m, p, 4)

	Searching for 5-tuples G(m, p, 4) so that the middle convolution will be rank 2 means we want the product to have an eigenvalue of multiplicity 3. A 4 × 4 matrix with an eigenvalue of multiplicity 3 can be viewed as a scalar times a reflection, so we want to explore the existence of [r_1, r_2, r_3, r_4, r_5] such that
	
    r_1r_2r_3r_4r_5 = λ r_6^-1
 where λ≠ 1 and r_6 is some reflection, not necessarily in G(m, p, 4). 
	
	
		If we have 5 reflections generating G = G(m , p, 4) such that 
		
    r_1r_2r_3r_4r_5 = λ r_6^-1
 
		then it must be that
		
			
  * r_6 ∈ G and G = G(m, m, 4) with order(λ) | m 
			
  * λ^4 = 1 
		
	
	
	
		Note first that  G(m, m, 4) ≤ G(m, p, 4) ≤ G(m, 1, 4) so r_i ∈ G(m, 1, 4) for i < 6 and so λ r_6^-1∈ G(m, 1, 4). Since this matrix has an eigenvalue of multiplicity 3, we know d = order(λ) is such that d divides 3 of the degrees in G(m, 1, 4) = {m, 2m , 3m, 4m}. This tells us that d | m as if it did not then d | 3m -2m = m. Since λ has order dividing m, λ I is in the center of G(m, 1, 4) and thus we can conclude r_6 ∈ G(m, 1, 4), though not yet necessarily in G(m, p, 4). This changes our problem into studying 6 reflections in G(m, 1, 4) satisfying
		
    r_1r_2r_3r_4r_5r_6 = λ I

		
		Next we use the homomorphism : G(m, 1, 4) ⟶ S_4. By Theorem <ref>, we know (r_1r_2r_3r_4r_5r_6) = (λ I) = e and since e is even, we know an even number of (r_i) must be transpositions.
		
		To conclude all 6 are transpositions, note that {r_i}_i < 6 are supposed to generate G(m, p, 4) which means {(r_i)}_i<6 must generate a transitive subgroup of S_4, hence S_4 itself. This removes the possibility of just 2 transpositions since we need at least 3 distinct transpositions to generate S_4.
		
		While it is possible to generate S_4 with 4 transpositions, our equation also requires that their product is the identity which is not possible with at least 3 distinct transpositions (as can be verified by hand). 
		
		Thus we know all 6 reflections must be Type 1 and so {r_i}_i < 6 can only generate G(m , m, 4), we conclude G = G(m , m ,4).
		
		Using that all reflections of Type 1 have nontrivial eigenvalue -1, taking determinant gives us that 
    1 = (-1)^ 6 = (r_1r_2r_3r_4r_5r_6) =  (λ I) = λ^4
 so we know lambda must be a 4th root of unity. Further, using that order(λ) ∈{2, 4 } must divide at least 3 degrees in G(m, m , 4) = {m, 2m, 3m, 4} we conclude that order(λ) | m. 
	
	
	 
		To study such tuples in G(m, m, 4) we first find the ways to decompose e in S_4 into a transitive product of 6 transpositions up to conjugacy. To make such a list, note that we can assume the first two transpositions are not disjoint, as if they are we can swap them around. This tells us their product is a 3-cycle and up to conjugation we may assume it is (132). We then found all the 4-tuples of transpositions that multiple to (123) up to simultaneous conjugation and generate all of S_4. Adjoining these options to the (12), (13) on the left, we get 27 options for transitive factorizations of the identity , and using computer code to partially compute the braid group orbit we were able to reduce the list further to two options (not necessarily distinct). 
		The tuples in S_4 are: 
		
			
  * [(12), (13), (12), (23), (24), (24)]
			
  * [(12), (13), (24), (24), (12), (23)]
		
		Using the Type 1 notation for the reflections in G(m, m, 4) as r_i =s(j_i, k_i; b_i) we can take the matrix product. The first tuple gives us the equation
		
    [               ζ_m^b_1 -b_3                          0                          0                          0;                          0 ζ_m^-b_1 +b_2-b_4 +b_5-b_6                          0                          0;                          0                          0         ζ_m^-b_2 +b_3 +b_4                          0;                          0                          0                          0              ζ_m^-b_5 +b_6 ]   = [ λ 0 0 0; 0 λ 0 0; 0 0 λ 0; 0 0 0 λ ]

		and the second tuple gives us the equation 
		
    [ ζ_m^b_1 +b_3 -b_4 -b_5                      0                      0                      0;                      0      ζ_m^-b_1 +b_2-b_6                      0                      0;                      0                      0     ζ_m^-b_2 +b_5 +b_6                      0;                      0                      0                      0          ζ_m^-b_3 +b_4 ]   = [ λ 0 0 0; 0 λ 0 0; 0 0 λ 0; 0 0 0 λ ]

	

	
		Nice 5-tuples exist in G(m, m, 4) if and only if m = 2 or m = 4 and in either case λ is of order m. 
	
	
		Suppose we had a nice 5-tuple in G(m,m, 4), so by the previous lemma we know 2 | m. We know it must be equivalent to one of the two factorizations in Remark <ref>. Considering the first factorization and looking at all transitive subsets of 4 reflections we can choose from the 6, we get the following possibilities for δ(X): 
		
    δ(X) = 
    				|b_1 -b_3| 
    
    				|b_6 -b_5| 
    
    				|b_4 +b_3-b_2| 
    
    				|b_4 +b_1 -b_2|

		The first three of these are such that ζ_m^δ(X) equals λ up to the absolute value on δ(X), so if any of those are coprime to m then λ is of order m. So the only way m ≠ 2, 4 is if the only option for δ(X) that is coprime to m is |b_4 +b_1-b_2|. 
		Writing m = 2^sk with (k, 2) =1 and s ≥ 1, we can write λ = ζ_m^k (s-j) for j = 1, 2 depending on the order of λ. The equations for b_i  modulo m then becomes: 
		
    b_1-b_3 ≡ k2^s-j
    
    			-b_1 + b_2-b_4+b_5 -b_6 ≡ k2^s-j
    
    			-b_2 +b_3 +b_4 ≡ k2^s-j
    
    			-b_5 +b_6 ≡ k2^s-j

		The second and fourth equation give us -b_1 +b_2-b_4 ≡ 2 k2^s-j which is not coprime to m. Thus, this option for δ(X) does not work so one of the first three options are coprime to m and hence λ is of order m. Since λ is a 4-th root of unity, we know m = 2 or m = 4. 
		
		A similar argument will give the same conclusion for the second factorization. 
		For m = 4, consider the tuple [s(1, 2; 1), s(1, 3; 0), s(1, 2; 0), s(2,3; 1), s(2, 4; 1)]. Then δ(X) = 1 and so these reflections will generate G(4, 4, 4). The product of this tuple will be i · s(2, 4; 2) as this corresponds to the first factorization.
		For m = 2, we can take the tuple [s(1, 2; 1), s(1, 3; 0), s(1, 2; 0), s(2,3; 1), s(2, 4; 0)]. Then δ(X) = 1 and so these reflections will generate G(2, 2, 4).  The product of this tuple will be -1 · s(2, 4; 1) as this corresponds to the first factorization.
	
		Note that with the examples above, we will have to take middle convolution with parameter λ^-1 since our equations are for the tuple product and not the inverse product.
	
	

 §.§ Nice n-Tuples in G(m, p, n) for n ≥ 5
 
	Recall that the only possible groups when nice n-tuples can exist are when p = 1 and p = m since these are the only well-generated imprimitive groups (See Definition <ref>).
	Fact: A connected graph of n vertices and n-1 edges must be a tree.
	
	
	 The product of n-1 transpositions is an n-cycle if and only if 
		the graph associated to a set of transposition in S_n is a tree, where the edge (a,b) is only present if (ab) is one of the n-1 transpositions. 
	
	
		By induction: if the product of n-1 transpositions is the product of a disjoint k-cycle and an (n-k)-cycle, then each of these correspond to  disjoint trees with k-1 and (n-k-1) edges respectively, so the remaining edge must connect the two trees, resulting in a tree with n-1 edges, hence the product is an n-cycle.
	
	
	
		There are no nice n-tuples in G(m, 1, n) for n ≥ 5. 
	
	
	
		In G(m, 1, n) any nice n-tuple consists of n-1 reflections of Type 1 and one reflection of Type 2, say r_n. Suppose we had r_1r_2⋯ r_n = A with [r_1, …, r_n] a nice tuple. Then by the claim(s) above, the graph associated to the transpositions {(r_i) }, with (r_n) = e must be a tree since the graph is connected by transitivity and therefore (A) is an n-cycle. Thus the characteristic polynomial 
    char_A(x) = (-1)^n (x^n - ζ)
 for some m-th root of unity ζ and therefore has n distinct roots so no A has no eigenvalues of multiplicity n-2 for n > 3.
	
	For G(m, m, n) we need the additional observations provided below.
	
	
		Let σ = (x_1x_2, · x_n) be an n-cycle in S_n and τ = (y_1y_2) be some transposition in S_n with y_1 <y_2. Letting c denote the distance between y_1 and y_2 in σ modulo the length of σ, we have that the product σ·τ is the product of a disjoint c-cycle and a disjoint (n-c)-cycle. 
	
	
		Since σ fixes no points, it is enough to show that multiplying with τ cannot fix two points in {1, …, n}. Letting y_1 = x_i and y_2 = x_j with i <j then we get 
    y_1  ↦ x_j+1↦ x_j+2↦…↦ x_i-1↦ x_i = y_1
 where the subscripts are read modulo n. The length of this cycle is precisely j-i  n. Doing a similar process for y_2 we get a cycle of length n -(j-i)  n. 
	
	Note that since σ is written in disjoint cycle notation the `distance' between, i.e. number of digits between y_1 and y_2 read in the order of multiplication (e.g. left to right if multiplication is left to right) is well defined. 
	
		In the notation of the claim above, if n > 2, then the product στ has at most one fixed point, with this occurring only if y_1 and y_2 are adjacent in the cycle σ. 
	
	
		If y_1 and y_2 are adjacent then c  = 1, so we are left with just an n-1 cycle which has exactly 1 fixed point when n-1 > 1 and 2 fixed points when n = 2. 
	
	
	
		There are no nice n-tuples in G(m, m, n) for n ≥ 5.
	
	
		In G(m, m, n), any nice tuple [r_1, r_2, …, r_n] consists of n reflections of Type 1. Again by transitivity, we know the graph associated to {(r_i)} is connected and since we only need n-1 edges to make a connected graph on n vertices, we know there is some transposition we can remove so that the associated graph is still connected. By cyclic permutation of the r_i via simultaneous conjugation, we can assume the redundant transposition corresponds to r_n, so (A · r_n^-1) is an n-cycle by the claim above. This tells us that (A) is the product of an n cycle and a transposition. 
		By the proofs above, we know (A) is either an (n-1)-cycle or the product of a c-cycle and an (n-c)-cycle. So either the characteristic polynomial is 
    char_A(x) = (-1)^n(x^n-1-ζ_1)(x-ζ_2)
 or 
    char_A(x) = (-1)^n (x^c -ζ_1)(x^n-c-ζ_2)

		We see that in both cases an eigenvalue of A can have atmost multiplicity 2, so when n > 4, there are no eigenvalues of multiplicity n-2, hence no such nice tuples can exist in G(m, m, n). 
	
	
	

§ PRIMITIVE REFLECTION GROUPS

	
	For each rank r ∈{3, 4}, we investigate the existence of nice T-tuples for T = r and T = r+1 in the primitive reflection groups, denoted G_i for i = 23, 24, …, 32. 
	For each group and each choice of T, we provide a table of the different types (up to inverse pairs), including the label for the type, the residues of the eigenvalues, as well as the nontrivial eigenvalues of the exemplar tuple under `Tuple Eigenvalues'.
	We also list the chosen exemplars 𝐀 = [A_1, A_2, …, A_T] for each type. For each exemplar we compute the middle convolution with the suitable choice(s) of parameter λ and this data is available in the appendices <ref>, <ref>, <ref>. These tables also include, for each choice of eigenvalue λ, the residue of λ in the `ξ' column as well as the notation used in Magma to identify the root of unity under `λ'.  We list the 2 × 2 matrices composing MC(𝐀) = [M_1, …, M_T] and record the order of the GL_2() subgroup generated by the these matrices under `S. Size'. Note that a zero in this column means the subgroup is infinite. 
	
	To get the induced tuple Â from MC_λ(𝐀) many choices of character will work, so we provide the one we use under the column `Character' or sometimes abbreviated as `Char.'. 
	In some cases, we may need to go to a larger field to construct a suitable character. We then compute and record the braid group orbit size of the induced tuple (under action of the positive powers of the braid group generators) and record it under `O. Size'. More information about how we compute the braid group orbit is provided in Section <ref>.
	For examples where the braid group orbits are the same size, we check if the signature of one lies in the orbit of the other.
	
		We say two braid group orbits are 𝐞𝐪𝐮𝐢𝐯𝐚𝐥𝐞𝐧𝐭 if the signature of one of the induced tuples lies in the orbit of another induced tuple.
	  
	This is one reason why our braid group code optionally can return the list of signatures in each orbit. 
	Note that this differs from the broader notion of equivalence in other literature such as Tykhyys's classification <cit.>. Note that the work of Lisovy-Tykhyy <cit.> and Tykhyy <cit.> only consider the case where matrices in the tuple do not have a shared eigenvector. In our case, since nice tuples give an irreducible representation and middle convolution preserves irreducibility, we know all our tuples cannot share a common eigenvector. Thus, we expect to identify which tuples in these classifications come from the complex reflection groups via middle convolution. Details on what parameters we use to compare our orbits with these classifications is provided below at the start of Section <ref> and <ref>.   
	
	For each distinct orbit we find, we provide data in the comparison tables on how our orbits relate to other classifications, as well as identify the finite SL_2() subgroups generated by the tuples. We list the subgroups of SL_2() in the Magma notation, given by order, number where order is the order of the finite group and number is the numbering of the group in the Small Groups Database. The numbering used by Magma is the same as that in the Small Group database implemented in GAP (see <https://magma.maths.usyd.edu.au/magma/handbook/text/779>, accessed on July 7th, 2024) and can be readily found online for all the groups we see. For infinite groups, the order is listed as `0', following the notation used in Magma. The notation SL(2, k) means 2× 2 matrices of determinant 1 over the finite field of _k.
	
		

 §.§ Nice 3-Tuples in Rank 3 Groups
 
	Nice tuples in this case will be tuples of 3 reflections generating the whole group such that their inverse product has some nontrivial eigenvalue of multiplicity 1. 
	After middle convolution, our the induced tuple will correspond to SL_2() local systems on the 4-punctured sphere and we can compare our braid group orbits to those in the literature. We compute the parameters (ω_X, ω_Y, ω_Z, ω_4) in the notation of <cit.> and found that the ω parameters for each of our orbits match (up to permutation) with one listed in the work of Lisovy-Tykhyy. Since Tykhyy's classification <cit.> further refines the original classification by Lisovy-Tykhy <cit.>, each tuple of ω parameters gives multiple orbits that Tykhyy identifies uniquely in terms of residues of the trace coordinates  (θ_1, θ_2, θ_3, θ_4) and (σ_12, σ_23, σ_13, σ_24) for some representative. In the nice 3-tuple case, using the ω parameters, we are able to easily identify how our orbit size relates the orbit size listed by Tykhyy for each ℳ^(4) (in Tykhyy's notation) and narrow down the possibilities for which of Tykhyy's orbits our tuple corresponds to. 
	In Tykhyy's classification,  tuples are considered equivalent up to:  multiplying any two matrices by -1, complex conjugation of all elements of all matrices, cyclically permuting the matrices in the tuple, and up to taking inverses and reversing the order of the tuple <cit.>. We perform all these operations and compute the residue signatures for the matrices in each orbit to find one that matches the example provided in Tykhyy's work. In most cases we found exact matches, and in other cases we provide the closest match, marking the differing coordinate with an asterisk (see Section <ref> and <ref>).   
	
	Note that we only list this data for orbits that are considered different by our definition and these may end up being the same orbit in the Lisovy-Tykhyy or Tykhyy classifications (<cit.>, <cit.>). 
	For each orbit we compute, we provide the identifying label used in both the classifications, as well as our orbit size and the orbit size they provided  (which is the same in both works). 
	
	
	

  §.§.§ G_23
 
	The group G_23 of order 120 is also denoted W(H_3) with degrees (G_23) = {2, 6, 10} <cit.>. All reflections in this group are conjugate with nontrivial eigenvalue -1. 
	For nice 3-tuples in G_23, the computation returned three possible sets of eigenvalues for the inverse product. 
	
	With q = ζ_5, we provide one example of a nice tuple such that the inverse product has eigenvalues corresponding to each type. 
	
	For Type A in G_23 one exemplar is:
	
    A_1    = [  1  0  0;  0  1  1;  0  0 -1 ]   A_2 = [            0           -1            0;           -1            0            0; -q^3-q^2 + 1   -q^3-q^2+1            1 ]   A_3 = [       1       0       0; q^3+q^2      -1      -1;       0       0      -1 ]

	Similarly for Type B, an exemplar is: 
	
    B_1 = [  1  0  0;  0  1  1;  0  0 -1 ]   B_2 =[        1 -q^3-q^2 -q^3-q^2;        0        0       -1;        0       -1        0 ]   B_3 = [       1       0       0; q^3+q^2      -1      -1;       0       0       1 ]
 	
	For Type C an exemplar is: 	
    C_1 = [  1  0  0;  0  1  1;  0  0 -1 ]   C_2 = [        1 -q^3-q^2        0;        0       -1        0;        0        1        1 ]   C_3 = [          0         -1          0;         -1          0          0; -q^3-q^3+1 -q^3-q^2+1          1 ]

	The data for all three types for G_23 is compiled in Table <ref>. For all three types, choosing a different exemplar would not have resulted in a new orbit. The two orbits of size 10 arising from Type A are the same, as are the two orbits of size 10 from Type B. However, these two orbits are distinct across the two types in the sense that their signatures did not show up in the others' orbit. For Type C, the two orbits of size 18 are the same. Information about how these 6 distinct orbits relate to the other classifications is provided in Table <ref>. In this group we also find nice 4-tuples, details provided in Section <ref>
	
	
		

  §.§.§ G_24
 
	The group G_24 of order 336 is also denoted W(J_3(4)) with degrees (G_24) = {4, 6, 14} <cit.>. All reflections in this group are conjugate with nontrivial eigenvalue -1. 
	Looking at the degrees, we first expand the base field to be the cyclotomic field of degree 84. There is one inverse pair of nice 3-tuples in this group, so we only provide the details for one in Table <ref>.
	
	
	Letting z = ζ_7, an exemplar of Type A is: 
	
    A_1 = [          1 -z^4-z^2-z -z^4-z^2-z;          0          0         -1;          0         -1          0 ]   A_2 = [  1  1  0;  0 -1  0;  0  1  1 ]   A_3 = [        -1         0         0;         1         1         0; z^4+z^2+z         0         1 ]

	The middle convolution data for G_24 is provided in Table <ref>. These three distinct orbits are the only orbits from Type A and we checked that in all three cases, the inverse of the exemplars do not produce any new orbits. The comparison data for G_24 is provided in <ref>.
	
	
	

  §.§.§ G_25
 
	The group G_25 of order 648 is also denoted W(L_3) with degrees (G_25) = {6, 9, 12} <cit.>. 
	G_25 has two conjugacy classes, consisting of matrices with nontrivial eigenvalue w or w^2 respectively, for w = ζ_3.
	Expanding the base field to be the cyclotomic field of order 36, we get 8 possible types, which actually split into 4 pairs of inverses. We only list one set of inverse eigenvalues for each pair in Table <ref>.
	
	
	With w = ζ_3 the exemplars for each type are provided below. 
	Exemplar of Type A: 
	
    A_1 = [   1   0   0;   0   1 w^2;   0   0   w ]   A_2 = [   w   0   0; w^2   1   0;   0   0   1 ]   A_3 = [   1   w   0;   0 w^2   0;   0   w   1 ]

	Exemplar of Type B: 
	
    B_1 = [   1   0   0;   0   1 w^2;   0   0   w ]   B_2 = [   -w    1  w^2;    1 -w^2    1;  w^2    1   -w ]   B_3 = [   1   w   0;   0 w^2   0;   0   w   1 ]

	Exemplar of Type C: 
	
    C_1 = [    1 -w^2    0;    0    w    0;    0 -w^2    1 ]   C_2 = [   w   0   0; w^2   1   0;   0   0   1 ]   C_3 = [    1    0    0; -w^2    w -w^2;    0    0    1 ]

	Note that for Type D, there is only two choices of eigenvalue to take midlde convolution with and only one of them will result in 2× 2 matrices. Exemplar of Type D: 
	
    D_1 = [    1 -w^2   -w;    0 -w^2    1;    0   -w    0 ]   D_2 = [   w   0   0; w^2   1   0;   0   0   1 ]   D_3 = [    0   -w  w^2;    0    1    0; -w^2   -1 -w^2 ]

	The details for the middle convolution of each type is provided in Table <ref>. For all three types, different exemplars do not produce new orbits nor do their inverses. For Type A, the two orbits of size 12 are distinct, as are all three orbits of size 36 from Type B. For Type C, we had to extend to a field containing ζ_72 to construct the character. The two orbits of size 4 from Type C are the same. Including the one from Type D, we get 8 distinct orbits from G_25. The comparison data is provided in Table <ref>. Comparing with the Lisovy-Tykhyy paper, all the ω parameters seemed to correspond to the family of orbits listed in Lemma 39 in <cit.>, so we have provided the specific values for the variables that work in each case. When comparing with Tykhyy's classification, there was only one orbit (from Type B) that we could not find an exact match for. We listed the closest match, which seemed to be Orbit 5 in <cit.> and distinguish the different coordinate with an asterisk. 
	In this group we also find nice 4-tuples, details provided in Section <ref>
	
	
	
	

  §.§.§ G_26
 
	The group G_26 of order 1296 is also denoted W(M_3) with degrees (G_26) = {6, 12, 18} <cit.>.
	In G_26 there are three conjugacy classes of reflections, corresponding to nontrivial eigenvalues in {ζ_3, ζ_3^2, -1}. In this group the are 6 sets of eigenvalues for the inverse product, which correspond to 3 inverse pairs, so we only list one from each pair.  
	
	The exemplars for this group are all provided over (ζ_36) with z = ζ_36.
	
	Here is an exemplar of Type A:
	
    A_1 = [       1       0       0;      -1  2· z^6     z^6;  -z^6+2 -3· z^6    -z^6 ]   A_2 = [      0    z^6      0;      1 -z^6+1      0;  z^6-2  z^6+1      1 ]   A_3 = [  1  0  0;  0  1  1;  0  0 -1 ]

	An exemplar for Type B: 
	
    B_1 = [  z^6-1 2· z^6    z^6;      0      1      0;      0      0      1 ]   	B_2 = [      0  z^6-1  z^6-1;    z^6      2      1; -z^6-1  z^6-2  z^6-1 ]   B_3 = [  1  0  0;  0  1  1;  0  0 -1 ]

	An exemplar for Type C: 
	
    C_1= [          1          0          0;       -z^6      z^6-1      z^6-1;      z^6+1 -3·(z^6+1) -2·(z^6-1) ]   C_2 = [      1 -z^6+1 -z^6+1;      0      1      0;      0 -z^6-1   -z^6 ]   C_3 = [  1  0  0;  0  1  1;  0  0 -1 ]

	
	In G_26 choice of different exemplars nor the inverses for any type produce new orbits. The middle convolution data for this group can be found in Table 
	<ref>. Further, all three orbits from Type A are distinct, as are all three orbits from Type B. When comparing with Tykhyy's classification, there was only one orbit (from Type B) that we could not find an exact match for. We listed the closest match, which seemed to be Orbit 5 in <cit.> and distinguish the differing coordinate with an asterisk. 
	
	In this group we also find nice 4-tuples, details provided in Section <ref>
	
		 
			λ     ω_X, Y, Z, 4      Lisovy-Tykhyy #     θ_1, 2, 3, 4     σ_12, 23, 13, 24       -90 Tykhyy #       -90 Their O. Size      -90 Our O. Size       -90 SL_2() Subgroup 
  [0pt]0pt0ex
 [0ex] 
			9|c|G_26 Type A Comparisons [0pt]0pt3ex
 
			z^3     (0, √(3), 0, 0)    Type II, X' = 0, X” = √(3)     ( 5/12, 1/4, 5/12, 3/4)     (1/2, 1/2, 5/6, 1/2)      4      2     24     0 [0pt]0pt4ex 
 [1ex] 
			-z^3     (0, -√(3), 0, 0)     Type II, X' = 0, X” = -√(3)     (1/4, 1/12, 1/4, 11/12)     (1/2, 1/2, 1/2 , 5/6)     4       2     24     0 [0pt]0pt4ex
 [1ex] -z^6+1     (0, -z^9+2z^3, 0, 0)     Type II, X' = 0, X” = -z^9+2z^3     (1/6, 1/2, 2/3, 1/2)     (1/2, 1/2, , 5/6, 1/2 )     3      2     24     24, 4 [0pt]0pt4ex
 [1ex] 9|c|G_26 Type B Comparisons [0pt]0pt3ex
 
			z^10     (-z^9+2z^3, -z^9+2z^3, 2, -1)     Type III, ω = -z^9+2z^3     (7/9,7/18, 7/18, 2/3)     ( 1/2, 1/3, 1/2, 5/6^*)     5^*     3     36     0 [0ex]0ex4ex 
 [1ex]
			-z^4     (z^9-2z^3, z^9-2z^3, 2, -1)     Type III, ω = z^9-2z^3     (1/9,1/18, 1/18, 2/3)     (1/2, 1/3, 1/2, 1/6)     5     3     36     0 [0ex]0ex4ex 
 [1ex]
			-z^16     (-z^9+2z^3, -z^9+2z^3, 2, -1)     Type III, ω = -z^9+2z^3     (5/9, 5/18, 5/18, 2/3)     (1/2, 1/3, 1/2, 1/6)     5     3     36     0 [0ex]0ex4ex 
 [1ex]
			9|c|G_26 Type C Comparisons [0pt]0pt3ex
 
			-z^6+1     (0, 0, 2, -1)     Type III, ω = 0     (1, 1/2, 1/2, 2/3)     ( 1/2, 1/3, 1/2, 1/2)     5     3     12     12, 1 [0ex]0ex4ex 
 [1ex]   Here w = ζ_3, z = ζ_36. The correct signature for Tykhyy's orbit 5 would have 3(7/18) ≡1/6 as σ_24 but such a signature was not in the orbit so we provide the closest match. The subgroups 24, 4 and 12, 1 are the dicyclic groups of order 24 and 12 respectively.
	
	

  §.§.§ G_27
	
	This group is of order 2160 and is also denoted W(J_3(5)).
	This group only has one conjugacy class of reflections with nontrivial eigenvalue -1 and (G_27) = {6, 12, 30}. The original base ring of this group in Magma is the cyclotomic field of order 15, and we expanded the base ring to the cyclotomic filed of order 60 to ensure all the eigenvalues are found. We found three distinct inverse pair types in this group.
	
	
	Writing z = ζ_60 and setting u = -z^10-z^8+z^2 and v = z^16 + z^4, we get that z^50v = -z^14+ z^6+ z^4 so we can write the exemplar for Type A as: 
	
    A_1 = [  z^50v      0   z^20;   -v+1      1    v+1;      v      0 -z^50v ]   A_2 = [        0        0    -z^20; u+z^10+2        1   v+z^10;     z^10        0        0 ]   A_3 = [  1  1  0;  0 -1  0;  0  1  1 ]

	Using the same notation, we can write the Type B exemplar as: 
	
    B_1 = [   -1    u    u;  v+1   -u -u-1; -v-1  u+1  u+2 ]   B_2 = [  z^50v      0   z^20;   -v+1      1    v+1;      v      0 -z^50v ]   B_3 = [  1  1  0;  0 -1  0;  0  1  1 ]

	Similarly, the Type C exemplar is:
	
    C_1 = [  z^50v      0   z^20;   -v+1      1    v+1;      v      0 -z^50v ]   C_2= [  1  1  0;  0 -1  0;  0  1  1 ]    C_3 = [    1    0    0; -v-1  u+1  u+2;  v+1   -u -u-1 ]

	In G_27 for all three types, different exemplars do not produce new orbits. Moreover the inverse types do not produce new orbits and we have checked that these 9 orbits are distinct. The middle convolution details for Type A, B and C are listed in Table <ref>, <ref>, and <ref> respectively.  The comparison data is provided in Table <ref>. In this group we also find nice 4-tuples, details provided in Section <ref>
	
	
	

 §.§ Nice 4-Tuples in Rank 3 Groups

	Nice tuples in this group are 4-tuples of reflections generating the whole group such that the inverse product has a nontrivial eigenvalye of multiplicity 2. If (ζ) denotes the trace field for some FCRG W, and we have a matrix A with eigenvalues {λ, λ, λ_1}, then we know (A) = 2λ + λ_1 ∈(ζ). If we further knew that λ∈(ζ) then we can conclude that λ_1 ∈(ζ) and so we do not need to extend the base field to ensure all characteristic polynomials of such matrices A split. By Theorem <ref> we know that any such λ of multiplicity 2 must have order dividing at least two of the degrees of W. Studying degrees, we see that in G_23 and G_24, λ must have order dividing 2, i.e. λ = -1 and so we do not need to expand the base ring. Similarly, for G_25, G_26, G_27 such  λ must have order dividing 6, which means we do not have to expand the base ring in these cases either. 
	Our search found nice 4-tuples in all the primitive rank 3 groups except for G_24. 
	
	In comparing with the existing classifications, we can only compare with Tykhyy's ℳ^(5) orbits <cit.> since our 4-tuples gives us local systems on the 5-punctured sphere. Unfortunately, without the ω parameters from the work of Lisovy-Tykhyy <cit.> like in the 3-tuple cases, it is more difficult to relate the our orbit sizes to Tykhyy's. We still computed all the signatures for the different possible equivalent representatives for each orbit and attempted to find a match, which was successful. 
	
	

  §.§.§ G_23
 
	We only found one type of nice tuple. 
	

	Letting q = ζ_5, here is an exemplar for Type A: 
	
    A_1 = [    q^3+q^2    q^3+q^2    q^3+q^2; -q^3-q^2+1 -q^3-q^2+1   -q^3-q^2;    q^3+q^2         -1          0 ]     A_2 = [ -q^3-q^2+1 -q^3-q^2+1   -q^3-q^2;    q^3+q^2    q^3+q^2    q^3+q^2;         -1    q^3+q^2          0 ]
     A_3 = [   -q^3-q^2          1          0;    q^3+q^2    q^3+q^2          0;   -q^3-q^2 -q^3-q^2+1          1 ]      A_4 = [       -1        0        0; -q^3-q^2        1        0;        0        0        1 ]

	The middle convolution details can be found in Table <ref>. This type is its own inverse. 
	For this tuple we could not compute the orbit size as it was too large, but we were able to find a signature in the orbit (up to Tykhyy's equivalence relations) that agreed with one listed in Tykhyy's paper. Because we cannot compute the full braid group orbit, we also cannot check if different exemplars would result in a different orbit after middle convolution. We provide the comparison details in Table <ref>. 
	
	
	

  §.§.§ G_25

	We found 6 different types which split into 3 inverse pairs, so we only provide the details for one type from each pair in Table <ref>. 
	
	
	For all the exemplars listed below, z = ζ_3. 
	Here is an exemplar for Type A: 
	
    A_1 = [   z   0   0; z^2   1   0;   0   0   1 ]   A_2 = [  -z   1  -z;   0   1   0;   z z^2   0 ]   A_3 = [   1   z   0;   0 z^2   0;   0   z   1 ]   A_4 = [  -z   1 z^2;   1 z+1   1; z^2   1  -z ]

	Here is an exemplar for Type B: 
	
    B_1 = [  -z   1   0; z+1   0   0;   1   z   1 ]   B_2 = [  -z   1  -z;   0   1   0;   z z^2   0 ]   B_3 = [   0 z^2   z;   0   1   0;  -z   1  -z ]   B_4 = [   1   z   0;   0 z^2   0;   0   z   1 ]

	Here is an exemplar for Type C:
	
    C_1 = [   1   z   1;   0   0 z+1;   0   1  -z ]   C_2 = [   z   0   0; z^2   1   0;   0   0   1 ]   C_3 = [ z+1  -1 z+1;   0   1   0; z^2  -z   0 ]   C_4 = [  -z   1   0; z+1   0   0;   1   z   1 ]

	The middle convolution details for G_25 can be found in Table <ref>.
	The orbits for Type A and Type B are the same. All three types do not split into new orbits and their inverses do not produce new orbits. The comparison details for the two distinct orbits can be found in Table <ref>.
	
	

  §.§.§ G_26

	In this group we found two inverse pairs of nice tuples and we provide the details for one type from each pair in Table <ref>. 
	
	With the notation z = ζ_3, here is an examplar for Type A: 
	
    A_1 = [     z   z+2     1;     0     1     0;   z+2 z^2-z    -z ]   A_2 = [   z   0   0; z^2   1   0;   0   0   1 ]   A_3 = [ -z+1 2z^2  z^2;  z^2   -1   -1;    0    0    1 ]   A_4 = [    -z    -1    -1;     z   z+2   z+1; z^2-z z^2-1   z^2 ]

	The exemplar for Type B: 
	
    B_1 = [    1  z^2    0;    0    z    0;    0 -z+1    1 ]   B_2 = [   -z   -1    0;  z^2    0    0; 2z+1  z+2    1 ]   B_3 = [   z   0   0; z+1   1   0;   0   0   1 ]   B_4 = [     1 z^2-z    -z;     0   z^2   z^2;     0  2z+1   z+1 ]

	The middle convolution details for G_26 can be found in Table <ref>. 
	Neither of these types split into multiple orbits nor do their inverse types produce new orbits. The finite group generated by the matrices in SL_2() are the same for both groups, Magma identifies this as SmallGroup<24, 4> which corresponds to the dicyclic group of order 24.
	
	
	

  §.§.§ G_27
 
	In this group we found one inverse pair, and we provide the details for one type of the pair. 
	
	
	Here is an exemplar of Type A, where z = ζ_15, u = z^4 + z, v = z^7-z^3+z^2-1: 
	
    A_1    = [    u  u-1   -1; -u-1    u -v-u;    0    0    1 ]   A_2 = [ -1  0  0;  1  1  0; -u  0  1 ]   A_3 = [          u+2v        v+z^10 -u+z^5(z^5-1);             0             1             0;          -u-1        -v-u-1         -2v-u ]
     A_4    = [ -v-u-z^5   -v-z^5      u+1;  v+u+z^5        v  -u+z^10;   -v-u+1  -v-z^10   u-z^10 ]

	
	The middle convolution details can be found in Table <ref>. In this case we could not compute the orbit size since it was too long. Similar to G_23, we were able to compute some signatures for tuples in the orbit and we found a match with a signature listed in Tykhyy's classification.
	
	
	

 §.§ Nice 4-Tuples in Rank 4 Groups

	In this case we are looking for nice tuples of 4 reflections generating the whole group such that their inverse product has a nontrivial eigenvalue of T-2 = 2. Of the five rank 4 reflection groups, we found nice 4-tuples in G_28, G_30 and G_32. In the case of G_32, the field containing all the eigenvalues would be (ζ_360) which is much too large to perform the computations over. For G_32, in Section <ref> we justify why it is enough to search for such tuples over (ζ_12). 
	
	After taking the middle convolution, multiplying with a suitable character and adjoining the inverse product, our tuples become SL_2() local systems on the 5-punctured sphere. We found that in some cases, one of the 5 matrices in the local system ends up being the identity, which means our tuple comes from some local system on the 4-punctured sphere. In the case where we have 4 nontrivial matrices, we can compute the ω parameters in the notation of Lisovy-Tykhyy <cit.> and identify the corresponding ℳ^(4) orbit in Tykhyy's paper <cit.>. When all 5 of the matrices are nontrivial, we can only compare with Tykhyy's ℳ^(5) orbit signatures from <cit.>.
	
	

  §.§.§ G_28
 
	The group G_28 of order 1152 is also denoted W(F_4) with degrees (G_28) = {2, 6, 8, 12} <cit.>. 
	All reflections in G_28 have nontrivial eigenvalue -1, and form two conjugacy classes of reflections. We only found one type of nice 4-tuples in this group, with two possible choices of λ parameters for middle convolution. 
	
	An exemplar for such a tuple of Type A is: 
	
    A_1 = [  2  3  4  2; -1 -2 -4 -2;  0  0  1  0;  0  0  0  1 ]   A_2 = [  1  2  2  0;  0 -1 -2  0;  0  0  1  0;  0  1  1  1 ]   A_3 = [ -1 -2 -4 -2;  2  3  4  2; -1 -1 -1 -1;  0  0  0  1 ]   A_4 = [ -1  0  0  0;  1  1  0  0;  0  0  1  0;  0  0  0  1 ]
	
	A different choice of exemplar does not produce a new orbit and the two orbits computed above are the same, so we only get one new orbit from this group. The middle convolution details can be found in Table <ref>. After putting the matrices in SL_2(), we see that the inverse product of the middle convolution of this tuple is the identity, so this tuple comes from a local system on the 4-punctured sphere. As in the case of 3-tuples, we provide the details comparing with the results of both Lisovy-Tykhyy and Tykhyy (<cit.>, <cit.>) in Table <ref>. The ω parameters appear to correspond to Orbit 31, but in fact this orbit is of Type III, as described in Lemma 39 in <cit.>.
	

	
	

  §.§.§ G_30
 
	The group G_30 of order 14400 is also denoted W(H_4) with degrees (G_30) = {2, 12, 20, 30} <cit.>.
	All reflections in G_30 have nontrivial eigenvalue -1 and are conjugate to each other. Looking at the degrees, we extend the base field to (ζ_60). We found three distinct types of nice 4-tuples, the details are in Table <ref>.
	
	For all of the exemplars below z = ζ_60 and u = z^14-z^6-z^4. 
	An exemplar of type A is given by: 
	
    A_1 = [  1  0  0  0;  0  1  0  0;  0  0  1  1;  0  0  0 -1 ]   A_2 = [     1     0     0     0;  2u-1  2u-1   u-1    -1; -2u+1 -2u+2  -u+2     1;  -u+1   -2u    -u    -u ]
     A_3 = [     u   u-1   u-1    -1; -2u+1 -2u+2 -2u+1    -u;  2u-1  2u-1    2u     u;    -u    -u    -u    -u ]   A_4 = [ -u+1 -u+1 -u+1   -u;    0    1    0    0;    u  u-1    u    u;   -1    u    u    0 ]
	
	An exemplar of type B is given by:
	
    B_1 = [  1  0  0  0;  0  1  0  0;  0  0  1  1;  0  0  0 -1 ]   B_2 = [    0   -1   -1   -1;   -u -u+1   -u   -u;    0    0    1    0;  u-1  u-1  u-1    u ]
     B_3 = [  1 -u -u -u;  0  0 -1 -1;  0  0  1  0;  0 -1 -1  0 ]   B_4 = [ -2u+2 -3u+2 -2u+1  -u+1;  2u-1  3u-1  2u-1   u-1;    -u  -u+1  -u+1     1;     u   u-1     u     0 ]

	An exemplar of type C is given by: 
	
    C_1 = [  1  0  0  0;  0  1  0  0;  0  0  1  1;  0  0  0 -1 ]   C_2 = [     1     0     0     0;  -u+1 -2u+1  -u+1    -u;   u-1    2u     u     u;  2u-1  2u-2  2u-1     u ]
     C_3 = [   u-1  2u-1    2u     u; -3u+1 -3u+3 -2u+2  -u+1;  3u-1  3u-2  2u-1   u-1;     0     0     0     1 ]   C_4 = [    0   -1   -1   -1;   -u -u+1   -u   -u;    0    0    1    0;  u-1  u-1  u-1    u ]

	The middle convolution details for each type is presented in Table <ref>. For all three types, different exemplars do not produce new orbits and since each type is their own inverse type, these are all the orbits from this group.  Further, the two orbits produced from Type A are distinct, as are the two from Type C. The two orbits from Type B are actually the same. In total G_30 gives us 5 different orbits. 
	
	For all 5 orbits, once we put the matrices in SL_2(), we see that the last matrix is the identity, hence these actually corresponds to orbits of 4 matrices multiplying to the identity. We compared these to the existing classification(s) by Lisovy-Tykhyy and Tykhyy (<cit.>, <cit.>) as we did for the rank 3 complex reflection groups. Within each type, the two orbits are equivalent to the same orbit of ℳ^(4) in Tykhyy's classification, so we only provide the details once in Table <ref>.
	
	

  §.§.§ G_32
 
	The group G_32 of order 155520 is also denoted W(L_4) <cit.>.
	In this group, the nontrivial eigenvalues of reflections are in {ζ_3, ζ_3^2 } corresponding to two conjugacy classes. 
	For G_32, (G_32) = {12, 18, 24, 30} so the eigenvalues of multiplicity 2 must be 12-th roots of unity. Expanding the field to (ζ_12) we found 5 distinct eigenvalue sets for the inverse product. Note that to ensure we found all examples, we would have to run the code over (ζ_360) since there may be matrices with one or two eigenvalues other than the 12-th roots which would not be found, however we claim it is enough to check over (ζ_12). 
	
		If A = A_1A_2A_3A_4, with [A_1, A_2, A_3, A_4] a nice tuple in G_32 so that A has a nontrivial eigenvalue of multiplicity 2, then the other two eigenvalues of A are both m-th roots of unity (not necessarily primitive) for the same m ∈{12, 18, 24, 30}.
	
	
		Let the eigenvalues of A be {λ, λ, λ_1, λ_2}. Taking residues, we write 
    Res(λ) = x/12,    Res(λ_1) = p_1/u_1· 6,    Res(λ_2) = p_2/u_2 · 6
 where u_i ∈{2, 3, 4, 5} which we know must be the case by the degrees of G_32. Suppose λ_i are not 24-th roots of unity. So u_i must be 3 or 5 and (p_i, u_i) = 1. 
		
		Using that (A_i) is a 3-rd root of unity, we know that (A) ∈{1, ζ_3, ζ_3^2}. This tells us that λ_1λ_2 = (λ^2)^-1(A) and so λ_1λ_2 must be a 6-th root of unity since λ^2 is a 6-th root.  
		In terms of the residues, this means 
		
    p_1/u_1 · 6 + p_2/u_2 · 6 = u_2p_1 + u_1p_2/6 · u_1u_2 = y/6
 for some 1≤ y ≤ 6.
		
		This equation tells us that u_1 | u_2 p_1 and since (p_1, u_1) = 1 we know u_1 | u_2. Similar arguemnt shows u_2 | u_1, so u_1 = u_2 if one of them is 3 or 5.
		
		Suppose now that Res(λ_1) = p_1/4 · 6, is a primitive 24-th root of unity, i.e. u_1 = 4, so (p_1, 24) = 1. We know u_2 is not 3 or 5, so it can only be 2 or 4. If u_2 = 4 and (p_2, 24) = 1 then we are done since they are both primitive 24th roots. Else λ_2 is a 12-th root of unity, which means λ_1 = (λ^2)^-1(A) λ_2^-1. Since the right hand side is a product of 12th roots, λ_1 must also be a 12th root, completing the proof.
	
	In the notation of the lemma above, we know that for nice 4-tuples, if the eigenvalues of the inverse product A is {λ, λ, λ_1, λ_2} then λ_1 + λ_2 = -2λ + (A) ∈(ζ_12) and similarly λ_1λ_2 = λ^2 (A) ∈(ζ_12). 
	With the result from the lemma, we do computer search in each of the fields (ζ_18), (ζ_24), (ζ_30) to find a pair of roots of unity (s, t) other than 12-th roots such that s +t, s· t ∈(ζ_12). For (ζ_18) and (ζ_30) no such pair could be found. In (ζ_24) one example that works is the pair (ζ_24, -ζ_24), however no pairs are found if we further require that s ≠ -t. This tells us the only cases we might be missing when searching over (ζ_12) are tuples whose inverse product has eigenvalues {λ, λ, λ_1, -λ_1} with λ a 12th-root (not nec. primitive) and λ_1 a primitive 24th root. In the notation of the lemma above, using that (A) is a 3rd root, we know -λ^2 ·λ_1^2 is also a 3rd root. Note that if λ_1 is a primitive 24th-root then -λ_1^2 is a primitive 12th-root. Again using a computer (or by hand) we can check that no primitive 12th root multiplied with the square of a 12th-root can be a 3rd root. Thus, we conclude it is enough to search for tuples over (ζ_12).
	
	Searching for nice tuples of in G_32 we find 3 inverse pairs. We list only one from each pair, noting that Type B is its own inverse.
	
	
	For all the exemplars listed below, w = ζ_3. 
	An exemplar of type A is given by: 
	
    A_1 = [    1    0    0    0; -w^2    w  w+2    w;   -w  w+2  -2w    1;    0    0    0    1 ]   A_2 = [   1   0   0   0;   0   1 w^2   0;   0   0   w   0;   0   0 w^2   1 ]
     A_3 = [    1    0    0    0; -w^2    w -w^2    0;    0    0    1    0;  w^2 -w+1  w^2    1 ]   A_4 = [      0      w  w^2-1     -w;   -w^2      2 -w+w^2      1;      w   -w^2      w    w^2;      0      0      0      1 ]
	
	An exemplar of type B is given by: 
	
    B_1 = [  -w   1  -w   0;   0   1   0   0;   w w^2   0   0; w^2   1  -w   1 ]   B_2 = [   1   0   0   0;   0   1 w^2   0;   0   0   w   0;   0   0 w^2   1 ]
     B_3 = [    1    w -w^2    w;    0   -w   -1  w^2;    0   -1 -w^2   -1;    0  w^2   -1   -w ]   B_4 = [    0   -w  w^2   -w;    0    1    0    0;    1    w  w+2    w; -w^2   -1    w    0 ]
	
	An exemplar of type C is given by: 
	
    C_1 = [      1      0      0      0;      0      1      0      0;     -1 -w+w^2     -1    w^2;    w^2   -w+1   2w^2   -w+1 ]   C_2 = [      0    w^2    w-1    w^2;     -w      2 -w+w^2      1;    w^2     -w    w^2     -w;      0      0      0      1 ]
     C_3 = [    1    w    1 -w^2;    0    0 -w^2    w;    0    0    1    0;    0   -w   -1   -w ]   C_4 = [   1   w   0   0;   0 w^2   0   0;   0   w   1   0;   0   0   0   1 ]
	
	
	The middle convolution details can be found in Table <ref>. Different exemplars nor the inverse types produce new orbits. The two orbits from Type B are distinct as are the two orbits from Type C.
	Of the five middle convolutions computed in G_32, after putting the matrices into SL_2() three of them (one from Type B and both orbits from Type C) produce tuples where the 5-th matrix is the identity. For these three, we compare with the classification(s) by Lisovy-Tykhyy and Tykhyy (<cit.>, <cit.>) to see which ℳ^(4) orbits they correspond. The orbits from Type C correspond to the same orbit in Tykhyy's classification so we only provide the details for it once. 
	The tuples that correspond to nontrivial local systems on the 5-punctured sphere are those coming from Type A and the other middle convolution from Type B. For the Type B tuple, after putting the matrices into SL_2(), we get that the product of the third and fourth matrix is not of finite order, hence we cannot compute its signature to compare with Tykhyy's table of ℳ^(5).
	The matrices in this tuple MC_-z^2+1(𝐁) are: 
	
    B̂_̂1̂   = [  z^3    z;    0 -z^3 ], B̂_̂2̂ = [  z^3-z      0; -z^3+z     -z ],
    		B̂_̂3̂ = [ -z^3   -z;    0  z^3 ],
    		B̂_̂4̂ = [      z      0;     -z -z^3+z ],
    		B̂_̂5̂ = [ z^2-1     0;     0  -z^2 ]

	
	
		
	
	
	

 §.§ Nice 5-Tuples in Rank 4 Groups

	In this case we are searching for tuples of 5 reflections generating the group such that their inverse product has a nontrivial eigenvalue of multiplicity 3. Searching all the rank 4 groups, we found that such tuples only existed in G_32. 
	

  §.§.§ G_32

	Since (G_32) = {12, 18, 24, 30}, we know that any eigenvalue of multiplicity three must be a 6-th root of unity. Using the trace, we then get that the 4-th eigenvalue must also be a 6-th root, so we do not need to expand the field to compute all the examples. 
	In G_32, there were too many possible tuples to perform an exhaustive search for all the nice tuples. We were able to find at least 4 types, which correspond to two inverse pairs, so we only provide the details for one type from each pair. 
	
	
	An exemplar of Type A is provided below with w = ζ_3: 
	
    A_1    = [    1    0    0    0;  w+1    w  w+1    0;    0    0    1    0; -w-1 -w+1 -w-1    1 ]   A_2 = [    1    0    0    0;    0    1 -w-1   -w;    0    0  w+1   -1;    0    0    w    0 ]   A_3 = [   1   w   1 w+1;   0   0 w+1   w;   0   0   1   0;   0  -w  -1  -w ]
     A_4    = [    w    0    0    0; -w-1    1    0    0;    0    0    1    0;    0    0    0    1 ]   A_5 = [    1    0    0    0;    1 2w+2 -w+1  w+1; -w-1 -w+1 -w-1   -w;    0    0    0    1 ]

	An exemplar of Type A is provided below with w = ζ_3:
	
    B_1    = [    1    w    1    w;    0    0  w+1   -1;    0  w+1 -w+1  w+1;    0   -1  w+1    0 ]   B_2 = [    w    0    0    0; -w-1    1    0    0;    0    0    1    0;    0    0    0    0 ]   B_3 = [   1   w   1   0;   0   0 w+1   0;   0   1  -w   0;   0 w+1  -w   1 ]
    
    		B_4    = [    1    0    0    0;    0    1 -w-1    0;    0    0    w    0;    0    0 -w-1    1 ]   B_5 = [    1  w+1   -w    1;    0  w+1    1  w+1;    0    0    1    0;    0 -w-1    w    0 ]

	The details of the middle convolution for both types is presented in Table <ref>. The SL_2() subgroups generated by both tuple types is the dicyclic group of order 24. Inspecting the 6 SL_2() matrices we get for both of these types, both tuples satisfy the description of `TYPE' A of ℳ^(6) belonging to the dihedral group in Tykhyy's classification <cit.>. In these cases we also could not check if different examplars/ inverses produce different orbits as we could not generate the list of all nice tuples. 
	plain
	 
	
	The notation for all the appendices are the same.
	The `ξ' column gives the residue of the root of unity in the `λ' column used to compute middle convolution. The `Character' column is the character we tensor our tuple with to get matrices in SL_2(). The column `O. Size' is the size of the SL_2() character variety orbit of the tuple (i.e after it is tensored with the character). In these tables `S. Size' denotes the size of the GL_2() subgroup generated by the 2 × 2  matrices  we get after middle convolution but before we tensor with the character. The SL_2() subgroup size after tensoring with the character is provided in each section for each group as part of the comparison tables. For infinite subgroups, the order is listed as `0', following the notation used in Magma.
	

§ MIDDLE CONVOLUTION TABLES FOR NICE 3-TUPLES
 
	

		 
			8|c|G_23 Middle Convolutions	[0pt]0pt3ex

			ξ     λ     M_1     M_2     M_3     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
			8|c|G_23 Type A Middle Convolutions [0pt]0pt3ex
 
			1/2     -1     [            1            0; -q^3-q^2 + 1            1 ]     [  1 -1;  0  1 ]     [ 1 0; 1 1 ]     (1, 1, 1, 1)     40     0 [0pt]0pt4ex 
 
			1/10     -q^3     [  q^3    0; -q^3    1 ]     [ q^3  -1;   0   1 ]     [   1   0; q^3 q^3 ]      (q, q, q, q^2)     10     600 [0pt]0pt4ex
 9/10     -q^2     [  q^2    0; -q^2    1 ]     [ q^2  -1;   0   1 ]     [   1   0; q^2 q^2 ]     (q^4, q^4, q^4, q^3)     10     600 [0pt]0pt4ex
 8|c| G_23 Type B Middle Convolutions [0pt]0pt3ex
 
			1/2     -1     [ 1 0; 1 1 ]     [  1 -1;  0  1 ]     [         1         0; q^3+q^2+2         1 ]     (1, 1, 1, 1)     40     0 [0pt]0pt4ex 
 
			3/10     -q^4     [        q^4          0; -q^4+q^3+1          1 ]     [ q^4  -1;   0   1 ]     [         1         0; q^4-q^3-1       q^4 ]     (q^3, q^3, q^3, q)     10     600 [0pt]0pt4ex
 7/10     -q     [       q       0; q^2-q+1       1 ]     [  q -1;  0  1 ]     [        1        0; -q^2+q-1        q ]     (q^2, q^2, q^2, q^4)     10     600 [0pt]0pt4ex
 8|c| G_23 Type C Middle Convolutions [0pt]0pt3ex
 
			1/2     -1     [  u-1 3u-5;    1 -u+3 ]     [  1 -1;  0  1 ]     [   1   0; u+1   1 ]     (1, 1, 1, 1)     72     0 [0pt]0pt4ex 
 
			1/6     z^5     [ -u-v-z^10      -z^5; u+2v+z^10       u+v ]     [ -z^5   -1;    0    1 ]     [        1        0; -u-v-z^5     -z^5 ]     -(z^5, z^5, z^5, -1)     18     360 [0pt]0pt4ex
 5/6     -z^10     [     z^5+v      z^10; -u-2v-z^5         v ]     [ z^10   -1;    0    1 ]     [      1      0; v+z^10   z^10 ]     (z^10, z^10, z^10, 1)     18     360 [0pt]0pt4ex
  tableHere q = ζ_5, z = ζ_30, u = z^7-z^3-z^2+1 and v = -z^4+z
	

		

			
				8|c|G_24 Middle Convolutions
				[0pt]0pt3ex
 ξ     λ     M_1     M_2     M_3     Character     O.Size     S. Size [0pt]0pt3ex
 [0.5ex]
				8|c|G_24 Type A Middle Convolutions [0pt]0pt3ex
 
				1/14     -z^4     [    z^5+z^4+z+1 -z^5+z^4+z^3+1;           -z^2         -z^5-z ]     [ z^4   1;   0   1 ]     [    1    0; -z^4  z^4 ]     (z^5, z^5, z^5, z^6)     28     0 [0pt]0pt4ex
 9/14     -z     [ -z^6-z^5-z^4    z^6-z^3+1;         -z^4     -z^3-z^2 ]     [ z 1; 0 1 ]     [  1  0; -z  z ]     (z^3, z^3, z^3, z^5)     28     0 [0pt]0pt4ex
 11/14     -z^2     [        -z^5-z^3-z -z^6 +z^5 +z^2 +1;                -z          -z^6-z^4 ]     [ z^2   1;   0   1 ]     [    1    0; -z^2  z^2 ]     -(z^6, z^6, z^6, z^3)     28     0 [0pt]0pt4ex 
 
				 tableHere z = ζ_7 
		

		
		
		
		
		
		
			 
				8|c|G_27 Type A Middle Convolutions [0pt]0pt3ex
 
				ξ     λ     Ã_̃1̃     Ã_̃2̃     Ã_̃3̃     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				2/60     z^2     [                  z^6u-z^2 2(z^50+z^4)+z^22+z^18-z^6;             (u+1)(z^12+1)                   -z^6u+1 ]     [ -z^2 z^50;    0    1 ]     [            1            0; z^24+2z^12+1         -z^2 ]     (z^14, z^14, z^14, z^18)     60      0 [0pt]0pt4ex 
 
				38/60     -z^8     [                z^24-z^20 (z^28+2z^4)u +2(z^8-z^2);             z^14+u-z^2-1              -z^24-u+z^2 ]     [   z^8 -z^20;     0     1 ]     [          1          0; -2z^18-z^6        z^8 ]     (z^26, z^26, z^26, z^42)      60     0 [0pt]0pt4ex
 50/60     -z^20     [   -u-z^20         u; -u-z^20-1       u+1 ]     [ -z^20  z^20;     0     1 ]     [           1           0; -z^24+z^6+1       -z^20 ]     (z^20, z^20, z^20, 1)     60     360 [0pt]0pt4ex
  Here z = ζ_60 and u = -z^10-z^8+z^2
			

			 	
				8|c|G_27 Type B Middle Convolutions [0pt]0pt3ex
 
				ξ     λ     B̃_̃1̃     B̃_̃2̃     B̃_̃3̃     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				5/60     z^5     1/3[       z^5f_1-2z^10+1  f_1+2z^15+3z^10-z^5;              f_1-g_1 -z^5f_1+2z^10-3z^5+2 ]     [         -z^5 -z^20f_1-u+2;            0            1 ]     [          1          0; z^25u+z^15       -z^5 ]      (q^25, q^25, q^25, q^45)     96     0 [0pt]0pt4ex 
 
				35/60     -z^5     1/3[     -z^5f_1-2z^10+1 f_1-2z^15+3z^10+z^5;             f_1+g_1 z^5f_1+2z^10+3z^5+2 ]     [          z^5 -z^20f_1-u+2;            0            1 ]     [          1          0; z^25u-z^15        z^5 ]     (q^55, q^55, q^55, q^75)     96     0 [0pt]-5pt4ex
 50/60     -z^20      [ f_1-u-1+2z^10     -f_1+2u+1;   f_1-u+2z^10   -f_1+u-z^20 ]     [      z^20 -f_1+2u+2;         0         1 ]     [       1       0; f_1-u+1    z^20 ]     (z^20, z^20, z^20, 1)      96     360 [0pt]-5pt4ex
  Here z = ζ_60, q = ζ_120, f_1 = -z^14-2z^10-z^8+z^6+z^4+z^2 and g_1 = 4z^15+3z^13-3z^11-3z^9-3z^7-5z^5+3z.
			

			 
				8|c|G_27 Type C Middle Convolutions [0pt]0pt3ex
 
				ξ     λ     C̃_̃1̃     C̃_̃2̃     C̃_̃3̃     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				10/60     z^10      [ z^50v-z^10-1 z^50v-u+z^10;     2z^50v-3     -z^50v+2 ]     [ -z^10   u+2;     0     1 ]     [                     1                     0; 1/2(3z^20v+u+2z^10+4)                 -z^10 ]     (z^10, z^10, z^10, -1)     60     360 [0pt]0pt4ex 
 
				34/60     -z^4     [           -z^8-z^6    z^18f_1+2z^12+4; z^14-z^50v+2z^10+u          u-z^50v+1 ]     [ z^4 u+2;   0   1 ]     [                          1                          0; 1/2(z^4f_1+3z^24-z^12+z^6)                        z^4 ]     (z^28, z^28, z^28, z^36)     60     0 [0pt]0pt4ex
 46/60     -z^16      [      z^14-z^4+z^2               f_2; -v+z^28v+z^10-z^8    (z^50+z^28)v-u ]     [ z^16  u+2;    0    1 ]     [                          1                          0; 1/2(z^22+2z^16u+z^10-3z^6)                       z^16 ]     (z^22, z^22, z^22, z^54)     60     0 [0pt]0pt4ex
  Here z = ζ_60, v = z^16+z^4, u = -z^10-z^8+z^2, f_1 = -z^14-2z^10-z^8+z^6+z^4+z^2 and f_2 = -4z^14-z^12+z^10+2z^6+3z^4-2z^2
		
		
		

§ MIDDLE CONVOLUTION TABLES OF NICE 4-TUPLES
 
		In these sections the notation is the same as above.
		
		
		

			
			 
				10|c|G_23 4-Tuple Middle Convolutions [0pt]0pt3ex
 
				Type     ξ     λ     Ã_̃1̃     Ã_̃2̃     Ã_̃3̃     Ã_̃4̃      Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				A     1/2     -1     [    u -u+1;  u-1 -u+2 ]     [  0  1; -1  2 ]     [   1 u+2;   0   1 ]     [  1  0; -1  1 ]     (1, 1, 1, 1, 1)     -     0 [0pt]0pt4ex 
 
				 tableHere q = ζ_5 and u = q^3 +q^2
		

		

			 
				10|c|G_25 4-Tuple Middle Convolutions [0pt]0pt3ex
 
				Type     ξ     λ     M̃_̃1̃     M̃_̃2̃     M̃_̃3̃     M̃_̃4̃     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				A      1/3     z     [  0 -z;  z -z ]     [ 1 1; 0 1 ]     [ 1 1; 0 1 ]     [  1  0; -1  1 ]     (z^2, 1, 1, 1, z)       45     0 [0pt]0pt4ex 
 
				B      1/3     z     [   1 z^2;   0   1 ]     [  1  0; -z  1 ]     [  1  0; -z  1 ]     [  1  0; -z  1 ]     (1, 1, 1, 1, 1)      45       0[0pt]0pt4ex 
 
				C     1/6     z+1     [ -z  0; -1  1 ]     [    -z    -1; z^2-1     z ]     [  -1 z+1;   0   1 ]     [  1  0;  1 -z ]     (u^7, u^3, u^3, u^7, u^4)      120      72 [0pt]0pt4ex 
 
				 tableHere z = ζ_3 and u = ζ_12 
		

		
			 
				10|c|G_26 4-Tuple Middle Convolutions  [0pt]0pt4ex

				Type     ξ     λ     M̃_̃1̃     M̃_̃2̃     M̃_̃3̃     M̃_̃4̃     Character     O. Size     S. Size [0pt]0pt4ex 

				A     1/6     z+1     [ z^2   0; z+2   1 ]     [ -1 -1;  0  1 ]     [   1   0; z^2  -z ]     [   1   0; z^2  -z ]     (u^8, -u^3, u^7, u^7, u^5)      120      72 [0pt]0pt4ex 
 
				B     1/6     z+1     [   -z  z+1; -z+1    z ]     [  1  0;  z -z ]     [ -1 -z;  0  1 ]     [    1    0; 2z+1  z^2 ]     (u^3, u^7, u^3, u^8, u^3)      240       72 [0pt]0pt4ex 
 
				 Here z = ζ_3 and u = ζ_12
			 
				10|c|G_27 4-Tuple Middle Convolutions [0pt]0pt3ex
 
				Type     ξ     λ     M̃_̃1̃     M̃_̃2̃     M̃_̃3̃     M̃_̃4̃     Character     O. S.     S. Size [0pt]0pt4ex
 [0.5ex] 
				A     5/6     -z^5     [   -f_2+1   1/4g_1;      g_2 z^5 +f_2 ]     [  f_1+2 1/2f_2;    g_3    g_4 ]     [    z^5 1/4g_5;      0      1 ]     [   1   0; g_6 z^5 ]     (z^5, z^5, z^5, z^5, z^10)      -     360 [0pt]0pt4ex 
 
				 Here z = ζ_15 f_1 = z^7-z^3+z^2-1, f_2 = z^5-z^4-z +2, g_1 = 2z^7-3z^5+3z^4-2z^3+2z^2+3z-2, g_2 = 2z^7-z^4-2z^3+2z^2-z+1, g_3 = -3z^7+3z^5-2z^4+3z^3-3z^2-2z^2-2z+3, g_4 = -z^7+z^5+z^3-z^2, g_5 =  -z^7-z^5+z^3-z^2-2, g_6 = -z^7+3z^5-z^4+z^3-z^2-z+2
			 
				9|c|G_28 Type A Middle Convolutions [0pt]0pt3ex
 
				ξ     λ     Ã_̃1̃     Ã_̃2̃     Ã_̃3̃     Ã_̃4̃     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				1/6     z^4     [    1    1;    0 -z^4 ]     [ -z^4+2 -z^4+1;     -2     -1 ]     [ -z^4   -1;    0    1 ]     [     1     0; -2z^4  -z^4 ]     (-z^4, -z^4, -z^4, -z^4, (-z^4)^2)       45     72 [0pt]0pt4ex 
 
				5/6     -z^4+1     [   1   1;   0 z^8 ]     [ z^4+1   z^4;    -2    -1 ]     [ z^8  -1;   0   1 ]     [    1    0; 2z^8  z^8 ]     (z^8, z^8, z^8, z^8,-z^4)       45     72 [0pt]0pt4ex 
 
				 Here z = ζ_24
		
		
			 
				ξ     λ     M_1     M_2     M_3     M_4     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				9|c|G_30 Type A Middle Convolutions [0pt]0pt3ex
 3/10     z^18     [        -z^6v          z^6; -z^24+z^12-1        -z^24 ]     [ -z^18+z^6     -z^18;       v^2        -v ]    [ -z^18    -1;     0     1 ]     [       1       0; -z^18+v   -z^18 ]     (z^6, z^6, z^6, z^6, z^36)      50     600 [0pt]0pt4ex 
 
				7/10     -z^12     [  z^12-v   -z^24; -z^18+v     z^6 ]     [ -z^24+z^12       z^12;  z^24-z^6v     z^24+1 ]     [ z^12   -1;    0    1 ]     [            1            0; -z^24+z^12-1         z^12 ]     (z^24, z^24, z^24, z^24, z^24)    50     600 [0pt]0pt4ex
 9|c|G_30 Type B Middle Convolutions [0pt]0pt3ex
1/6     z^10      [ -z^20-z^16 -z^16-z^10;     z^16-1       z^16 ]    [ -z^10     0;  z^10     1 ]     [ -z^10   u-1;     0     1 ]     [     1     0; -z^10 -z^10 ]     -(z^10, z^10, z^10, z^10, -z^20)     90     360 [0pt]0pt4ex 
 
				5/6     -z^10+1       [   y+z^20 y+z^10-2;       -y     -y+1 ]     [  z^20     0; -z^20     1 ]     [ z^20  u-1;    0    1 ]     [    1    0; z^20 z^20 ]     (z^20, z^20, z^20, z^20, -z^10)     90     360 [0pt]-5pt4ex
 9|c|G_30 Type C Middle Convolutions [0pt]0pt3ex

				1/10     z^6      [    1 -u-1;    0 -z^6 ]     [      -v -z^12+v;       u       0 ]     [ -z^6  u+1;    0    1 ]     [       1       0; -z^12-1    -z^6 ]     (z^12, z^12, z^12, z^12, z^12)     50     600 [0pt]0pt4ex 
 
				9/10     -z^24       [    1 -u-1;    0 z^24 ]     [ z^24+1   z^6v;      u      0 ]     [ z^24  u+1;    0    1 ]     [      1      0; u+z^12   z^24 ]     (z^18, z^18, z^18, z^18, z^48)     50     600 [0pt]-5pt4ex
  Here z denotes a 60th root of unity, u = z^14-z^6-z^4 and v = z^6 -1 and y = z^10 +z^8-z^2.
			 
				ξ     λ     M_1     M_2     M_3     M_4     Character     O. Size     S. Size [0pt]0pt4ex
 [0.5ex] 
				9|c|G_32 Type A Middle Convolutions [0pt]0pt3ex
 5/6     -z^2 +1     [  0 -1; -1  0 ]     [   1   0;   0 z^2 ]     [ z^2   0;   0   1 ]     [   1   0;   0 z^2 ]      (-z^3, z^5, z^5, z^5, 1)      40     72 [0pt]0pt4ex 

				9|c|G_32 Type B Middle Convolutions [0pt]0pt3ex
 1/6     z^2     [  1 -w;  0 -w ]     [  w^2  w-1;   -1 -w^2 ]     [ -w  w;  0  1 ]     [  1  0; -w -1 ]      (z, z^3, z, z^3, z^2-1)      60     72 [0pt]0pt4ex 
 
				5/6     -z^2 +1     [  1 -w;  0 -1 ]     [    1    0;   -1 -w^2 ]     [ -1  w;  0  1 ]     [  1  0; -1 -w ]     (z^3, z^5, z^3, z, 1)         0 [0pt]0pt4ex
 9|c|G_32 Type C Middle Convolutions [0pt]0pt3ex
 1/12     z     [ -z^3    0; z^11    1 ]     [    1  z^4;    0 -z^3 ]     [ -z^3 -z^4;    0    1 ]     [     1     0; -z^11  -z^3 ]      (q^3, q^3, q^3, q^3, -1)      20     96 [0pt]0pt4ex 
 
				7/12     -z     [   z^3     0; -z^11     1 ]     [   1 z^4;   0 z^3 ]     [  z^3 -z^4;    0    1 ]     [    1    0; z^11  z^3 ]     (q^9, q^9, q^9, q^9, -1)    20     96 [0pt]0pt4ex
  Here z = ζ_12, w = ζ_3 and q = ζ_24
		
		
		

§ MIDDLE CONVOLUTION TABLES FOR NICE 5-TUPLES
 
		The notation here is the same as above.

		
		


	
	
	
	
	
	


 Paper Title: Automatic meromorphy in non-archimedean geometry.


Parallel ergotropy: Maximum work extraction via parallel local unitary operations
    Donato Farina
    August 5, 2024
=================================================================================





The purpose of this text is to prove the following theorem and its corollary. 




Let k be a complete, non-archimedean field and let X be a reduced k-analytic space. Let U
be a dense Zariski-open subset of X.
Every analytic function on U whose zero-locus is closed 
inside X
(, an element of 𝒪(U)^×) extends to a meromorphic function on X.



Let X be a reduced scheme of finite type over an affinoid 
algebra 
and let f be an analytic function on X. Assume that (at least)
one of the two following conditions is fulfilled: 



  * f is invertible ; 

  * X is separated and the zero-locus of f is compact. 

Then f 
is algebraic; , it belongs to (the image of) 𝒪_X(X).


[remark]Comments
Our framework is that of Berkovich spaces. 
We refer the reader to the first chapters of <cit.>
and <cit.> for the basic definitions,  <cit.>
and <cit.>
for the “commutative
algebra properties"  (like being normal, reduced, etc.), <cit.>
for the Zariski topology, <cit.> and <cit.> 
for the notion of quasi-smoothness, <cit.>
for the normalization, <cit.>
for the meromorphic functions, and <cit.>
for the analytification of a scheme. 



On a reduced analytic space, two meromorphic
functions that coincide on a Zariski-dense open subset
actually coincide. This will
ensure the uniqueness of the 
meromorphic
extension of
f in  Theorem <ref>, and this will allow us to argue
G-locally on X: since G-local meromorphic extensions
will be canonical, we will be able to glue them. 





Theorem <ref>
exhibits a typical non-archimedean feature. Indeed, it fails definitely 
in the complex-analytic setting, as well as
Corollary <ref>, as witnessed by the exponential function on 𝐀^1,an_𝐂. 



The reducedness assumption is necessary for Theorem <ref>
and
 Corollary <ref>
 to hold, at least when k is not trvially valued. 
Indeed, assume that
k^×≠{1} and choose
f=∑ a_it^i∈ k[[t]] with infinite radius
which is not a polynomial.  Then 
1+ϵ f is an invertible analytic function
on the k-analytic
space 𝐀^1,an_k[ϵ] with
ϵ nilpotent of positive index. And f is not 
algebraic and does not admit any meromorphic extension 
to 𝐏^1,an_k[ϵ]. 

If k is trivially valued, one can still exhibit a 
counter-example to the Theorem, by considering the former
counter-example over K:=k((s)) (equipped with any s-adic
absolute value); indeed, 𝐀^1,an_K[ϵ]
can be seen as a 2-dimensional (non-strict) k-analytic space. 
But one cannnot exhibit any counter-example to the Corollary, since
in the trivially valued case, every analytic function on the
analytification of a (possibly non-reduced) k-scheme  of finite
type is algebraic – one checks it on affine schemes by reducing to
the case of the n-dimensional affine space, for which this
is obvious. 




[remark]About the proof of the Theorem
It consists of five steps. 
Steps 1 and 2 are only needed over a non-perfect field
to overcome some possible pathologies. 
If k is perfect, reducedness implies generic quasi-smoothness, 
and every normal space is geometrically normal, which allows
to start from Step 3. 

Roughly speaking, the proof from Step 3 to the end
consists in reducing 
by using 
non-archimedean extension 
theorems through Zariski-closed
subsets (for holomorphic and meromorphic functions)
to the case where X is smooth and
where U is the complement of a smooth hypersurface S,
and then,
by looking at the local structure
of the embedding S↪ X, to the 
the case where X is a product Y×Δ
for Δ some closed disc
and Y smooth and irreducible, where S=Y×{0},
and where f is invertible. 
Then using a “generic fiber"
one more or less reduces to
the case where Y is a point, where the Theorem follows
from the explicit description of invertible analytic functions on an
annulus. The key fact is that such a function has
a “dominant monomial"; this is essentially the only specifically
non-archimedean ingredient of our proof, which
definitely prevents it to be
adapted to the archimedean case (see Remark <ref>). 



We can argue G-locally on X, so we can assume that X is affinoid, say X=ℳ(A). 
We denote by p the characteristic exponent of k. 



  §.§.§ Step 1

Let n be an integer and set L=k^1/p^n. 
Assume that the theorem holds over the field L for (X_L,red,U_L,red). Then it holds
for (X,U). 

Indeed, let f be an analytic invertible function on U; we still denote by f its image in 𝒪(U_L,red). 
Since the theorem holds for  (X_L,red,U_L,red) there exist g and h in A_L:=A⊗_k L
with h regular (, not a zero divisor)
in A_L,red such that f=g/h in 𝒪(U_L,red). We have then fh=g+N in 
𝒪(U_L) for some nilpotent function N on U_L (the sheaf of locally nilpotent functions on U_L is the restriction of
the coherent sheaf on X_L associated with the nilradical of the noetherian ring A, so its sections are actually nilpotent). It follows that
for m≥ n large enough one has f^p^mh^p^m=g^p^m in 𝒪(U_L); but since
m≥ n all functions
involved in this equality belong to the subring 𝒪(U) of 𝒪(U_L), so f^p^mh^p^m=g^p^m  should be understood as
an equality between analytic functions on U.

Set H=h^p^m and G=g^p^m, so that we have f^p^mH=G
on U. Both G and H belong to A. Let us show that 
H is a regular element of A. 
As X is reduced, it suffices to show that the zero-locus of H does not contain any irreducible component of X. But since h
is a regular element of A_L,red, the zero-locus of h in X_L, which is the same as the 
zero-locus of H, does not contains any irreducible component of X_L, and we conclude by noticing that X_L→ X induces 
a homeomorphism for the Zariski topologies on the source and the target.

Since f^p^mH=G one has (fH)^p^m=GH^p^m-1 on U; therefore the analytic function fH is bounded on U. By the non-archimedean version of Riemann's extension theorem 
(essentially due to Bartenwerfer, see Proposition  <ref> (1) of the Appendix for more details), 
fH extends to a holomorphic function on the normalization X' on X. Therefore fH extends to a meromorphic function on X
and so does f since H is regular. 




  §.§.§ Step 2

If n is large enough, a result by Conrad <cit.>, see also <cit.>,
ensures that the normalization of X_k^1/p^n is geometrically normal. 
In view of Step 1, it suffices to prove our theorem for (X_k^1/p^n,red, U_k^1/p^n,red). As a consequence, 
we may assume that the normalization X' of X is geometrically normal
(and then X'_L is the normalization of X_L for every complete extension L of k, see
<cit.>). Let U' be the pre-image of U in X'.

Another result
by Conrad <cit.>; see also <cit.>, ensures that for large enough n, every reduced irreducible
component of X'_k^1/p^n∖ U'_k^1/p^n is geometrically reduced. So using again Step 1, we may moreover assume that 
every reduced irreducible component of X'∖ U' is geometrically reduced. 

Finally we can replace X with X' and U with U', that is, assume that X is normal. And then by arguing componentwise we can 
assume that X is moreover integral. Let n denote its dimension. As U is dense, X∖ U is a Zariski-closed subset of X
of dimension ≤ n-1. 



  §.§.§ Step 3
 
Let K be the completion of an algebraic closure of k. 
Let Y denote the complement of the quasi-smooth locus of X. The space 
X_K is normal, therefore Y_K≤ n-2, hence Y≤ n-2. 
Let Z be a reduced irreducible component of X∖ U. 
The espace Z_K is reduced, thus its quasi-smooth locus is non-empty; therefore
the quasi-smooth locus of Z is non-empty. 

Let us denote by T be the union of Y, of all irreducible
components of X∖ U of
dimension
≤ n-2, and of the non-quasi-smooth locus
of all reduced irreducible components of X∖ U of dimension n-1. 
By construction, T is a Zariski-closed subset of X of dimension 
≤ n-2. As X is  reduced, it follows from an extension theorem
essentially due to Bartenwerfer
(see Proposition <ref> (2) of the Appendix for more details)
that every meromorphic function on  X∖ T extends to a meromorphic function on X. 
It is therefore sufficient to prove that f|_V∩ U extends to a meromorphic function on V for every
affinoid domain V of X∖ T. Hence we
have reduced to the case where X is quasi-smooth and U is the complement
of a quasi-smooth hypersurface S. 



  §.§.§ Step 4

Let x∈ S and let I be the ideal defining 
the smooth hypersurface S. The x-vector space Ω_S/k⊗ x 
of dimension n-1, and is the quotient of  the n-dimensional
space Ω_X/k⊗ x by the subspace
generated by the g̣⊗ 1 for g∈ I. Therefore this subspace is one-dimensional 
and generated by g̣⊗ 1 for some g. Let S' be the Zariski closed subspace of X 
defined by the equation g=0. It contains x
and is everywhere of dimension at least
n-1 by the Hauptidealsatz, 
and Ω_S'/k⊗ x is of dimension n-1 by construction. Therefore S' is quasi-smooth of dimension
n-1 at x, so there exists an affinoid neighborhood V of x such that S'∩ V is quasi-smooth and irreducible
of dimension n-1. As a purely (n-1)-dimensional 
closed analytic subspace of S'∩ V,
the intersection S∩ V
(which is non-empty since it contains x)
is equal to S'∩ V. Otherwise said S∩ V is defined by the single equation g=0. 

Set g_1=g and pick
analytic functions g_2, …, g_n on V such that the g̣_i⊗ 1 generate 
Ω_S/k⊗ x. Then g̣_1⊗ 1,…, g̣_n⊗ 1 generate Ω_X/k⊗ x. The family (g_1,…, g_n) defines a map p from V to 𝐀^n,an_k, which 
by compactness takes value in D^n for some suitable closed disc D. Since S∩ V is described by the equation g=0,
one has S∩ V=p^-1
({0}× D^n-1). The maps p V→ D^n and p|_S∩ V S→{0}× D^n-1≃ D^n-1
are quasi-étale at x by 
construction and in view of <cit.>, so we can shrink V around x so that
both maps are quasi-étale. 
Set q:=Id× p|_S∩ V D× (S∩ V)→ D^n. The map q is quasi-étale.
By construction, p^-1({0}× D^n-1) and q^-1({0}× D^n-1) are isomorphic as
quasi-étale spaces over the closed analytic subspace {0}× D^n-1 of D^n (they both identify to
S∩ V[rr,"p|_S∩ V"']      {0}× D^n-1). 
It follows by <cit.>
(which itself relies on the henselian property of a Berkovich space along a Zariski-closed subspace, 
see <cit.>)
that there exists an analytic neighborhood U of {0}× D^n-1 in D^n such that p^-1(U) and q^-1(U) are
U isomorphic. Up to shrinking U we can 
always assume that it is of the form Δ× D^n-1 where 
Δ is the closed disc of radius r for some suitable r. 
We thus get an isomorphism between 
{v∈ V, g(v)≤ r} and S×Δ 
such that: 

 

  * the second component of this isomorphism is given by g ; 

  * the induced isomorphism between g^-1(0)=S and S×{0} is the obvious one. 





  §.§.§ Step 5

For proving the theorem we may once again argue locally on
X. The theorem obviously holds on X∖ S , 
and if x is a point of S it admits by Step 4 an affinoid
neighborhood V such that (V,S∩ V) is isomorphic
to ((S∩ V)×Δ, (S∩ V)×{0}) for some 
closed one-dimensional disc Δ. Moreover since the
zero-locus of f is closed in X we can shrink V
(and lower the radius of Δ) so that
f|_V∖ S 
is invertible and the smooth space S
∩ V is connected, hence irreducible (and reduced). 

Therefore we can assume that X=Y×Δ for some
irreducible
and reduced analytic space
Y and some closed
disc Δ, that S=Y×{0}
and that the analytic
function f
on Y× (Δ∖{0})
is invertible.
(The quasi-smoothness of S≃ 
Y was useful
for reducing to this
product situation, but will not be used anymore; the fact that Y
is irreducible and reduced
will  be sufficient.)

Let t be the coordinate function on Δ. 
By hypothesis, f is an analytic
(invertible)
function defined on Y× (Δ∖{0}), 
so it can be written ∑_i∈𝐙b_i t^i with b_i∈𝒪(Y).
Choose x∈ Y lying over the generic point
of Spec 𝒪(Y).
The  fiber of Y× (Δ∖{0})
over x is canonically isomorphic to 
Δ_ x∖{0}
and the restriction 
of f to this fiber is ∑ b_i(x)t^i. 
By the explicit description of invertible
functions on an annulus, there exists some integer j
such that
the inequality b_i(x)ρ^i<b_j(x)ρ^j 
holds
for every j≠ i and every ρ∈ (0,r] for
r the radius of Δ. By letting ρ tend to zero, 
we see that b_i(x)=0 for every i<j. Since x
is Zariski-generic on the reduced, irreducible space Y,
this implies that b_i=0 for every i<j. 
Thus f=∑_i≥ jb_i t^i extends to a meromorphic function on
X.



We want to prove that if either (a) or (b) is
fulfilled then f belongs to 𝒪_X(X). 
The latter property is local on X, and invertibility 
is preserved by restriction to Zariski-open subsets; so for dealing
with situation (a) we can moreover assume that X is separated, in which 
case (a) is a particular case of (b); it is therefore
sufficient to handle (b).
By Nagata's compactification theorem 
the separated A-scheme of tinite type
X admits a proper compactification X, 
which can be taken reduced. Being compact in X, the zero-locus of f
is closed in the compact space X. 
By Theorem <ref>, the  analytic function f on X
“is" then a meromorphic function on X. By GAGA for meromorphic function
(see <ref> of the Appendix below), f “is" a meromorphic function 
on X. Then f is a meromorphic function on the scheme X
inducing a holomorphic function on X. By Lemma <ref>
in the Appendix below, f∈𝒪_X(X). 





§ APPENDIX: EXTENSION THEOREMS AND GAGA THEOREMS FOR MEROMORPHIC FUNCTIONS





 §.§ Extension theorems


For proving Theorem <ref>
we have made use of the following non-archimedean analogues of classical complex-analytic
results, both due to Bartenwerfer in the strict case. 


Let X be a k-analytic space and let T be 
a Zariski-closed subset of X with empty interior; set U=X∖ T. 



  * If X is normal any bounded holomorphic function on U extends to a holomorphic function on 
X. 

  * If X is reduced and T is everywhere of codimension ≥ 2 in X every meromorphic
function on U extends to a meromorphic function on X. 

  * If X is normal and T is everywhere of codimension ≥ 2 in X every holomorphic
function on U extends to a holomorphic  function on X. 




If k is non-trivially valued and X is strictly k-analytic, 
(1) is <cit.>
and (2) and (3) are particular cases of
<cit.>. We are just simply
going to explain how to extend these statements 
when
X is not assumed to be strict. Everything being G-local on X,
we can assume that X is affinoid; and since for proving (2) one can replace X
with its normalization, we can assume that X is normal and,  by arguing componentwise, 
irreducible. Let A be the algebra of analytic functions on X, and let r be a
k-free
 polyradius such that
A_r:=A⊗k_r is k_r-strict. The ring A_r is still a normal integral 
domain – this follows for instance from
<cit.>, but 
an elementary proof, 
can be found in <cit.>.
Then Bartenwerfer's statements apply on the k_r-analytic space X_r, 
and for deducing them on X it suffices to prove the following: 



  * if f=∑ a_I T^I is an element of A_r whose
restriction to U_r belongs to 𝒪_X(U) then f∈ A. 

  * If g=∑ b_I T^I and h=∑ c_IT^I are two elements
of A_r with h≠ 0 such that the restriction of g/h to U_r
belongs to 𝒦_X(U) (where 𝒦_X is the sheaf of
meromorphic functions on X), then g/h belongs to the ring of fractions
of A. 


Let us prove (a). If V is an affinoid domain of U then by assumption
the element ∑ (a_I|_V)T^I of 𝒪_X(V)_r belongs to 𝒪_X(V), 
which means that a_I|_V=0 as soon as I≠ 0. Since this holds for arbitrary V
we see that a_I|_U=0 as soon as I≠ 0. As X is reduced and U is dense
one has a_I=0 as soon as I≠ 0, which proves that f∈ A. 

Let us prove (b). Let 
E
denote the set of indices I with c_I≠ 0 (it is non-empty by assumption). 
Let V be a connected and non-empty affinoid domain of U.
As X is normal 𝒪_X(V)
is an integral domain, and by assumption there exist s and 
t in 𝒪(V) with t≠ 0 and g/h=s/t as meromorphic functions
on V_r, which amounts to the equality gt=hs in 𝒪_X(V)_r. 
As a consequence, if I∈ E then s/t=b_I/c_I as meromorphic
functions on V (note that c_I|_V≠ 0 since
V is non-empty and X is irreducible
and reduced) and b_I|_V=0 if I∈ E. 
Therefore all meromorphic functions
b_I/c_I for I∈ E coincide on every irreducible affinoid domain V of the normal space U, 
then coincide on the whole of U, and then of X since the latter is reduced ; let u denote the 
common value of the b_I/c_I for I∈ E, as meromorphic function on X. 
If I∉ E then b_I=0 on every irreducible affinoid domain V of the normal space U, 
then on the whole of U, and then on X since the latter is reduced. Hence g/h=u, which ends
the proof. 





 §.§ GAGA results for meromorphic
functions

In our proof of Corollary <ref> 
we have used  GAGA results for meromorphic functions on a proper scheme
in non-archimedean geometry. 
They are certainly
well-known
but due to the lack of reference in the litterature we will state and prove them here. 

Let A be an affinoid algebra and let
X be an A-scheme of finite type. Let us denote by
𝒦_X and 𝒦_X the sheaves of
meromorphic functions on X and X, respectively. 
Let 𝒮, 𝒯, 
be the subsheaf of 𝒪_X, 𝒪_X, consisting of those functions
whose germ at every point is regular. The structure map 
π X→ X is a faithfully flat map of locally ringed spaces. This implies that 
the natural arrow 𝒪_X→π_*𝒪_X is injective, maps 𝒮
into π_*𝒯, and induces an injective map of presheaves 𝒮^-1𝒪_X
→𝒯^-1𝒪_X. The presheaf 𝒯^-1𝒪_X is separated, 
thus it embedds ito its sheafification 𝒦_X, whence an injective map 
from
𝒮^-1𝒪_X↪ to π_*𝒦_X and
eventually by sheafifyfing once again
an injective map 𝒦_X↪π_*𝒦_X. 


Let f∈𝒦_X(X) and let ℐ⊂𝒪_X
be its sheaf of denominators. 
Then ℐ is the sheaf of denominators of f viewed
as a meromorphic function on X. 



If 𝒥 denotes the sheaf of denominators of f viewed as 
a meromorphic function on X, it is clear that  ℐ⊂𝒥. To show that this inclusion is actually an isomorphism, one may
argue locally on X, and thus assume that X is affine. One can then write
f=g/h with g and h in 𝒪_X(X) and h regular. 
Then ℐ is 
the kernel of the map
𝒪_X→𝒪_X/(h) induced by multiplication by g,
and 𝒥
is te kernel of the map
𝒪_X→𝒪_X/(h) induced by multiplication by g,
hence 𝒥=ℐ.



One has 𝒪_X(X)=
𝒦_X(X)∩𝒪_X(X).



The direct inclusion is obvious. For the converse one, let f be an element of 
𝒦_X(X)∩𝒪_X(X) and let ℐ⊂𝒪_X be
the sheaf of denominators of f viewed as a meromorphic
function on X.  By Lemma <ref>
above, ℐ is the sheaf of denominators of f viewed as a
meromorphic function on X. The fact that f∈𝒪(X) means that 
1∈ℐ(X), which we can reformulate by saying that 1 is zero in (𝒪_X/
ℐ)(X). But by
faithfull flatness of X→ X, the map
(𝒪_X/ℐ)(X)→ (𝒪_X/ℐ)(X) is injective. 
Therefore 1 is zero in (𝒪_X/ℐ)(X), so
1∈ℐ(X) and f∈𝒪_X(X). 



Assume that X is proper over A. 
The natural embedding
𝒦_X(X)↪𝒦_X(X) is an isomorphism. 



Let f be a meromorphic function on
X. By GAGA for coherent sheaves (see <cit.>), the sheaf denominators of f is 
is of the form ℐ for a (uniquely determined) sheaf of ideals ℐ on 
X. Multiplication by f defines a morphism from ℐ to 𝒪_X, which once again
by GAGA comes from a unique map 
from ℐ to 𝒪_X.
In order to prove that f belongs to 𝒦(X), it then suffices to prove that on every
affine open subscheme
U of X,
the sheaf ℐ
has a regular  section h (for if one denotes by g the image of h in in 𝒪(U) one will have
f=g/h on U). But this is proved exactly like in <cit.>
(where we show the analogous statement
on an affinoid space). 




§ ACKNOWLEDGEMENTS

The starting point of this work was a discussion with Marco Maculan who drew my attention to the fact that
invertible analytic functions on analytifications of curves are algebraic. This is what led me to Theorem <ref>
and Corollary <ref>, and I would like to thank him warmly for that. 



smfalpha





 Paper Title: Infinitesimal 2-braidings from 2-shifted Poisson structures.




Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems

    Lise-Marie Imbert-GérardDepartment of Mathematics, University of Arizona, USA  (mailto:lmig@arizona.edulmig@arizona.edu) ,
Andrea MoiolaDepartment of Mathematics, University of Pavia, Italy       (mailto:andrea.moiola@unipv.itandrea.moiola@unipv.it) ,
Chiara PerinatiDepartment of Mathematics, University of Pavia, Italy (mailto:chiara.perinati01@universitadipavia.itchiara.perinati01@universitadipavia.it) , 
Paul StockerFaculty of Mathematics, University of Vienna, Austria (mailto:paul.stocker@univie.ac.atpaul.stocker@univie.ac.at)
    August 5, 2024
===========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================



§ ABSTRACT 
It is shown that every 2-shifted Poisson structure on a finitely generated semi-free commutative differential graded algebra A defines a very explicit infinitesimal 2-braiding on the homotopy 2-category of the symmetric monoidal dg-category of finitely generated semi-free A-dg-modules. This provides a concrete realization, to first order in the deformation parameter ħ, of the abstract deformation quantization results in derived algebraic geometry due to Calaque, Pantev, Toën, Vaquié and Vezzosi. Of particular interest is the case when A is the Chevalley-Eilenberg algebra of a higher Lie algebra, where the braided monoidal deformations developed in this paper may be interpreted as candidates for representation categories of `higher quantum groups'.






  
*Keywords: derived algebraic geometry, shifted Poisson structures, L_∞-algebras, quantization, braided monoidal 2-categories





  
*MSC 2020: 14A30, 17B37, 18N10, 53D55












§ INTRODUCTION AND SUMMARY


Quantum groups are interesting and important mathematical objects
which enjoy a wide range of applications in representation
theory, topological quantum field theory
and integrability, see e.g. <cit.> for surveys.
There exist two complementary main approaches to the study of quantum groups:
The algebraic approach based on quasi-triangular Hopf algebras
and the categorical approach studying linear braided monoidal categories.
The relationship between these approaches is given by the observation 
that a quasi-triangular Hopf algebra has a linear braided monoidal category
of representations from which it can be recovered by Tannakian reconstruction.
An important class of examples of quantum groups are the quantized 
universal enveloping algebras U_q() which are defined
as quasi-triangular Hopf algebra deformations of the enveloping algebra U()
of a Lie algebra . In the case of formal deformations q=e^ħ, 
their linear braided monoidal representation categories 𝐑𝐞𝐩_ħ() can also be obtained 
directly as braided monoidal deformations of the linear symmetric monoidal representation
category 𝐑𝐞𝐩() of the Lie algebra , see
e.g. <cit.> and <cit.>.


Given the importance of quantum groups, it is justified and natural to search 
for a generalization of such structures to higher category theory.
Such `higher quantum groups' are then expected to have applications
in higher-categorical representation theory, higher-dimensional
topological quantum field theory and higher-dimensional integrability.
There have been various attempts to develop a theory
of `higher quantum groups' based on `higher Lie algebras' such as Lie 2-algebras, 
reaching from algebraic approaches,
see e.g. <cit.>, to more categorical ones,
see e.g. <cit.>. These attempts are distinct from 
the approach to `higher quantum groups' via `categorification' in the sense 
of <cit.> where 
`higher quantum groups' are constructed based on classical Lie algebras.


In the present paper, we initiate a new approach to `higher quantum groups'
which is rooted in recent developments in derived algebraic geometry,
in particular the theory of shifted symplectic and shifted Poisson structures on 
derived stacks <cit.>. Our inspiration 
comes from Safronov's results in <cit.> which show that the semi-classical
data associated with quantum groups, namely Lie bialgebra structures
and invariant symmetric tensors, arise naturally as 1 and 2-shifted Poisson structures
on the classifying stack BG=[pt/G] of an algebraic group G,
or its infinitesimal analogue given by the formal classifying stack 
B=[pt/] of the Lie algebra  of G.


Given an n-shifted Poisson structure on a derived stack X, for an integer 
n≥ 1, the symmetric monoidal dg-category  𝐏𝐞𝐫𝐟(X) 
of perfect modules over X can be quantized using the 𝔼_n-monoidal 
deformation quantization results from <cit.>. 
In the case where X = B, one has an equivalence
of symmetric monoidal dg-categories
𝐏𝐞𝐫𝐟(B) ≃𝐝𝐠𝐑𝐞𝐩()
identifying perfect modules over B with (homotopy-coherent) 
representations of  on cochain complexes. Hence, every 
2-shifted Poisson structure on B, which by <cit.>
is an invariant symmetric tensor in (^2)^,
defines a braided monoidal deformation 𝐏𝐞𝐫𝐟_ħ (B)
≃𝐝𝐠𝐑𝐞𝐩_ħ() which is expected to generalize
the constructions in <cit.> and <cit.> 
to representations of  on cochain complexes. In Corollary <ref>
we provide a concrete realization of this braided monoidal deformation to all orders of ħ.


Our working hypothesis is that the above picture should generalize
to the case where the Lie algebra  is replaced by a `higher Lie algebra', 
i.e. an L_∞-algebra L. Inspired by <cit.>, 
we will define the semi-classical `higher quantum group' data
on L to be given by the 2-shifted Poisson structures
on the formal classifying stack BL = [pt/L],
and propose the braided monoidal deformation 𝐏𝐞𝐫𝐟_ħ (BL)
≃𝐝𝐠𝐑𝐞𝐩_ħ(L) from <cit.> as a model for
the representation category of the `higher quantum group U_q(L)' with q=e^ħ.



It is worthwhile to emphasize that the existence results for deformations in <cit.> 
are quite implicit as they rely on formality of the 𝔼_n+1-operads for n≥ 1, 
see also <cit.>. Hence, it is a priori not evident how these abstract methods
relate to the more explicit deformation techniques for ordinary Lie algebras 
from <cit.> and <cit.>, which are based on the simple but powerful concept 
of an infinitesimal braiding, controlling the first-order deformation in ħ,
and its lift to a full deformation via a choice of Drinfeld associator.
The main goal of this paper is to display an explicit relationship between the 
abstract deformation results of <cit.> and infinitesimal braidings. This is done 
by adopting a more concrete and computational approach to derived algebraic geometry
which is rooted in Pridham's works <cit.>.


We will now explain our results by outlining the content of this paper.
In Section <ref>, we recall some basic aspects of derived algebraic geometry
that are needed in our work. In particular, we provide a brief discussion of the crucial concept of an 
n-shifted Poisson structure (Definition <ref>)
on a finitely generated semi-free commutative differential graded algebra 
(CDGA) A. By restricting this discussion to the stacky-affine case, 
the treatment remains self-contained in the language of CDGAs and their dg-modules. 



In Section <ref>, we study deformations of the symmetric monoidal dg-category 
_A^fgsf of finitely generated semi-free
A-dg-modules into a braided monoidal category.
More precisely, in order to have good control on 
the tower of homotopy coherences which can enter such deformations,
we restrict our attention to the simpler problem of deforming the homotopy 2-category

    _A := 𝖧𝗈_2(_A^fgsf)

of this symmetric monoidal dg-category.
The homotopy 2-category _A has the same underlying objects, 
i.e. finitely generated semi-free A-dg-modules, and its morphisms
are given by the good truncations to cohomological degrees {-1,0} of the 
morphism complexes in _A^fgsf, see Appendix <ref>.
We show in Proposition <ref> that, 
to first order in the deformation parameter ħ, such deformations
are governed by infinitesimal 2-braidings as in Definition <ref>. 
Our concept of infinitesimal 2-braidings is compatible with the one proposed
by Cirio and Faria Martins <cit.> in the context of categories enriched over 2-vector spaces.


Our main results are presented in Subsection <ref>,
where we show that every 2-shifted Poisson structure on a finitely generated semi-free CDGA
A defines, by a very explicit construction, a γ-equivariant 
infinitesimal 2-braiding on _A, see in particular Theorem <ref>. This means that
every 2-shifted Poisson structure on A defines an explicit first-order
braided monoidal deformation of the homotopy 2-category of the symmetric monoidal
dg-category _A^fgsf, which realizes concretely 
the abstract deformation results from <cit.> to first order in ħ.


In Subsection <ref>, we explore the behavior
of our first-order deformations at higher orders in the deformation parameter ħ.
Following the conventional theory of infinitesimal braidings from
<cit.> and <cit.>,
we observe in Proposition <ref> that
a γ-equivariant infinitesimal 2-braiding defines 
an all-order in ħ braided monoidal deformation of the homotopy 2-category _A 
provided that certain identities which are similar to Drinfeld associators hold true strictly.
We observe in Proposition <ref> that, in general, these identities
do not hold true strictly, but only up to homotopies which we 
construct explicitly. This suggests that lifting our first-order deformation construction
to all orders in ħ requires a homotopy-coherent generalization
of the concept of Drinfeld associators, involving suitable
hexagonators and pentagonators,  which we however currently 
do not know how to develop.


In Section <ref>, we apply our deformation construction 
from Section <ref> to explicit examples 
associated with L_∞-algebras. 
Following <cit.>, we describe
the formal classifying stack BL = [pt/L] of an L_∞-algebra L 
by its associated Chevalley-Eilenberg algebra ^∙(L).
For finite-dimensional L, this is a finitely generated semi-free CDGA,
hence our techniques from the previous sections apply to this case.
In particular, our main Theorem <ref>
shows that every 2-shifted Poisson structure
on ^∙(L) defines an explicit first-order braided monoidal deformation
of the homotopy 2-category 

    _^∙(L) ≃ 𝖧𝗈_2(𝐝𝐠𝐑𝐞𝐩(L))

of the symmetric monoidal dg-category of representations of the L_∞-algebra L
and their ∞-morphisms. In Subsection <ref>, we will specialize this result
to the case where L= is an ordinary
Lie algebra, which leads to a homotopical generalization of
the traditional constructions in <cit.> and <cit.>,
and in Subsection <ref> to the case where L=_κ is the string Lie 2-algebra
obtained from an ordinary Lie algebra  and a 3-cocycle κ:⋀^3→.


In the technical Appendix <ref>, we develop and present
a suitable higher-categorical framework in which our deformation
constructions in this paper are carried out. The main concept defined here
is the symmetric monoidal 3-category _R^[-1,0],ps from 
Proposition <ref> whose objects
are categories enriched over 2-term cochain complexes concentrated
in degrees -1 and 0, morphisms are enriched functors,
2-morphisms are enriched pseudo-natural transformations
and 3-morphisms are enriched modifications. Working with pseudo-natural transformations provides 
the required flexibility for the deformation constructions of this paper. Indeed, 
the homotopy 2-categories _A := 𝖧𝗈_2(_A^fgsf)
and their braided monoidal deformations from Section <ref>
are symmetric or, respectively, braided monoidal category objects in 
the symmetric monoidal 3-category _R^[-1,0],ps.








§ BASIC DERIVED ALGEBRAIC GEOMETRY




 §.§ Cochain complexes of modules


Throughout this work, we fix a field  of characteristic 0.
Let R be a commutative, associative and unital -algebra.
We denote the category of (not necessarily bounded) cochain complexes 
of R-modules by _R. Concretely, an object V=(V,_V)∈_R is a pair consisting of 
a family V={V^i}_i∈ of R-modules labeled by their degrees i∈
and a family _V = {_V^i : V^i → V^i+1}_i∈ of degree-increasing
R-linear maps (called the differential) which squares to zero, i.e. _V _V =0. A morphism f :V→ W in _R
is a family f={f^i : V^i→ W^i}_i∈ of degree-preserving R-linear maps
which commutes with the differentials, i.e. _W f = f _V.


The category _R is closed symmetric monoidal with respect to the following structures:
The tensor product V⊗_R W∈_R of two cochain complexes V,W∈_R is defined by
the relative tensor product of R-modules


(V⊗_R W)^i := ⊕_j∈ (V^j ⊗_R W^i-j)  ,

for all i∈, and the differential 

_V⊗_R W(v⊗_R w) :=  _V(v)⊗_R w + (-1)^|v| v⊗_R _W(w)  ,


for all homogeneous v∈ V and w∈ W, where | v|∈ denotes the degree of v.
The monoidal unit is R∈_R concentrated in degree 0 with trivial differential _R=0.
The symmetric braiding is defined by the Koszul sign rule

γ_V,W^ :  V⊗_R W ⟶ W⊗_R  V ,   v⊗_R w  ⟼ (-1)^|v| |w| w⊗_R v  ,

for all homogeneous v∈ V and w∈ W. The internal hom _R(V,W)∈_R between 
two cochain complexes V,W∈_R is defined by the internal hom of R-modules


_R(V,W)^i :=  ∏_j∈hom_R(V^j,W^j+i)  ,

for all i∈, and the differential 

__R(V,W)(f) := _W f - (-1)^|f | f _V  ,


for all homogeneous f∈_R(V,W).


In the special case where R= is the base field, we shall simply write
(_,⊗,,) for the closed symmetric monoidal structure, 
suppressing the subscript _ from our notations.


Given any integer n∈ and any cochain complex V∈_R, we define the n-shifted 
cochain complex V[n]∈_R by V[n]^i:= V^i+n, for all i∈, and _V[n]:= (-1)^n _V.
Using the identification V[n]≅ R[n]⊗_R V, we shall write elements of the n-shifted cochain
complex as s^-n v∈ V[n], where s^-n∈ R[n] denotes the degree -n element represented
by the unit of R.




 §.§ Semi-free CDGAs and dg-modules


A commutative differential graded algebra over the field  (in short, a CDGA) is a commutative
algebra object A = (A,_A,μ_A,η_A) in the closed symmetric monoidal category _.
More explicitly, a CDGA is a cochain complex A=(A,_A)∈_ 
endowed with _-morphisms μ_A : A⊗ A→ A (called multiplication)
and η_A : → A (called unit) which satisfy the usual associativity,
unitality and commutativity axioms internal to _.

A CDGA A= (A,_A,μ_A,η_A) is called semi-free
if its underlying -graded commutative algebra A^♯ = (A,μ_A,η_A) is free, 
i.e. 

A^♯ ≅ (V^♯)

is isomorphic to the symmetric algebra of a -graded vector space V^♯.
A semi-free CDGA is called finitely generated if the total dimension
(V^♯) := ∑_i∈(V^i)<∞ of the -graded 
vector space V^♯ is finite.


Associated with any CDGA A is its dg-category _A of (left) A-dg-modules.
More explicitly, an object M = (M,_M,ℓ_M) in this category is
a cochain complex M=(M,_M)∈_ endowed with a _-morphism 
ℓ_M : A⊗ M→ M (called left action) which satisfies the 
usual left module axioms internal to _. The cochain complex
of morphisms between two objects M,N∈_A is defined as the subcomplex


_A(M,N) ⊆  (M,N)

of the internal hom in _ containing the homogeneous elements
f∈(M,N) which commute with the left actions, i.e.
f(ℓ_M(a⊗m)) = (-1)^|f| |a| ℓ_N(a⊗f(m))  ,


for all a∈ A and m∈ M. The dg-category _A is closed symmetric monoidal
with respect to the relative tensor product

M⊗_A N := (
@C=8em
M⊗N       @<1ex>[l]^-𝕀_M⊗ℓ_N@<-1ex>[l]_-(ℓ_M⊗𝕀_N) (γ_M,A^⊗𝕀_N)M⊗A⊗N

) ∈ _A  ,

the monoidal unit A=(A,_A,μ_A)∈_A, the symmetric braiding

γ_M,N^ :  M⊗_A N  ⟶ N⊗_A M ,  m⊗_A n ⟼ (-1)^|m| |n| n⊗_A m  ,

for all homogeneous m∈ M and n∈ N, and the internal hom

_A(M,N) ∈ _A
given by the cochain complex (<ref>) and the left action 
A⊗_A(M,N)→_A(M,N) , a⊗ f ↦ a· f
defined by (a· f)(m) := ℓ_N(a⊗ f(m)), for all a∈ A,
f∈_A(M,N) and m∈ M.

An A-dg-module M = (M,_M,ℓ_M) is called semi-free
if its underlying A^♯-module M^♯ = (M,ℓ_M) is free, i.e. 

M^♯ ≅ A^♯⊗W^♯
for a -graded vector space W^♯. 
A semi-free A-dg-module is called finitely generated if the total dimension
(W^♯) := ∑_i∈(W^i)<∞ of the -graded 
vector space W^♯ is finite. We denote by

_A^fgsf ⊆  _A
the full dg-subcategory whose objects are all finitely generated 
semi-free A-dg-modules.



The closed symmetric monoidal structure on _A
restricts to a closed symmetric monoidal structure
on the full dg-subcategory _A^fgsf⊆_A.
Indeed, the monoidal unit A∈_A is finitely generated semi-free and,
given any two finitely generated semi-free A-dg-modules
M,N∈_A^fgsf with 
M^♯≅ A^♯⊗ W^♯ and N^♯≅ A^♯⊗ U^♯, 
we have isomorphisms


(M⊗_A N)^♯ ≅ 
A^♯⊗(W^♯⊗U^♯)

and

_A(M,N)^♯ ≅ A^♯⊗(U^♯⊗(W^♯)^∗)  ,


which imply that M⊗_A N∈_A^fgsf and
_A(M,N)∈_A^fgsf are finitely generated semi-free.
Here (W^♯)^∗ denotes the dual of the -graded vector space W^♯,
which exists since W^♯ is, by hypothesis, of finite total dimension.





 §.§ Derivations, Kähler differentials and shifted Poisson structures


We recall some basic aspects of the geometry of a finitely generated semi-free CDGA A, i.e.A^♯≅ (V^♯) for some -graded vector space V^♯ of finite total dimension.
The dg-module of derivations 𝖳_ A∈_A of A is defined as the subcomplex


𝖳_ A ⊆  (A,A)

of the internal hom in _ containing the homogeneous elements D∈(A,A) which satisfy the
Leibniz rule

D(a a^') =  D(a) a^'+ (-1)^|D|  |a| a D(a^')  ,


for all homogeneous a,a^'∈ A. The A-dg-module structure 
A⊗𝖳_ A→𝖳_ A , a⊗ D↦ a· D is given by
(a· D)(a^'):= a D(a^'), for all a,a^'∈ A and D∈𝖳_ A.
Since A is finitely generated semi-free, each derivation is completely 
specified by its action on the generators V^♯, which yields an isomorphism

(𝖳_ A)^♯ ≅ A^♯⊗(V^♯)^∗  ,

where (V^♯)^∗ denotes the dual of the -graded vector space V^♯.
Hence, 𝖳_ A∈_A^fgsf is a finitely generated semi-free A-dg-module.


The underlying cochain complex of 𝖳_ A can be endowed with the Lie algebra structure
given by the commutator

[ · , · ] : 𝖳_ A⊗𝖳_ A   ⟶  𝖳_ A
 ,  D⊗D^' ⟼ [D,D^'] := 
D D^'- (-1)^|D| |D^'| D^' D  ,

for all homogeneous D,D^'∈𝖳_ A.
The Lie algebra and A-dg-module structures on 𝖳_ A satisfy the compatibility condition

[D,a·D^'] =  D(a)·D^'+ (-1)^|D| |a| a·[D,D^']  ,

for all homogeneous D,D^'∈𝖳_ A and a∈ A.


As a consequence of our hypothesis that A is a finitely generated semi-free CDGA, 
the dg-module of derivations 𝖳_ A∈_A^fgsf is dualizable. We define
the dg-module of Kähler differentials by the internal hom


Ω_A := _A(𝖳_ A , A ) ∈  _A^fgsf  ,

which is finitely generated semi-free as a consequence of Remark <ref> and

(Ω_A)^♯ ≅ A^♯⊗(V^♯)^∗∗  ≅ A^♯⊗V^♯  .


We denote the duality pairings between Ω_A and 𝖳_ A by


 ·  ·  : Ω_A⊗_A 𝖳_ A ⟶ A ,  
ω⊗_A D ⟼ ωD := ω(D)

and 

 ·  ·  : 𝖳_ A ⊗_AΩ_A ⟶ A ,  
D⊗_A ω ⟼ Dω := (-1)^|D| |ω| ω(D)  ,


for all homogeneous ω∈Ω_A and D∈𝖳_ A. The de Rham differential
_ : A→Ω_A is defined by

D_(a) := D(a)  ,

for all a∈ A and D∈𝖳_ A. From this definition one directly
checks that _(a a^') = _(a)· a^' + a·_(a^'),
for all a,a^'∈ A, and that _Ω_A _ = _ _A.


A crucial preliminary concept for the definition of shifted Poisson structures
is that of shifted polyvectors.

Let n∈ be an integer. The CDGA of n-shifted polyvectors on
a finitely generated semi-free CDGA A is defined as the relative symmetric algebra

(A,n) :=  _A(𝖳_ A[-n-1]) =  ⊕_m≥0_A^m(𝖳_ A[-n-1])

of the (-n-1)-shift of the A-dg-module of derivations 𝖳_ A. The non-negative integer
m∈^≥ 0 in the direct sum decomposition in terms of symmetric powers is called
the weight of polyvectors.


The CDGA of n-shifted polyvectors (A,n) can be endowed with a canonical ℙ_n+2-algebra
structure (called Schouten-Nijenhuis bracket)

{ · , · } ∈ ((A,n)⊗(A,n),(A,n))^-n-1

which is defined on the generators by


{a,a^'}    := 0  ,

{s^n+1D,a}    := D(a)  ,

{s^n+1D,s^n+1D^'}    := (-1)^|D| (n+1) s^n+1[D,D^']  ,


for all homogeneous a,a^'∈ A and D,D^'∈𝖳_ A, where we recall that
D(a)∈ A denotes the evaluation of the derivation D on a∈ A 
and [D,D^']∈𝖳_ A denotes
the Lie bracket (<ref>) on 𝖳_ A. The bracket
{ · , · } is then extended to (A,n) by the defining
properties of a ℙ_n+2-algebra structure, i.e.


  (i) Antisymmetry: For all homogeneous P,Q∈(A,n),

{P,Q} = -(-1)^(n+1) (-1)^|P| |Q| {Q,P}  .



  (ii) Derivation property: For all homogeneous P,Q,R∈(A,n),

{P,Q R} = {P,Q} R + (-1)^(|P|-n-1)  |Q| Q {P,R}  .



  (iii) Jacobi identity: For all homogeneous P,Q,R∈(A,n),

{ P,{Q,R} } = (-1)^(|P|-n-1) (n+1) {{P,Q} ,R } + (-1)^(|P|-n-1) (|Q|-n-1) {Q,{P,R}}  .



Note that the Schouten-Nijenhuis bracket decreases the polyvector weight by 1, i.e. for P,Q∈(A,n)
of weight m_P and m_Q, the weight of {P,Q}∈(A,n) is m_P + m_Q -1.


The definition of n-shifted Poisson structures in 
<cit.> and <cit.> uses a completion
of the ℙ_n+2-algebra (A,n)
in order to avoid bounds on the weights of polyvectors.
The relevant definition is as follows.

The ℙ_n+2-algebra of completed n-shifted polyvectors
on a finitely generated semi-free CDGA A is defined by

(A,n) :=  ∏_m≥0 ^m_A(𝖳_ A[-n-1])  .

A completed n-shifted polyvector is thus a formal sum P = ∑_m≥ 0 P^(m)
of homogeneous weight components P^(m)∈^m_A(𝖳_ A[-n-1]).
The ℙ_n+2-algebra structure on (A,n) is given by


P Q    :=  ∑_m≥0(∑_k+l=m P^(k) Q^(l))  ,

{P,Q}    := ∑_m≥0(∑_k+l-1=m {P^(k),Q^(l)})  ,


which is well-defined because the weight is bounded from below and hence
the sums in the parentheses are finite.


We now define the key concept of an n-shifted Poisson structure following
<cit.>, see also <cit.>.

An n-shifted Poisson structure on a finitely generated semi-free CDGA A 
is a completed n-shifted polyvector

π = ∑_m≥2 π^(m) ∈ (A,n)^n+2

of degree n+2 and weight ≥ 2 which satisfies the Maurer-Cartan equation

_(π) + 12 {π,π} = 0  .



Decomposing the Maurer-Cartan equation into homogeneous weight components
one obtains a tower of conditions

_(π^(2))    = 0  ,

_(π^(3))+12 {π^(2),π^(2)}    = 0  ,

    ⋮

_(π^(m))+12 ∑_k+l-1=m{π^(k),π^(l)}    = 0  ,

for all m≥ 3. The first condition states that the bivector component
π^(2) is closed with respect to the differential _, i.e. it
defines an (n+2)-cocycle in ^2_A(𝖳_ A[-n-1]).
The second condition provides a homotopical relaxation of the usual condition
that a Poisson bivector has a trivial Schouten-Nijenhuis bracket with itself,
with the trivector π^(3) playing the role of a homotopy witnessing this condition. 
The higher weight components π^(m) 
for m≥ 4 describe a coherent tower of higher homotopies for this relaxation.








§ INFINITESIMAL 2-BRAIDINGS


The aim of this section is to study a class of first-order deformations,
into a braided monoidal category, of the symmetric monoidal category _A^fgsf
of finitely generated semi-free dg-modules over a finitely generated
semi-free CDGA A over the field . The precise context
in which our deformation constructions take place is given by the symmetric monoidal
3-category _R^[-1,0],ps from Proposition <ref>,
consisting of categories, functors, pseudo-natural transformations
and modifications that are suitably enriched over the symmetric monoidal category
^[-1,0]_R of 2-term cochain complexes concentrated in degrees -1 and 0. 
Our starting point is thus given by the homotopy 2-category

_A := 𝖧𝗈_2(_A^fgsf) ∈  ^[-1,0],ps_
from Appendix <ref>, which we endow with the symmetric monoidal
structure that is induced from the standard one on _A^fgsf given in
(<ref>) and (<ref>).


The effect of forming the homotopy 2-category
is that the cochain complex _A(M,N)∈_ of morphisms 
between any two dg-modules M,N∈_A^fgsf from (<ref>) 
gets replaced by its good truncation _A(M,N):=τ^[-1,0](_A(M,N))∈_^[-1,0]
to degrees -1 and 0. Concretely, this means that the homotopy 2-category 
_A = 𝖧𝗈_2(_A^fgsf)  describes only 
degree 0 morphisms between A-dg-modules which commute 
with the differentials and 1-homotopies (modulo 2-homotopies) between such morphisms,  
but it does not include information about higher homotopies. Our
motivation behind this truncation of the dg-category _A^fgsf 
is that the homotopy 2-category _A = 𝖧𝗈_2(_A^fgsf) 
is a 2-categorical object whose deformations can be described very explicitly 
in the context of the 3-category _R^[-1,0],ps from Proposition <ref>.
Most importantly, every tower of homotopy coherence data will automatically truncate beyond the level of 1-homotopies.
Our main result is that every 2-shifted Poisson structure on the CDGA A as in Definition <ref>
defines a very explicit first-order deformation of the braiding on _A.




 §.§ Setup and definitions


To set up a framework for studying first-order deformations
of the symmetric monoidal category object _A from (<ref>) in the symmetric 
monoidal 3-category _R^[-1,0],ps from Proposition <ref>,
we introduce a deformation parameter ħ which squares to zero ħ^2=0.
Performing a change of base along the 
commutative -algebra morphism →[ħ]/(ħ^2) defines
an object

_A^ħ :=  _A⊗[ħ]/(ħ^2) ∈  _[ħ]/(ħ^2)^[-1,0],ps  ,

which we endow with the symmetric monoidal structure that is induced from the one on _A.
More concretely, _A^ħ is the ^[-1,0]_[ħ]/(ħ^2)-enriched
category with the same objects as _A, i.e.  finitely generated
semi-free A-dg-modules, and whose morphisms 
are given by _A^ħ(M,N) = _A(M,N)⊗[ħ]/(ħ^2)∈^[-1,0]_[ħ]/(ħ^2),
for all M,N∈_A^ħ. Following the notation from Appendix <ref>,
we shall denote the symmetric braiding on _A^ħ by

    _A^ħ⊠_A^ħ      _A^ħ
       _A^ħ⊠_A^ħ[""name=0, anchor=center, inner sep=0, "⊗_A", from=1-1, to=1-3]
    	["τ"', from=1-1, to=2-2]
    	["⊗_A"', from=2-2, to=1-3]
    	["γ", shorten <=6pt, shorten >=3pt, Rightarrow, from=0, to=2-2]
      ,

where τ is an abbreviation for the component τ__A^ħ,_A^ħ
of the symmetric braiding in (<ref>) of the ambient 3-category 
_[ħ]/(ħ^2)^[-1,0],ps. 
When written in terms of the 3-categorical composition operations
from Appendix <ref>,
the symmetric braiding reads as γ: ⊗_A⇒⊗_A ∗τ.
Note that, by construction, this is a strict 
^[-1,0]_[ħ]/(ħ^2)-enriched natural transformation.


In the context of braided monoidal category
objects in _[ħ]/(ħ^2)^[-1,0],ps
from Definition <ref>, we would like to study deformations of
the braiding γ: ⊗_A⇒⊗_A ∗τ of _A^ħ
which keep all the other structures (i.e. the tensor product, monoidal unit, associator and unitors) 
fixed. The general ansatz for a first-order deformation of the 
braiding γ: ⊗_A⇒⊗_A ∗τ of _A^ħ
is obtained by considering a ^[-1,0]_[ħ]/(ħ^2)-enriched pseudo-natural transformation
of the form

γ^ħ :=  γ∘(+ ħ2  t) : 
⊗_A ⟹ ⊗_A ∗τ : _A^ħ⊠_A^ħ ⟶ _A^ħ
which satisfies the two hexagon identities



.9!@C=2em@R=2em   ⊗_A∗ (𝕀⊠⊗_A)@=>[r]^-γ^ħ∗   ⊗_A∗ (⊗_A⊠𝕀) ∗τ_23∗τ_12@=>[dr]^-     α∗∗   
⊗_A∗ (⊗_A⊠𝕀) @=>[ru]^-α @=>[dr]_-∗ (γ^ħ⊠)              ⊗_A∗ (𝕀⊠⊗_A) ∗τ_23∗τ_12
   ⊗_A∗ (⊗_A⊠𝕀) ∗τ_12@=>[r]_-α∗   ⊗_A∗ (𝕀⊠⊗_A) ∗τ_12@=>[ru]_-     ∗ (⊠γ^ħ)∗


and


.9!@C=2em@R=2em   ⊗_A∗ (⊗_A⊠𝕀)@=>[r]^-γ^ħ∗   ⊗_A∗ (𝕀⊠⊗_A)∗τ_12∗τ_23@=>[dr]^-     α^-1∗∗   
⊗_A∗ (𝕀⊠⊗_A) @=>[ru]^-α^-1 @=>[dr]_-∗ (⊠γ^ħ)              ⊗_A∗ (⊗_A⊠𝕀)∗τ_12∗τ_23   ,
   ⊗_A∗ (𝕀⊠⊗_A) ∗τ_23@=>[r]_-α^-1∗   ⊗_A∗ (⊗_A⊠𝕀) ∗τ_23@=>[ru]_-     ∗ (γ^ħ⊠)∗



where α denotes the associator and we use 
the typical short-hand notations τ_12:= τ⊠𝕀 and 
τ_23:=𝕀⊠τ. Note that in the top horizontal
arrows of these diagrams we applied the 
properties (<ref>) and (<ref>)
of the symmetric braiding τ.
The composition of the  ^[-1,0]_[ħ]/(ħ^2)-enriched 
pseudo-natural transformations along the hexagons (<ref>) is given by the 
∘-composition from Appendix <ref> and these diagrams are required to
commute strictly, i.e. up to equality of enriched pseudo-natural transformations. 



Leveraging the fact that ħ^2=0, the hexagon identities are equivalent
to two linear relations for the deformation t, which can be regarded as 
the datum of a _^[-1,0]-enriched pseudo-natural
transformation (without change of base along →[ħ]/(ħ^2))
of the form t : ⊗_A ⇒⊗_A for the relative tensor
product on _A. 

The first hexagon identity (<ref>) for the deformed braiding (<ref>) 
is equivalent to the identity


    α^-1∘(t∗)∘α = ∗(t⊠) 
    
    + (∗(γ⊠)^-1)∘((α^-1∘(∗(⊠ t))∘α) ∗)∘(∗(γ⊠))

of _^[-1,0]-enriched 
pseudo-natural transformations 
⊗_A∗ (⊗_A⊠𝕀)⇒⊗_A∗ (⊗_A⊠𝕀).
The second hexagon identity (<ref>) for the deformed braiding (<ref>) 
is equivalent to the identity

    α∘(t∗)∘α^-1 = ∗(⊠ t) 
    
    + (∗(⊠γ)^-1)∘((α∘(∗(t⊠))∘α^-1) ∗)∘(∗(⊠γ))


of _^[-1,0]-enriched 
pseudo-natural transformations 
⊗_A∗ (𝕀⊠⊗_A)⇒⊗_A∗ (𝕀⊠⊗_A).


The proof follows directly by computing and simplifying the order ħ 
terms of the hexagon identities. The key observation is 
that the pseudo-functor coherences (<ref>)
which describe the interchange law between the ∗ and ∘ compositions
are trivial in all instances appearing in this calculation. Let us illustrate this
through an example: The order ħ term corresponding to the upper path of (<ref>)
involves the pseudo-natural transformation (γ∘ t)∗, which arises
from a particular composition of the pasting diagram

    _A𝐂⊠_A𝐂⊠_A𝐂      _A𝐂⊠_A𝐂      _A𝐂
             ["id⊠⊗_A", curve=height=-30pt, from=2-1, to=2-3]
    	["id⊠⊗_A"', curve=height=30pt, from=2-1, to=2-3]
    	[""name=0, anchor=center, inner sep=0, "id⊠⊗_A"description, from=2-1, to=2-3]
    	["⊗_A", curve=height=-30pt, from=2-3, to=2-5]
    	["⊗_A∗τ"', curve=height=30pt, from=2-3, to=2-5]
    	[""name=1, anchor=center, inner sep=0, "⊗_A"description, from=2-3, to=2-5]
    	["Id "', shorten <=7pt, shorten >=7pt, Rightarrow, from=1-2, to=0]
    	["t "', shorten <=7pt, shorten >=7pt, Rightarrow, from=1-4, to=1]
    	["Id "', shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=3-2]
    	["γ "', shorten <=7pt, shorten >=7pt, Rightarrow, from=1, to=3-4]
        .

The pseudo-functor coherences (<ref>) for this pasting diagram are trivial
because of the identity t_(M,N⊗_A L),(M, N⊗_A L)(𝕀_(M,N⊗_A L)) =0,
for all (M,N,L)∈_A𝐂⊠_A𝐂⊠_A𝐂,
given by property (2) of the Definition <ref>
of _^[-1,0]-enriched pseudo-natural transformations.
It then follows that 

(γ∘t)∗ = (γ∘t)∗(∘)  = (γ∗)∘(t∗)   ,

which is the key step to identify the left-hand side of (<ref>). 
In addition to arguments of this form, one also has to use that 
the pseudo-functor coherences (<ref>) are trivial
whenever ζ^' is strictly natural, i.e. ζ^'_d,d^'=0 for all d,d^'∈.


This result justifies the following 

An infinitesimal 2-braiding on the symmetric monoidal category object
_A=𝖧𝗈_2(_A^fgsf) in the symmetric monoidal 3-category
^[-1,0],ps_ from Proposition <ref>
is a _^[-1,0]-enriched pseudo-natural transformation 
t: ⊗_A⇒⊗_A : _A⊠_A→_A which satisfies 
the two infinitesimal hexagon identities in (<ref>).



As a consequence of Proposition <ref>, the datum of an infinitesimal
2-braiding t is equivalent to a first-order deformation
γ^ħ = γ∘ ( +ħ2 t) : ⊗_A⇒⊗_A∗τ of the symmetric braiding
on _A to a braiding on _A^ħ = _A⊗[ħ]/(ħ^2).
Note that the deformed braiding is not necessarily symmetric because

(γ^ħ∗)∘γ^ħ = + ħ2 ( t + (γ∗)∘(t∗)∘γ) 
 :  ⊗_A ⟹ ⊗_A

is in general different from .



Using the explicit composition formulas from Appendix <ref>, we can 
express the infinitesimal hexagon identities (<ref>)
in terms of the components of the _^[-1,0]-enriched pseudo-natural transformations involved.
(Recall also Definition <ref>.) In this calculation,
various simplifications arise from the fact that all transformations but t
are strictly natural, i.e. their homotopy data given by the double-indexed components
(<ref>) is trivial. To ease notations, we suppress 
the components of the associator α in what follows. One then finds that the first
infinitesimal hexagon identity (<ref>) is equivalent to


t_M,N⊗_A L =  t_M,N⊗_A 𝕀_L +  
(γ_N,M⊗_A𝕀_L) (𝕀_N⊗_A t_M,L) (γ_M,N⊗_A𝕀_L)
  ,

for all M,N,L∈_A, and

    t_(M,N⊗_A L),(M^',N^'⊗_A L^')(h⊗(k⊗_A l)) = 
    t_(M,N),(M^',N^')(h⊗k)⊗_A l 
    
    +(γ_N^',M^'⊗_A𝕀_L^') (k ⊗_A t_(M,L),(M^',L^')(h⊗ l))
     (γ_M,N⊗_A𝕀_L)
      ,


for all M,N,L,M^',N^',L^'∈_A, h∈_A(M,M^')^0, k∈_A(N,N^')^0
and l∈_A(L,L^')^0.
The second infinitesimal hexagon identity (<ref>) is equivalent to


t_M⊗_A N,L =  𝕀_M⊗_A t_N,L + 
(𝕀_M⊗_Aγ_L,N) (t_M,L⊗_A𝕀_N)
 (𝕀_M⊗_Aγ_N,L)
  ,

for all M,N,L∈_A, and

    t_(M⊗_A N,L),(M^'⊗_A N^',L^')((h⊗_A k)⊗l) = 
    h⊗_A t_(N,L),(N^',L^')(k⊗l) 
    
    +(𝕀_M^'⊗_Aγ_L^',N^') (t_(M,L),(M^',L^')(h⊗l)⊗_A k)
     (𝕀_M⊗_Aγ_N,L)
      ,


for all M,N,L,M^',N^',L^'∈_A, h∈_A(M,M^')^0, k∈_A(N,N^')^0
and l∈_A(L,L^')^0.


Our examples of infinitesimal 2-braidings in Subsection <ref>
will satisfy a useful additional property that simplifies their analysis.

An infinitesimal 2-braiding t: ⊗_A⇒⊗_A is 
called γ-equivariant if it satisfies the identity

γ∘t = (t∗)∘γ
of _^[-1,0]-enriched pseudo-natural transformations ⊗_A⇒⊗_A∗τ.



It is important to emphasize that γ-equivariance does not imply
that the associated deformed braiding γ^ħ is symmetric. Indeed,
recalling the calculation in Remark <ref>, we find that

(γ^ħ∗)∘γ^ħ = + ħ  t  ,

for every γ-equivariant infinitesimal 2-braiding t.



Let t: ⊗_A⇒⊗_A be any _^[-1,0]-enriched pseudo-natural transformation
which satisfies the γ-equivariance property (<ref>).
Then the two infinitesimal hexagon identities (<ref>) for t are equivalent.


This can be verified directly by using the equivalent component-wise
description of the infinitesimal hexagon identities from Remark <ref>
and the fact that the γ-equivariance property reads in components as


γ_M,N t_M,N  =  t_N,M γ_M,N  ,

for all M,N∈_A, and

γ_M^',N^' t_(M,N),(M^',N^')(h⊗k)  = 
t_(N,M),(N^',M^')(k⊗h) γ_M,N  ,


for all
M,N,M^',N^'∈_A, h∈_A(M,M^')^0 and k∈_A(N,N^')^0.






 §.§ Construction from 2-shifted Poisson structures


Suppose that π = ∑_m≥ 2π^(m)∈(A,2)^4
is a 2-shifted Poisson structure (recall Definition <ref>)
on the finitely generated semi-free CDGA A. 
In this subsection, we will
present an explicit construction which assigns to
this datum a γ-equivariant infinitesimal 2-braiding t:⊗_A⇒⊗_A
in the sense of Definitions <ref>
and <ref>
on the homotopy 2-category _A =𝖧𝗈_2(_A^fgsf)∈_^[-1,0],ps of the dg-category of finitely generated semi-free A-dg-modules.
We will illustrate later in Section <ref>
that our construction is a non-trivial homotopical generalization
of the standard construction of an infinitesimal braiding on the representation category
𝐑𝐞𝐩() of a Lie algebra  from an invariant symmetric tensor in (^2)^, see
e.g. <cit.> and <cit.>.


Our infinitesimal 2-braiding t is built as a composite of 
more primitive _^[-1,0]-enriched pseudo-natural transformations
which we are going to define now. Let us denote by

Ω_A[1]⊗_A( · )  :  _A ⟶ _A
the _^[-1,0]-enriched functor of taking the relative tensor product 
with the 1-shift Ω_A[1]∈_A of the dg-module of Kähler differentials
from (<ref>). We define a 
_^[-1,0]-enriched pseudo-natural transformation


ξ :  𝕀 ⟹ Ω_A[1]⊗_A( · ) :  _A ⟶ _A
from the _^[-1,0]-enriched  identity functor 𝕀 : _A→_A
by specifying its components as in Definition <ref>:
For each object M∈_A, the component 

(ξ_M : M→Ω_A[1]⊗_A M) ∈  _A(M,Ω_A[1]⊗_A M)^0

is the A-dg-module map 
which is defined on the generators ⊗ w∈ M^♯≅ A^♯⊗ W^♯ 
of M by

ξ_M(⊗w) :=  s^-1 (_⊗𝕀)(_M(⊗w))  ,

where we recall our notation from Subsection <ref>
that s^-1∈[1] denotes the degree -1 element represented by the unit of .
For each pair of objects M,M^'∈_A and each A-dg-module map 
(h: M→ M^')∈_A(M,M^')^0,
the component 

(ξ_M,M^'(h) : M→Ω_A[1]⊗_A M^') ∈  
_A(M,Ω_A[1]⊗_A M^')^-1

is the homotopy which is defined on the generators
⊗ w∈ M^♯≅ A^♯⊗ W^♯ 
of M by

ξ_M,M^'(h)(⊗w) := - s^-1 (_⊗𝕀)(h(⊗w))  .



The components in (<ref>) define a _^[-1,0]-enriched 
pseudo-natural transformation ξ : 𝕀⇒Ω_A[1]⊗_A( · ).


Before we verify the three properties (1), (2) and (3) from Definition <ref>,
let us first comment on the well-definedness of the components of ξ.
Concerning (i) from Definition <ref>, 
we have to show that __A(ξ_M) := _Ω_A[1]⊗_A M ξ_M - ξ_M _M 
= 0, i.e. that ξ_M commutes with the differentials, which can be verified by picking a basis
{w_i∈ W^♯} and evaluating on the associated generators ⊗ w_i∈ M. 
This allows us to write _M(⊗ w_i) =: M_i^ j⊗ w_j
and hence ξ_M(⊗ w_i) = s^-1 _(M_i^ j)⊗ w_j, 
with M_i^ j∈ A and summations over repeated indices understood. We then compute

__A(ξ_M)(⊗w_i)    = 
_Ω_A[1]⊗_A M(ξ_M(⊗w_i)) - 
ξ_M(_M(⊗w_i) )

    =  -s^-1_Ω_A⊗_A M(_(M_i^ j)⊗w_j)
- M_i^ j s^-1 _(M_j^ k)⊗w_k

     = -s^-1 (_(_A(M_i^ j)) ⊗w_j
+ (-1)^|M_i^ j|  _( M_i^ j M_j^ k)⊗w_k )

    = 0  ,

where the last identity follows from expressing the square-zero condition _M_M(⊗ w_i)=0
in our index notation. Concerning (ii) from Definition <ref>, 
we note that ξ_M,M^'(h) as defined 
in (<ref>) is clearly -linear in the argument h.


Property (2) from Definition <ref> follows immediately
from _=0. To verify the first property in (1),
we use also the index notations h(⊗ w_i) =:h_i^ j⊗ w^'_j
and _M^'(⊗ w^'_i)=:M_i^' j⊗ w^'_j,
where {w^'_i∈ W^'♯} is a choice of basis for the generators
of M^', and compute

__A(ξ_M,M^'(h))(⊗w_i)    = 
_Ω_A[1]⊗_A M^'(ξ_M,M^'(h)(⊗w_i))
+ ξ_M,M^'(h)(_M(⊗w_i))

     = s^-1 (_Ω_A⊗_A M^'(_(h_i^ j)⊗w^'_j)
- M_i^ j _(h_j^ k)⊗w^'_k)

     = s^-1 (_(_A(h_i^ j))⊗w^'_j
+ (-1)^|h_i^ j| _(h_i^ j) M_j^' k⊗w^'_k

           - M_i^ j _(h_j^ k)⊗w^'_k)

    = s^-1(- (-1)^|h_i^ j| h_i^ j _(M_j^' k)⊗w^'_k
+ _(M_i^ j) h_j^ k⊗w^'_k)

    =  (𝕀_Ω_A[1] ⊗_A h)(ξ_M(⊗w_i)) - 
ξ_M^'(h(⊗w_i))  ,

where the fourth step follows from expressing
the cocycle condition __A(h)(⊗ w_i)=0 in our index notation.
To verify the second property in (1), one performs a similar calculation for 
(k: M→ M^')∈_A(M,M^')^-1 of degree -1 and finds that 

   (𝕀_Ω_A[1] ⊗_A k)(ξ_M(⊗w_i)) - 
ξ_M^'(k(⊗w_i)) -ξ_M,M^'(__A(k))(⊗w_i)

      =  s^-1(_(_A(k_i^ j))⊗w_j^'+ 
(-1)^|k_i^ j|  _(k_i^ j)  M_j^' k⊗w_k^'+ 
(-1)^|M_i^ j|  M_i^ j _(k_j^ k)⊗w^'_k)  .

The right-hand side can be identified as the differential __A(q)(⊗ w_i) of the 
2-homotopy q∈_A(M,Ω_A[1]⊗_A M^')^-2 which is defined on generators by
q(⊗ w) = -s^-1 (_⊗𝕀)(k(⊗ w)),
hence the right-hand side is zero in our truncated cochain complex _A(M,Ω_A[1]⊗_A M^').


It remains to verify property (3) from Definition <ref>.
Given (h: M→ M^')∈_A(M,M^')^0 and (h^' : M^'→ M^'')
∈_A(M^',M^'')^0, we introduce one more index notation
h^'(⊗ w_i^')=: h_i^' j⊗ w_j^'', where 
{w^''_i∈ W^''♯} is a choice of basis for the generators
of M^'', and compute

ξ_M,M^''(h^' h)(⊗w_i)    = 
-s^-1 _(h_i^ j h_j^' k)⊗w_k^''

    = -s^-1(_(h_i^ j) h_j^' k⊗w_k^'' +
h_i^ j _(h_j^' k)⊗w_k^'')

    = ξ_M^',M^''(h^')(h(⊗w_i)) + 
(𝕀_Ω_A[1] ⊗_A h^')(ξ_M,M^'(h)(⊗w_i))  .

This concludes the proof.


In addition to the _^[-1,0]-enriched pseudo-natural transformation
ξ from (<ref>), we require two more strict
_^[-1,0]-enriched natural transformations to build our infinitesimal 2-braiding.
First, using the original symmetric braiding γ, we obtain a family of isomorphisms


𝕀⊗_Aγ_M,Ω_A[1]⊗_A𝕀 : Ω_A[1]⊗_A M ⊗_A Ω_A[1]⊗_A N
 ⟶  Ω_A[1]⊗_A Ω_A[1]⊗_A M ⊗_A N

in _A, for all M,N∈_A, which 
defines a strict _^[-1,0]-enriched natural isomorphism

⊗_A ∗((Ω_A[1]⊗_A( · ))⊠(Ω_A[1]⊗_A( · )) )
 ≅⟹ (Ω_A[1]⊗_AΩ_A[1]⊗_A( · ))∗⊗_A  .


Second, consider the extension 
⟨ · , · ⟩ :_A^2(𝖳_ A[-1])⊗_A Ω_A[1]⊗_AΩ_A[1]
→ A of the duality pairing (<ref>) defined by

    ⟨ sD sD^', s^-1ω⊗_A s^-1ω^'⟩ := 
    (-1)^|ω| + | D| ⟨ D, ⟨ D^',ω⟩·ω^'⟩
    
    + (-1)^(| D| +1) (| D^'| +1) (-1)^|ω| + | D^'| ⟨ D^', ⟨ D,ω⟩·ω^'⟩  ,
 
for all homogeneous D, D^'∈𝖳_ A 
and ω,ω^'∈Ω_A.
Considering the weight 2 component 
π^(2)∈_A^2(𝖳_ A[-3])^4≅_A^2(𝖳_ A[-1])^0
of the 2-shifted Poisson structure (recall that this is a cocycle, see Remark <ref>),
we obtain a family of morphisms


⟨π^(2), · ⟩⊗_A𝕀⊗_A𝕀 :  Ω_A[1]⊗_AΩ_A[1]⊗_A M⊗_A N ⟶ A⊗_A M⊗_A N  ≅ 
M⊗_A N

in _A, for all M,N∈_A, which defines a 
strict _^[-1,0]-enriched natural transformation

ι_π :  (Ω_A[1]⊗_AΩ_A[1]⊗_A( · ))∗⊗_A ⟹ 
⊗_A  .


Using the ∘-composition of 
_^[-1,0]-enriched pseudo-natural transformations from Appendix <ref>,
we define the _^[-1,0]-enriched pseudo-natural transformation 


.9!@C=0.1em@=[d]⊗_A@==>[rr]^-t          ⊗_A
⊗_A∗(𝕀⊠𝕀) @=>[dr]_-∗ (ξ⊠ξ)              (Ω_A[1]⊗_AΩ_A[1]⊗_A( · ))∗⊗_A@=>[u]_-ι_π
     ⊗_A ∗((Ω_A[1]⊗_A( · ))⊠(Ω_A[1]⊗_A( · ))) @=>[ru]_-  ≅


as our candidate for an infinitesimal 2-braiding on _A.
Note that this candidate depends only on the bivector component
π^(2) of the 2-shifted Poisson structure π = ∑_m≥ 2π^(m)∈(A,2)^4.

Given any 2-shifted Poisson structure π∈(A,2)^4 on a finitely generated
semi-free CDGA A over the field , the _^[-1,0]-enriched pseudo-natural transformation t 
from (<ref>) defines a γ-equivariant
infinitesimal 2-braiding in the sense of Definitions <ref>
and <ref>
on the homotopy 2-category _A =𝖧𝗈_2(_A^fgsf)∈_^[-1,0],ps of the dg-category of finitely generated semi-free A-dg-modules.


The most straightforward proof of this result is to compute
the components of the _^[-1,0]-enriched pseudo-natural transformation t
from (<ref>), and then to verify directly that 
these components satisfy the γ-equivariance property from (<ref>)
and the infinitesimal hexagons from Remark <ref>.
Using the composition operations from Appendix <ref>, one finds
by a straightforward computation that



@C=6.5em@R=3em
M⊗_A N [d]_-ξ_M⊗_A ξ_N [r]^-t_M,N     M⊗_A N 

Ω_A[1]⊗_A M⊗_A Ω_A[1]⊗_A N  [r]_-𝕀⊗_Aγ_M,Ω_A[1]⊗_A 𝕀     
Ω_A[1]⊗_A Ω_A[1]⊗_A M⊗_A N [u]_-⟨π^(2), · ⟩⊗_A𝕀⊗_A𝕀

    ,

for all M,N∈_A, and

.9!@C=6.5em@R=4em
M⊗_A N [d]_-ξ_M,M^'(h)⊗_A (𝕀⊗_A k)ξ_N
+ ξ_M^'h⊗_A ξ_N,N^'(k)[r]^-t_(M,N),(M^',N^')(h⊗k)     M^'⊗_A N^'
Ω_A[1]⊗_A M^'⊗_A Ω_A[1]⊗_A N^'[r]_-𝕀⊗_Aγ_M^',Ω_A[1]⊗_A 𝕀     Ω_A[1]⊗_A Ω_A[1]⊗_A M^'⊗_A N^'[u]_-⟨π^(2), · ⟩⊗_A𝕀⊗_A𝕀    ,


for all M,N,M^',N^'∈_A, h∈_A(M,M^')^0 and k∈_A(N,N^')^0.


The γ-equivariance property (<ref>)
for the degree 0 components t_M,N follows immediately by using 
symmetry of the extended pairing (<ref>), i.e.⟨π^(2), · ⟩ γ  = ⟨π^(2), · ⟩,
and the hexagon identities for the original symmetric braiding γ.
To prove the γ-equivariance property (<ref>)
for the degree -1 components t_(M,N),(M^',N^')(h⊗k),
one has to use additionally that

   γ_Ω_A[1]⊗_A M^', Ω_A[1]⊗_A N^' (ξ_M,M^'(h)⊗_A (𝕀⊗_A k)ξ_N+ ξ_M^'h⊗_A ξ_N,N^'(k)) γ_N,M

         =  (𝕀⊗_A k)ξ_N ⊗_A ξ_M,M^'(h) + 
ξ_N,N^'(k) ⊗_A ξ_M^'h

         = ξ_N^' k ⊗_A ξ_M,M^'(h) + 
ξ_N,N^'(k) ⊗_A (𝕀⊗_A h) ξ_M + __A(ξ_N,N^'(k)⊗_Aξ_M,M^'(h))

         = ξ_N^' k ⊗_A ξ_M,M^'(h) + 
ξ_N,N^'(k) ⊗_A (𝕀⊗_A h) ξ_M   ,

where in the second step we used property (1) of the Definition <ref>
of _^[-1,0]-enriched pseudo-natural transformations and in the last step
we used that the morphism complexes of _A are by definition truncated to degrees {-1,0},
hence the 2-homotopy in the last term gets identified with 0.


It remains to verify the first infinitesimal hexagon identity (<ref>),
since the second one (<ref>) then 
follow from Proposition <ref> and γ-equivariance.
For this we recall the definition of the components of ξ from (<ref>)
and observe that


ξ_N⊗_A L  =  ξ_N⊗_A𝕀+ (γ_N,Ω_A[1]⊗_A𝕀) (𝕀⊗_Aξ_L)  ,

for all N,L∈_A, and

ξ_N⊗_A L,N^'⊗_A L^'(k⊗_A l) =  
 ξ_N,N^'(k)⊗_A l + (γ_N^',Ω_A[1]⊗_A𝕀) (k⊗_Aξ_L,L^'(l))  ,


for all N,L,N^',L^'∈_A, k∈_A(N,N^')^0 and l∈_A(L,L^')^0.
Using also the hexagon identities for the original symmetric braiding γ,
one verifies the infinitesimal hexagon identity (<ref>).



For later use in our discussion of examples in 
Section <ref>, let us expand the components
t_M,N and t_(M,N),(M^',N^')(h⊗k)
of the infinitesimal 2-braiding (<ref>)
in terms of a choice of bases for the finitely generated
semi-free A-dg-modules M,N,M^',N^'∈_A.
Using similar notations as in the proof of Lemma <ref>,
we denote by {w_i∈ W^♯} a basis for M^♯≅ A^♯⊗ W^♯, 
by {u_q∈ U^♯} a basis for N^♯≅ A^♯⊗ U^♯,
by {w_i^'∈ W^'♯} a basis for M^'♯≅ A^♯⊗ W^'♯, 
and by {u_q^'∈ U^'♯} a basis for N^'♯≅ A^♯⊗ U^'♯.
Making use of these bases, we write


    _M(⊗ w_i)     = M_i^ j⊗ w_j  ,    _N(⊗ u_q)     = N_q^ r⊗ u_r  ,
    _M^'(⊗ w_i^') 
        = M_i^' j⊗ w_j^'  ,    _N^'(⊗ u_q^')
        = N_q^' r⊗ u_r^'  ,

for the differentials, 
and for the A-linear maps h∈_A(M,M^')^0 and 
k∈_A(N,N^')^0
we write 

h(⊗w_i)  =  h_i^ j⊗w_j^'  ,          k(⊗u_q)   =  k_q^ r⊗u_r^'  ,


where again all summations over repeated indices
are suppressed. Evaluating the diagram (<ref>), we then find the explicit basis
expression


   t_M,N(⊗w_i⊗u_q) =  (t_M,N)_iq^  jr⊗w_j⊗u_r  

        = (-1)^|w_j| (|N_q^ r|-1) 
⟨π^(2), s^-1_(M_i^ j)⊗_A s^-1_(N_q^ r)⟩⊗w_j⊗u_r  .

Furthermore, evaluating the diagram (<ref>), we find

   t_(M,N),(M^',N^')(h⊗k)(⊗w_i⊗u_q) = 
(t_(M,N),(M^',N^')(h⊗k))_iq^  jr⊗w_j^'⊗u_r^'

        = -(-1)^|w_j^'| (|N_q^ s|+ |k_s^ r|-1) 
⟨π^(2), s^-1_(h_i^ j) ⊗_A s^-1_(N_q^ s) k_s^ r⟩⊗w_j^'⊗u_r^'

            -(-1)^|h_i^ k|+ |w_j^'| (|k_q^ r|-1)
 ⟨π^(2), s^-1h_i^ k _(M_k^' j)⊗_A s^-1_(k_q^ r)⟩⊗w_j^'⊗u_r^'  .


Given a concrete example for the bivector
π^(2)∈_A^2(𝖳_ A[-3])^4≅_A^2(𝖳_ A[-1])^0,
one can evaluate these expressions further by using the definition (<ref>)
of the duality pairing.




 §.§ Towards higher-order deformations


A powerful result in the 1-categorical theory of infinitesimal braidings
is that, upon the choice of a Drinfeld associator <cit.>, every infinitesimal
braiding lifts to a formal deformation (i.e. to all orders in ħ)
of the given symmetric monoidal category into a braided monoidal one.
See e.g. <cit.> and <cit.> for the details.
The aim of this subsection is to explore which aspects of this construction admit 
a generalization to our concept of infinitesimal 2-braidings on _A.


To study formal deformations to all orders in ħ, we introduce
the commutative -algebra [[ħ]] of formal power series and perform
a base change 

_A[[ħ]] := _A⊗[[ħ]] ∈ _[[ħ]]^[-1,0],ps

along the commutative -algebra morphism →[[ħ]].
The goal is then to deform the symmetric monoidal structure on _A[[ħ]]
into a braided monoidal structure. In order to simplify notations,
we suppress in what follows the original associator α and simply write

⊗_A^2  :=  ⊗_A∗(⊗_A⊠𝕀)  ≅  ⊗_A∗(𝕀⊠⊗_A)  :  
_A[[ħ]]⊠_A[[ħ]]⊠_A[[ħ]] ⟶  _A[[ħ]]

for the 2-times iterated tensor product, and similarly
⊗_A^n : _A[[ħ]]^⊠(n+1)→_A[[ħ]] for the n-times iterated tensor product.
Given any γ-equivariant infinitesimal 2-braiding t : ⊗_A⇒⊗_A,
we denote by


t_ij : ⊗_A^n ⟹ ⊗_A^n :   _A[[ħ]]^⊠(n+1) ⟶  _A[[ħ]]

the _[[ħ]]^[-1,0]-enriched pseudo-natural transformation which evaluates
t on the i-th and j-th tensor factor, for all i,j∈{1,…,n+1} with i≠ j.
Explicitly, the components of t_ij read as


@C=13em
[d]_-≅⊗_k=1^n+1 M_k [r]^-(t_ij)_M_1,…,M_n+1      ⊗_k=1^n+1 M_k

M_i⊗_A M_j⊗_A ⊗_k=1
k≠i,j^n+1 M_k[r]_-t_M_i,M_j⊗_A⊗_k𝕀_M_k     
M_i⊗_A M_j⊗_A ⊗_k=1
k≠i,j^n+1 M_k[u]_-≅

  ,

for all M_1,…,M_n+1∈_A[[ħ]], and


@C=13em
[d]_-≅⊗_k=1^n+1 M_k [r]^-(t_ij)_(M_1,…,M_n+1),(M_1^',…,M_n+1^')(h_1⊗⋯⊗h_n+1)      ⊗_k=1^n+1 M^'_k

M_i⊗_A M_j⊗_A ⊗_k=1
k≠i,j^n+1 M_k[r]_-t_(M_i,M_j),(M_i^',M_j^')(h_i⊗h_j)⊗_A⊗_k h_k     
M^'_i⊗_A M^'_j⊗_A ⊗_k=1
k≠i,j^n+1 M^'_k[u]_-≅

  ,


for all M_1,…,M_n+1,M^'_1,…,M^'_n+1∈_A[[ħ]]
and h_k∈_A[[ħ]](M_k,M_k^')^0, for k=1,…,n+1,
where the unlabeled vertical isomorphisms are given by the original symmetric braiding γ.
Let us note that there is an obvious generalization
of the above construction which assigns to every pair 
I=(i_1⋯ i_I) and J=(j_1⋯ j_J) of tuples of pairwise distinct indices in {1,…,n+1}
the _[[ħ]]^[-1,0]-enriched pseudo-natural transformation 
t_IJ : ⊗_A^n ⟹ ⊗_A^n which evaluates t on the 
tensor product of the I-factors and the tensor product of the J-factors.
In this compact notation, the infinitesimal hexagons from (<ref>) 
and the γ-equivariance property from Definition <ref>
read as

t_i(jk) = t_ij + t_ik  ,    t_(ij)k =  t_ik + t_jk  ,    t_ij  = t_ji   ,

for all pairwise distinct i,j,k∈{1,…,n+1}.
Furthermore, one verifies that 

[t_ij,t_kl] := t_ij∘t_kl - t_kl∘t_ij = 0  ,

for all pairwise distinct i,j,k,l∈{1,…,n+1}, by explicitly computing the components.


Following the same strategy as in <cit.> and <cit.>, 
we consider a candidate for a deformed associator and braiding
on _A[[ħ]] which is given by the ansatz


α^ħ :=  Φ(t_12,t_23) : ⊗_A^2 ⟹ ⊗_A^2 :  
_A[[ħ]]⊠_A[[ħ]]⊠_A[[ħ]] ⟶ _A[[ħ]]

and

γ^ħ :=γ∘e^ħ/2 t :  
⊗_A ⟹ ⊗_A∗τ :  
_A[[ħ]]⊠_A[[ħ]] ⟶ _A[[ħ]]  ,


where Φ(𝔞,𝔟) is some formal power series in 
two non-commutative variables 𝔞 and 𝔟 satisfying Φ(0,0)=1.
In these expressions, both Φ and the exponential function e, which we describe by its formal Taylor series,
are evaluated by using the ∘-composition of pseudo-natural transformations from 
Appendix <ref>, hence α^ħ and γ^ħ are by construction
_[[ħ]]^[-1,0]-enriched pseudo-natural transformations.

The tuple (_A[[ħ]],⊗_A,A,α^ħ,λ,ρ,γ^ħ),
with deformed associator α^ħ and braiding γ^ħ 
from (<ref>), defines a semi-strict braided monoidal category 
object in _[[ħ]]^[-1,0],ps  in the sense of Definition <ref>
if and only if all of the following hold true:


  (1) The identity

Φ(t_12,t_23+ t_24)∘Φ(t_13+t_23,t_34) = 
Φ(t_23,t_34)∘Φ(t_12+t_13,t_24+t_34)∘Φ(t_12,t_23)

holds true in the algebra (PsNat(⊗_A^3,⊗_A^3),∘,_⊗_A^3)
of _[[ħ]]^[-1,0]-enriched pseudo-natural transformations with multiplication ∘.


  (2) The identities


Φ(t_23,t_13)∘e^ħ/2(t_12+t_13)∘Φ(t_12,t_23)
 =  e^ħ/2t_13∘Φ(t_12,t_13)∘e^ħ/2t_12

and

Φ(t_13,t_12)^-1 ∘e^ħ/2(t_13+t_23)∘Φ(t_12,t_23)^-1
 =  e^ħ/2t_13∘Φ(t_13,t_23)^-1∘e^ħ/2t_23


hold true in the algebra (PsNat(⊗_A^2,⊗_A^2),∘,_⊗_A^2)
of _[[ħ]]^[-1,0]-enriched pseudo-natural transformations with multiplication ∘.

Assuming additionally that Φ satisfies the inversion property 
Φ(𝔞,𝔟)^-1=Φ(𝔟,𝔞), then the two identities
in (<ref>) are equivalent.


The triangle identity holds true automatically as a consequence of Φ(0,0)=1
and the fact that all components of t involving the monoidal unit A∈_A[[ħ]] at least once 
are necessarily trivial. (The latter is a consequence of (<ref>).) Using (<ref>)
and arguments for strictness of the interchange law for ∘ and ∗
which are similar to the ones in the proof of Proposition <ref>, one shows that (1) is equivalent 
to the pentagon identity for α^ħ. With the same kind of computation,
one shows that (2) is equivalent to the two hexagon identities for γ^ħ and α^ħ.
Finally, under the additional assumption Φ(𝔞,𝔟)^-1=Φ(𝔟,𝔞),
one easily shows that (<ref>) and (<ref>) are identified
by permuting the indices according to 123↦ 321 and using once again (<ref>).



From a superficial perspective, the identities (1) and (2) in Proposition <ref>
take exactly the same form as the defining properties of a Drinfeld associator Φ,
see e.g. <cit.> or <cit.>.
However, there is the following caveat: The  analogous identities for a Drinfeld associator
are formulated in the universal enveloping algebra of the appropriate Drinfeld-Kohno Lie algebra
where the symbols t_ij satisfy the infinitesimal braid relations
t_ij = t_ji, [t_ij,t_kl]=0 and [t_ij,t_ik+t_jk]=0, for all pairwise distinct indices i,j,k,l.
In contrast to this, our identities
(1) and (2) must hold true in the appropriate algebra of 
_[[ħ]]^[-1,0]-enriched pseudo-natural transformations with multiplication ∘.
From (<ref>) and (<ref>),
we know that our t_ij's satisfy the first two types of infinitesimal braid relations,
i.e. t_ij = t_ji and [t_ij,t_kl]=0 for all pairwise distinct indices i,j,k,l,
but as we shall illustrate below they do not necessarily satisfy the third
type of relation [t_ij,t_ik+t_jk]=0 strictly. 


So the conclusion at this moment is as follows:
Assuming that our γ-equivariant infinitesimal 2-braiding t satisfies 
the remaining infinitesimal braid relations [t_ij,t_ik+t_jk]=0 strictly, then the choice of 
a Drinfeld associator Φ (in the traditional sense) provides
via Proposition <ref> a semi-strict
braided monoidal deformation (to all orders in ħ) 
of the symmetric monoidal category object (_A,⊗_A,A,α,λ,ρ,γ).


In light of the previous remark, we make the following observation.

The pseudo-naturality structure of the γ-equivariant infinitesimal 2-braiding
t:⊗_A⇒⊗_A induces _[[ħ]]^[-1,0]-enriched modifications


Γ_ijk : [t_ij,t_ik+t_jk]      0 :  ⊗_A^n ⟹ ⊗_A^n : 
_A[[ħ]]^⊠(n+1) ⟶ _A[[ħ]]  ,

for all pairwise distinct i,j,k∈{1,…,n+1}, which enforce weakly the remaining infinitesimal braid relations.
The components (recall Definition <ref>) of these modifications are given explicitly by

.9!@C=17em[d]_-≅⊗_l=1^n+1 M_l [r]^-(Γ_ijk)_M_1,…,M_n+1     ⊗_l=1^n+1 M_l

M_i⊗_A M_j⊗_A M_k⊗_A ⊗_l=1
l≠ i,j,k^n+1 M_l
[r]_-t_(M_i⊗_A M_j,M_k),(M_i⊗_A M_j,M_k) (t_M_i,M_j⊗𝕀)⊗_A𝕀     
M_i⊗_A M_j⊗_A M_k⊗_A ⊗_l=1
l≠ i,j,k^n+1 M_l[u]_-≅  ,


for all M_1,…,M_n+1∈_A[[ħ]]. 


Recalling (<ref>), one observes that 

[t_ij,t_ik+t_jk]  =  [t_ij,t_(ij)k]  =  t_ij∘t_(ij)k -t_(ij)k∘t_ij  .

Using the composition formulas from Appendix <ref>, the component expressions
(<ref>) and pseudo-naturality of t_(ij)k in the left entry, one computes 
the components of this _[[ħ]]^[-1,0]-enriched pseudo-natural 
transformation and finds that 

(t_ij∘t_(ij)k -t_(ij)k∘t_ij)_M_1,…,M_n+1  = __A((Γ_ijk)_M_1,…,M_n+1)

with Γ_ijk defined in (<ref>). By a routine calculation
one then shows that Γ_ijk satisfies the properties of a 
_[[ħ]]^[-1,0]-enriched modification from Definition <ref>.



The reader might have noticed that there are potential ambiguities
in defining the _[[ħ]]^[-1,0]-enriched modifications Γ_ijk.
These arise from the multiple equivalent ways to write the expression [t_ij,t_ik+t_jk]
by using the identities in (<ref>). 
Such ambiguities do not seem to be present for our γ-equivariant infinitesimal 2-braidings.
Indeed, one observes that

Γ_ijk  = Γ_jik

is symmetric in the first two indices, hence there are no ambiguities
arising from rewriting [t_ij,t_ik+t_jk] = [t_ji,t_jk+t_ik] by using
the symmetry property t_ij =t_ji.
Furthermore, rewriting [t_ij,t_ik+t_jk] = [t_ij,t_(ij)k] = [t_ij,t_k(ij)]
by using the identities (<ref>), one finds 
from the pseudo-naturality structure of t_k(ij) in the right entry
a modification which agrees with Γ_ijk.



For the complete description of a categorified version
of the infinitesimal braid relations, one would also 
require suitable coherence conditions for the modifications Γ_ijk.
One could then attempt to weaken the strict identities (1) and (2) 
from Proposition <ref> to identities which hold true up
to suitable modifications, playing the role of pentagonators and hexagonators
for a weakened variant of the semi-strict braided monoidal category objects from 
Definition <ref>. To the best of our knowledge, this 
is relatively unexplored territory, with the notable exception given by 
exploratory works of Cirio and Faria Martins <cit.>.
In our present paper, we will not attempt to address these coherence issues.



For our infinitesimal 2-braidings t in Theorem <ref>,
which are constructed from 2-shifted Poisson structures π on the CDGA A, we can use
the formulas from Remark <ref> in order to provide 
a more explicit basis expression for the modifications Γ_ijk
in Proposition <ref>. Denoting M=M_i, N=M_j and L=M_k,
we then find for the bottom horizontal arrow in (<ref>)

    t_(M⊗_A N,L),(M⊗_A N,L) (t_M,N⊗𝕀)(⊗w_i⊗u_q⊗v_s)

        =  -(-1)^|w_j| (|L_s^ t|-1) 
⟨π^(2),
s^-1_((t_M,N)_iq^  jr)⊗_A s^-1_(L_s^ t)
⟩⊗w_j⊗u_r⊗v_t  ,

where the components (t_M,N)_iq^  jr are defined in 
(<ref>) and 
the second term in (<ref>) vanishes because k=𝕀:L→ L
is the identity in the present case.


To conclude this subsection, we carry out briefly an explicit study of the 
identities (1) and (2) in Proposition <ref>
modulo ħ^3 by taking as ansatz the traditional Drinfeld associator

Φ(𝔞,𝔟) = 1 + ħ^224 [𝔞,𝔟]  mod ħ^3  ,

see e.g. <cit.>. Using the 
identity (<ref>), one immediately checks 
that the identity (<ref>) holds true strictly modulo ħ^3, 
i.e. without the need for any modifications.
In contrast to this, the identities (<ref>) do not hold true strictly modulo ħ^3.
Concretely, bringing all terms in (<ref>) to the left-hand side, one finds that

ħ^224([t_12, t_13+ t_23] - [t_13,t_12+ t_32])
  ħ^224(Γ_123-Γ_132)  0
 
only holds true weakly up to a certain linear combination of the _[[ħ]]^[-1,0]-enriched modifications
from Proposition <ref>. Similarly, bringing all terms in (<ref>) to the left-hand side,
one finds that

ħ^224([t_23, t_21+ t_31] - [t_13,t_12+ t_32])
  ħ^224(Γ_231-Γ_132)  0

only holds true weakly up to a certain linear combination of the _[[ħ]]^[-1,0]-enriched modifications
from Proposition <ref>. This illustrates that the strict identities
from Proposition <ref>, and the associated concept of semi-strict
braided monoidal category objects from Definition <ref>, is highly restrictive
even at low orders in the deformation parameter ħ and hence they should be relaxed to weaker concepts.






§ EXAMPLES


The aim of this section is to study explicit examples of our 
infinitesimal 2-braidings from Theorem <ref>.
The examples we consider are based on L_∞-algebras,
which are a generalization of differential graded Lie algebras
for which the Jacobi identities hold true only up to homotopy coherence data.
More precisely, an L_∞-algebra consists of -graded vector space L^♯,
which we always assume to be of finite total dimension (L^♯)=∑_i∈(L^i)<∞,
together with a family of brackets ℓ_n : L^♯⊗ n→ L^♯
of degree 2-n, for all n≥ 1, which satisfy the homotopy Jacobi identities.
In particular, the unary bracket ℓ_1 defines a differential on L^♯,
the binary bracket ℓ_2 defines a Lie bracket on the cochain complex
(L^♯,ℓ_1), and the ternary bracket ℓ_3
provides a homotopy witnessing the Jacobi identity for ℓ_2.
See e.g. <cit.> for the complete definition of the concept of
L_∞-algebras and an excellent introduction to their basic theory.


Associated to an L_∞-algebra L= (L^♯,{ℓ_n}) 
is its Chevalley-Eilenberg CDGA ^∙(L). The underlying 
-graded commutative algebra of ^∙(L) is given by

^♯(L) := (L^♯∗[-1])

and the differential _ :=∑_n≥ 1ℓ_n^∗ [-1] is defined on the generators L^♯∗[-1] by dualizing and shifting
the brackets ℓ_n : L^♯⊗ n→ L^♯.
The Chevalley-Eilenberg CDGA ^∙(L) of an L_∞-algebra
of finite total dimension is finitely generated and semi-free, hence it satisfies
our hypotheses from Section <ref>.
The Chevalley-Eilenberg CDGA ^∙(L) admits a geometric interpretation as the CDGA of functions
on the formal quotient stack BL:=[pt/L], which is an infinitesimal
analogue of the classifying stack BG:=[pt/G] of a group G.


The symmetric monoidal dg-category _^∙(L)^fgsf
of finitely generated semi-free ^∙(L)-dg-modules
describes representations of the L_∞-algebra L
and their ∞-morphisms.
Our result in Theorem <ref> then implies that,
given any 2-shifted Poisson structure on the CDGA ^∙(L),
one obtains a first-order deformation of the homotopy 2-category
𝖧𝗈_2(_^∙(L)^fgsf) 
of the representation category of L into a braided monoidal category.
Drawing analogies with the standard deformation theory 
of the (strict) representation category 𝐑𝐞𝐩() of an ordinary Lie algebra  
and its relationship to the quantum group U_q, see
e.g. <cit.> and <cit.>, one may interpret 
our deformation constructions as a first step towards an
L_∞-algebraic generalization of quantum groups.



 §.§ Ordinary Lie algebras


Let us start with the simplest
example which is given by the case where the L_∞-algebra L
is an ordinary (finite-dimensional) Lie algebra =(,[ · , · ]). This means that 
the underlying -graded vector space is , concentrated in degree 0,
and all brackets but the Lie bracket ℓ_2:=[ · , · ] : ⊗→ are trivial.
To simplify our calculations below, we shall choose a basis {x_a∈:a=1,…,()}
and denote the structure constants by [x_a,x_b]=f^c_ab x_c. The Chevalley-Eilenberg
CDGA ^∙() is then given by the -graded commutative algebra


^♯() = [{θ^a}]

which is freely generated by ()-many generators θ^a
of degree 1, for a=1,…,(), together with the differential

_(θ^a) = -12  f^a_bc θ^b θ^c


which is determined by the structure constants. Here and in what follows we 
shall always use the standard summation convention according to which summations over repeated indices are suppressed.
Note that the square-zero condition _ _(θ^a)=0 follows from the Jacobi identity
of the Lie bracket after expressing the latter in terms of the structure constants.


Since the input datum for Theorem <ref> is a 
2-shifted Poisson structure on ^∙() in the sense
of Definition <ref>, we first have to 
understand and characterize the possible choices for such data.
For the ^∙()-dg-module of derivations 𝖳_^∙()
we choose the basis {∂_a∈𝖳_^∙() : a=1,…,()}
which is defined by ∂_a(θ^b)=δ_a^b, for all a,b. Note that
each basis element ∂_a has degree -1 since all θ^b are of degree 1.
The underlying -graded commutative algebra of the completed 2-shifted polyvectors from Definition
<ref> is then given by

(^∙(),2)^♯ = [{θ^a}][[{s^3 ∂_a}]]  ,

where [[ · ]] denotes formal power series. Note that the generators 
s^3∂_a are of degree 3-1=2 in (^∙(),2).

Any 2-shifted Poisson structure π=∑_m≥ 2π^(m)∈(^∙(),2)^4 
on ^∙() is of the form


π = π^(2) = 12 π^ab  s^3∂_a  s^3∂_b  , 

where π^ab∈  are coefficients which
must satisfy the -invariance property

f_cd^a  π^d b +f_cd^b π^a d = 0  .




Using that θ^a is of degree 1 and that s^3∂_a is of degree 2, 
one finds by a simple degree-counting argument that (<ref>) is
the only possible form for a completed 2-shifted polyvector of degree 4 and weight ≥ 2.
The Maurer-Cartan equation from Definition <ref> for such π,
see also Remark <ref>, is equivalent to the two individual equations
_(π)=0 and {π,π}=0. The second equation holds true automatically
because (<ref>) has constant coefficients, while the first equation
is equivalent to the -invariance property (<ref>).
To see the latter point explicitly, one uses the Leibniz rule
for the differential _ and 
_(s^3∂_a) = -f_ab^c θ^b s^3∂_c.



Note that the 2-shifted Poisson structures π from 
Proposition <ref> 
are canonically identified with 
invariant symmetric tensors π =12 π^ab x_a x_b∈ (^2)^.
Hence, our results are compatible with the results of Safronov <cit.>
who has computed the 2-shifted (and also the 1-shifted) Poisson
structures on the classifying stack BG = [pt/G] of an algebraic group G.


Consider now the homotopy 2-category
_^∙()= 𝖧𝗈_2(_^∙()^fgsf)
of the symmetric monoidal dg-category of finitely generated semi-free ^∙()-dg-modules,
which is the object of focus in our deformation construction from Theorem <ref>.
Let us first illustrate the sense in which this category describes 
weak representations of the Lie algebra . By definition, given any object M∈_^∙(),
one has a -graded vector space of generators W^♯ such that M^♯≅^♯()⊗ W^♯.
Choosing any basis {w_i∈ W^♯} for the generators, the differential on M
is completely characterized by specifying _M(⊗ w_i)∈ M, for all i,
which when expanded in terms of the chosen bases reads as

_M(⊗w_i) =  M_i^ j⊗w_j  = ∑_n=0^∞1n! M_a_1…a_n i^a_1…a_n ij
 θ^a_1⋯θ^a_n⊗w_j  ,

where M_a_1… a_n i^a_1… a_n ij∈ are coefficients. 
The square-zero condition _M_M(⊗ w_i)=0 gives a tower of compatibility conditions
between the coefficients M_a_1… a_n i^a_1… a_n ij and the structure constants f^a_bc 
of the Lie algebra , which express a concept of Lie algebra representation up to coherent homotopies.
Strict representations of  on a cochain complex W are recovered in the special case
where _M(⊗ w_i) = M_i^ j⊗ w_j + M_ai^  j θ^a ⊗ w_j
is of polynomial degree ≤ 1 in θ^a. The first term in this expression defines
a differential _W on W^♯ and the second term defines a strict Lie algebra action map 
⊗ W → W which is a cochain map for the differential of W = (W^♯,_W).


In addition to weak representations of the Lie algebra , the homotopy 2-category
_^∙()= 𝖧𝗈_2(_^∙()^fgsf)
also describes a concept of ∞-morphisms
between such weak representations, i.e. weakly -equivariant maps.
(The 2-morphisms in _^∙() are homotopies between such ∞-morphisms.)
Indeed, a morphism h : M→ M^' in _^∙() is by definition
a cochain map which commutes with the left ^∙()-actions.
Hence, by choosing a basis {w_i∈ W^♯} for M^♯≅^♯()⊗ W^♯ 
and a basis {w_i^'∈ W^'♯} for M^'♯≅^♯()⊗ W^'♯,
we can expand 

h(⊗w_i) = h_i^ j⊗w_j^' =  ∑_n=0^∞1n! h_a_1…a_n i^a_1…a_n ij
 θ^a_1⋯θ^a_n⊗w_j^'  ,

where h_a_1… a_n i^a_1… a_n ij∈ are coefficients.
The cochain map condition _M^'(h(⊗ w_i)) = h(_M(⊗ w_i)) 
gives a tower of compatibility conditions
between the coefficients h_a_1… a_n i^a_1… a_n ij 
and the structure constants f^a_bc of the Lie algebra , 
which express a concept of -equivariance up to coherent homotopies.
Strict morphisms h : M→ M^' between strict -representations 
on cochain complexes W and W^' are recovered in the special case
where h(⊗ w_i) = h_i^ j⊗ w^'_j is constant in
θ^a. 


The following result is a direct consequence of Theorem <ref>
and Proposition <ref>.

Every invariant symmetric tensor 
π =12 π^ab x_a x_b∈ (^2)^
defines via Proposition <ref> 
and Theorem <ref> a γ-equivariant infinitesimal 
2-braiding t on the homotopy 2-category _^∙() of the
symmetric monoidal dg-category of finitely generated semi-free ^∙()-dg-modules.



Let us recall that explicit formulas for the components of the 
γ-equivariant infinitesimal 2-braiding t 
are given in (<ref>). Note that the 
morphism components t_M,N in (<ref>)
involve the de Rham differential

(_⊗𝕀)(_M(⊗w_i))    =  
∑_n=1^∞1(n-1)! M_a_1…a_n i^a_1…a_n ij
 θ^a_1⋯θ^a_n-1_(θ^a_n)⊗w_j

of (<ref>), as well as (_⊗𝕀)(_N(⊗ u_q)).
Hence, the components t_M,N in (<ref>) are in general 
non-constant polynomials in θ^a, unless both M and N are strict -representations.
Similar, the homotopy components t_(M,N),(M^',N^')(h⊗k) 
in (<ref>) involve the de Rham differential

(_⊗𝕀)(h(⊗w_i))    =  
∑_n=1^∞1(n-1)! h_a_1…a_n i^a_1…a_n ij
 θ^a_1⋯θ^a_n-1_(θ^a_n)⊗w^'_j

of (<ref>), as well as
(_⊗𝕀)(k(⊗ u_q)).
Hence, the homotopies t_(M,N),(M^',N^')(h⊗k)  
in (<ref>) are in general non-trivial,
unless both h and k are strict morphisms.


As a direct consequence of Proposition <ref>
and Proposition <ref>, we obtain the following result.

Every invariant symmetric tensor 
π =12 π^ab x_a x_b∈ (^2)^
defines via Corollary <ref> and
Proposition <ref>
a braided monoidal deformation to all orders in ħ
of the full subcategory of _^∙()∈_[[ħ]]^[-1,0],ps 
which consists of all ^∙()-dg-modules corresponding to strict -representations.


Combining our observations from
Example <ref> and Remark <ref>,
we obtain that the modifications Γ_ijk in Proposition <ref>
are all trivial for strict -representations because in this case
(t_M,N)_iq^  jr are constant polynomials in θ^a, hence
_((t_M,N)_iq^  jr)=0. This implies that
the hypotheses of Proposition <ref> are satisfied,
which yields the desired braided monoidal deformation to all orders in ħ.



Our result in Corollary <ref> is a homotopical
generalization of the deformation constructions in
<cit.> and <cit.>. We would like to stress that our generalization 
is non-trivial because, even though we restrict ourselves to strict -representations,
the morphisms in the full subcategory from Corollary <ref>
are general (i.e. non-strict) ∞-morphisms between strict -representations. 
Recalling our observations from Remark <ref>, this implies in particular
that the infinitesimal braiding t is a genuine pseudo-natural transformation,
i.e. it has non-trivial homotopy components t_(M,N),(M^',N^')(h⊗k),
and so are the associator and braiding of the deformed category from
Proposition <ref> and Corollary <ref>.





 §.§ String Lie 2-algebras


String Lie 2-algebras are simple examples of L_∞-algebras
which arise as shifted central extensions 
determined by an ordinary (finite-dimensional) Lie algebra =(,[ · , · ]), concentrated
in degree 0, and a 3-cocycle κ:⋀^3→ on .
Such L_∞-algebras appeared first in <cit.>, 
however the name “string Lie 2-algebras” has only been established later.
The resulting L_∞-algebra _κ
is given by the -graded vector space defined by _κ^0:=,
_κ^-1:= and _κ^i:=0, for all i∈∖{-1,0},
and the non-trivial brackets read as

ℓ_2(x,y)  := [x,y]  ,    ℓ_3(x,y,z) :=  κ(x,y,z)  ,

for all x,y,z∈. Let us choose again a basis {x_a∈} and denote the structure constants
by [x_a,x_b]=f^c_ab x_c and κ(x_a,x_b,x_c) = κ_abc.


The Chevalley-Eilenberg CDGA ^∙(_κ)
of the string Lie 2-algebra _κ
is then given by the -graded commutative algebra


^♯(_κ) = [{θ^a},ν]

which is freely generated by ()-many generators θ^a
of degree 1, for a=1,…,(), and a single generator ν
of degree 2, together with the differential

_(θ^a) = -12  f^a_bc θ^b θ^c  ,    _(ν) = -13! κ_abc θ^a θ^b θ^c


which is determined by the structure constants. 
Note that the square-zero condition _ _(θ^a)=0 follows from the Jacobi identity
of the Lie bracket and _ _(ν)=0 
follows from the 3-cocycle condition of κ.


Let us now characterize the
2-shifted Poisson structures on ^∙(_κ) in the sense
of Definition <ref>.
For the ^∙(_κ)-dg-module of derivations 𝖳_^∙(_κ)
we choose the basis {{∂_a},∂_ν}
which is defined by ∂_a(θ^b)=δ_a^b, ∂_a(ν)=0,
∂_ν (θ^a)=0 and ∂_ν(ν)=1, for all a,b. Note that
each basis element ∂_a has degree -1 and ∂_ν has degree -2.
The underlying -graded commutative algebra of the completed 2-shifted polyvectors from Definition
<ref> is then given by

(^∙(_κ),2)^♯ = [{θ^a},ν][[{s^3 ∂_a},s^3∂_ν]]  .

where [[ · ]] denotes formal power series. Note that the generators 
s^3∂_a are of degree 3-1=2 
and that s^3∂_ν is of degree 3-2=1 
in (^∙(_κ),2).

Any 2-shifted Poisson structure 
π=∑_m≥ 2π^(m)∈(^∙(_κ),2)^4 
on ^∙(_κ) is of the form


π = π^(2) =  12 π^ab  s^3∂_a  s^3∂_b + π̃_b^a θ^b s^3∂_a s^3∂_ν  , 

where π^ab∈ and π̃_b^a∈ are coefficients
which must satisfy the properties

f_cd^a π^db + f_cd^b π^ad    = 0  , 

f_bc^d π̃^a_d - f_bd^a π̃^d_c + f_cd^a π̃^d_b -κ_bcd π^da     = 0  ,


π^ac π̃_c^b + π^bc π̃_c^a     =0  . 




The general form (<ref>) for a 
completed 2-shifted polyvector of degree 4 and weight ≥ 2
follows from a simple degree counting argument. (Note that 
products of the form s^3∂_ν s^3∂_ν
vanish in the CDGA (^∙(_κ),2) 
because s^3∂_ν has degree 1, hence s^3∂_ν s^3∂_ν = 
-s^3∂_ν s^3∂_ν =0.)
The Maurer-Cartan equation from Definition <ref> for such π,
see also Remark <ref>, is equivalent to the two individual equations
_(π)=0 and {π,π}=0. 
The first equation _(π)=0 can be evaluated using
the Leibniz rule for the differential _ and

_(s^3∂_a) = -f_ab^c θ^b s^3∂_c +
12 κ_abc θ^b θ^c s^3∂_ν  ,    _(s^3∂_ν) = 0  .

From this one finds the first two properties (<ref>) and (<ref>).
The second equation {π,π}=0 can be evaluated
using the definition (<ref>) and properties (<ref>)
of the Schouten-Nijenhuis bracket, which leads to the third property (<ref>).



As a consequence of (<ref>), the first term in (<ref>) is equivalent to the
datum of an invariant symmetric tensor π =12 π^ab x_a x_b∈(^2)^
on the underlying Lie algebra  of the string Lie 2-algebra _κ.
The second term in (<ref>) is equivalent to a linear map 
π̃:→ , x_a↦π̃_a^b x_b
on the underlying Lie algebra. Property 
(<ref>) then demands that the linear map π̃ annihilates
the invariant symmetric tensor π according to


(π̃⊗𝕀_)(π) + (𝕀_⊗π̃)(π) = 0  .

The remaining property (<ref>) can be written equivalently as
 
π̃([x,y]) =  [π̃(x),y] + [x,π̃(y)] + 
(κ(x,y,-)⊗𝕀_)(π) + (𝕀_⊗κ(x,y,-))(π)    ,


for all x,y∈, which means that the linear map π̃:→ 
is a derivation of the underlying Lie algebra , up to corrections arising from
the 3-cocycle κ:⋀^3→ and the invariant symmetric tensor π∈(^2)^.


Our 2-shifted Poisson structures on ^∙(_κ)
are similar, but as it seems not identical, to the quasi-invariant symmetric
tensors on crossed modules of Lie algebras from <cit.>.
A direct comparison between these two structures is difficult because 
our preferred model for the string Lie 2-algebra _κ is given by
a finite-dimensional L_∞-algebra with a non-trivial ℓ_3 bracket (<ref>), 
while <cit.> use a different model which is given by an
infinite-dimensional crossed module of Lie algebras 
that encodes the cocycle κ less directly.
Our suspicion that these two concepts are slightly different
will be strengthened by the following example.



In order to illustrate Proposition <ref>,
let us consider the example where = 𝔰𝔩_2 
is the special linear Lie algebra of order 2. We work with
the usual basis {x_+ ,x_- ,x_3∈𝔰𝔩_2} 
whose Lie bracket relations are

[x_+,x_-]  = x_3  ,    [x_3,x_±] = ±2 x_±  .

Using the rescaled Killing form 
( · , · ):𝔰𝔩_2⊗𝔰𝔩_2→
whose non-vanishing values are given by (x_3,x_3) =1 and (x_+,x_-)=(x_-,x_+)=12,
we define the 3-cocycle κ(x,y,z):= (x,[y,z]). In terms of our basis,
this 3-cocycle is given by

κ(x_+,x_-,x_3) = 1

and total antisymmetry in its arguments.
This defines the string Lie 2-algebra (𝔰𝔩_2)_κ.


Since 𝔰𝔩_2 is a simple Lie algebra, there exists a unique (up to scale)
invariant symmetric tensor, which is concretely given by

π =  λ ( x_+   x_- +14 x_3^2)  ∈  (^2 𝔰𝔩_2)^𝔰𝔩_2  ,

where λ∈ is any parameter. 
Assuming for the moment that λ≠ 0 is non-zero, the general solution
for the linear map π̃ : 𝔰𝔩_2→𝔰𝔩_2
to the equation (<ref>), or 
the equivalent equation (<ref>), is given by

π̃(x_+) = a x_+ + b x_3  ,  π̃(x_-) = -a x_- + c x_3  ,  π̃(x_3) = -2 c x_+ -2 b  x_-  ,

where a,b,c∈ are arbitrary parameters. Evaluating the equivalent form (<ref>)
of the remaining equation (<ref>) on x=x_+ and y=x_-, one finds 
that it does not admit any solutions for λ≠ 0. This implies that, 
for every 2-shifted Poisson structure on ^∙((𝔰𝔩_2)_κ) as in
Proposition <ref>, the 
invariant symmetric tensor (<ref>) must necessarily be trivial π=0.
In this case the equations in (<ref>),
see also (<ref>) for their equivalent forms, simplify to the single condition
that π̃:𝔰𝔩_2→𝔰𝔩_2 is a Lie algebra derivation,
i.e. π̃([x,y]) = [π̃(x),y] + [x,π̃(y)],
for all x,y∈𝔰𝔩_2. Hence, the 2-shifted Poisson structures on ^∙((𝔰𝔩_2)_κ)
are classified by the Lie algebra derivations π̃ on 𝔰𝔩_2.
This behavior is drastically different to the case of ordinary Lie algebras  
from Proposition <ref> and the
quasi-invariant symmetric tensors on crossed modules of Lie algebras
from <cit.> and <cit.>.


The following result is a direct consequence of Theorem <ref>,
Proposition <ref> and Remark <ref>.

Every pair (π,π̃) consisting of a 
invariant symmetric tensor π =12 π^ab x_a x_b∈ (^2)^
and a linear map π̃ : → , x_a↦π̃_a^b x_b
which satisfies the properties in (<ref>)
defines via Proposition <ref>
and Theorem <ref> a γ-equivariant infinitesimal 
2-braiding t on the homotopy 2-category _^∙(_κ) of the
symmetric monoidal dg-category of finitely generated semi-free ^∙(_κ)-dg-modules.



Recalling the explicit formulas for the components of the 
γ-equivariant infinitesimal 2-braiding t 
from (<ref>), one immediately observes 
by using also (<ref>) that the morphism components t_M,N 
are in general non-constant polynomials in θ^a 
in the case where π̃^a_b≠ 0 is non-trivial. 
(As illustrated in Example <ref>, this seems to be the most relevant case.)
Hence, the modifications Γ_ijk from Proposition <ref> and Example
<ref> are in general non-trivial. This even holds true for
representation of the string Lie 2-algebra _κ which are strict
in the sense that the expansions as in (<ref>) 
of the differential _M are polynomials in θ^a and ν of degree ≤ 1.
As a consequence, we currently do not have a meaningful generalization 
of the all-order deformation result from Corollary <ref>
to the case of ^∙(_κ). Hence, further work as outlined in
Remark <ref> is needed in order to extend
our result in Corollary <ref> to a braided monoidal
deformation up to all orders in ħ.







§ ACKNOWLEDGMENTS

We would like to thank Jon Pridham for useful discussions.
C.K. is supported by an EPSRC doctoral studentship. 
A.S. gratefully acknowledges the support of 
the Royal Society (UK) through a Royal Society University 
Research Fellowship (URF\R\211015)
and Enhancement Grants (RF\ERE\210053 and 
RF\ERE\231077).










§ HIGHER-CATEGORICAL FRAMEWORK


In this appendix, we introduce a suitable 
higher-categorical framework in which the deformation constructions
from Section <ref> take place. 
The basic objects of our interest are categories 
which are enriched over the symmetric monoidal category _R^[-1,0] 
of 2-term cochain complexes (concentrated in degrees -1 and 0) 
of modules over a commutative -algebra R. 
We will show below that these objects assemble
into a suitably weakened kind of 3-category _R^[-1,0],ps whose 1-morphisms
are _R^[-1,0]-enriched functors, 2-morphisms are 
_R^[-1,0]-enriched pseudo-natural transformations and 
3-morphisms are _R^[-1,0]-enriched modifications. This 3-category carries a symmetric monoidal
structure, which allows one to define a concept of _R^[-1,0]-enriched
braided monoidal categories whose associators, unitors and braidings are 
enriched pseudo-natural transformations.
This increased flexibility arising from pseudo-naturality is essential 
for our construction of infinitesimal 2-braidings in Section <ref>.
We assume that the reader has some familiarity with 
enriched category theory, see e.g. <cit.> for a comprehensive treatment, 
or <cit.> and <cit.> for more concise introductions.




 §.§ 3-categories with pseudofunctorial composition


Let us recall that a strict 3-category can be defined as a category which is
enriched over the category Cat^str of strict 2-categories and strict 2-functors.
We require a slightly weakened variant of this concept where the enrichment
category is generalized to the category Cat^ps of strict 2-categories
and pseudo-functors. See e.g. <cit.> for a detailed
description of this category and, in particular, of its strictly associative and unital composition.
The category Cat^ps admits finite products and a terminal object,
given by the 2-category 1∈Cat^ps consisting of a single 0-cell
and its identity 1-cell and 2-cell, hence it is symmetric monoidal with respect to the 
Cartesian symmetric monoidal structure.

We denote by 

Cat := Cat^ps
the 2-category whose


  * objects are Cat^ps-enriched categories,

  * 1-morphisms are Cat^ps-enriched functors, and

  * 2-morphisms are Cat^ps-enriched natural transformations.



To illustrate how an object ∈Cat admits an interpretation
in terms of a weakened kind of 3-category, let us spell out, in some detail, 
the data defining . By definition of an enriched category,  has a class 
of objects and, for each pair of objects c,c^'∈, a strict 2-category
(c,c^')∈Cat^ps of morphisms. Since the 2-category
(c,c^') consists of 0-cells, 1-cells and 2-cells, there are three layers
of morphisms in ∈Cat, as required for a 3-category. The enriched category 
comes further with identity morphisms 𝕀_c : 1→(c,c), or equivalently
0-cells 𝕀_c∈(c,c), and composition morphisms

∗ :  (c^',c^'')×(c,c^')  ⟶ (c,c^'')

in Cat^ps, which in our case are pseudo-functors
between strict 2-categories. The compositions ∗ must be strictly associative 
and strictly unital with respect to the identities 𝕀_c.
Note that the relaxation to pseudo-functors makes our variant of 3-categories 
from Definition <ref> more general than strict 3-categories, 
for which the analogous composition morphisms ∗ must be strict 2-functors.


The 2-category Cat carries a symmetric monoidal structure
which is determined by the Cartesian symmetric monoidal structure of the enrichment category Cat^ps,
see e.g. <cit.> for the details. The monoidal product ×∈Cat
of ,∈Cat is the Cat^ps-enriched category
whose objects are pairs (c,d)∈× of objects c∈ and d∈
and whose morphisms (×)((c,d),(c^',d^')):=(c,c^')×(d,d^')
∈Cat^ps are given by the Cartesian symmetric monoidal structure on Cat^ps.
The monoidal unit ℐ∈Cat is the Cat^ps-enriched category
with a single object, say ⋆∈ℐ, and morphisms ℐ(⋆,⋆):=
1∈Cat^ps the terminal object. The symmetric braiding


flip_,^ : × ⟶ ×
is the Cat^ps-enriched functor which acts on objects by flipping 
(c,d)↦ (d,c) and on morphisms through the symmetric braiding on Cat^ps

(×)((c,d),(c^',d^')) =  (c,c^')×(d,d^') 
≅⟶  (d,d^')×(c,c^') =  (×)((d,c),(d^',c^'))  .


Making use of this symmetric monoidal structure on Cat,
one can define a concept of symmetric monoidal category objects in Cat.

A symmetric monoidal category object in Cat is a tuple 
(,⊠,u,α,λ,ρ,τ)
consisting of the following data:


  (i) an object ∈Cat,

  (ii) a 1-morphism ⊠ : ×→ in Cat (called monoidal product),

  (iii) a 1-morphism u : ℐ→ in Cat (called monoidal unit),

  (iv) invertible 2-morphisms

    𝒞×𝒞×𝒞      𝒞×𝒞
    
    𝒞×𝒞      𝒞["⊠×id_𝒞", from=1-1, to=1-3]
    	["id_𝒞×⊠"', from=1-1, to=3-1]
    	["α"', shorten <=12pt, shorten >=12pt, Rightarrow, from=1-3, to=3-1]
    	["⊠", from=1-3, to=3-3]
    	["⊠"', from=3-1, to=3-3]
           𝒞
    ℐ×𝒞      𝒞×ℐ
       𝒞×𝒞["≅"', from=1-2, to=2-1]
    	["≅", from=1-2, to=2-3]
    	["u×id_𝒞"', from=2-1, to=3-2]
    	["id_𝒞× u", from=2-3, to=3-2]
    	[""name=0, anchor=center, inner sep=0, "⊠"description, from=3-2, to=1-2]
    	["λ", shorten <=5pt, shorten >=9pt, Rightarrow, from=2-1, to=0]
    	["ρ"', shorten <=5pt, shorten >=9pt, Rightarrow, from=2-3, to=0]

in Cat (called associator, left unitor and right unitor),


  (v) an invertible 2-morphism 

    𝒞×𝒞      𝒞
       𝒞×𝒞[""name=0, anchor=center, inner sep=0, "⊠", from=1-1, to=1-3]
    	["flip_𝒞,𝒞^"', from=1-1, to=2-2]
    	["⊠"', from=2-2, to=1-3]
    	["τ", shorten <=6pt, shorten >=3pt, Rightarrow, from=0, to=2-2]

in Cat (called symmetric braiding).

These data have to satisfy the usual coherence conditions for symmetric monoidal categories
internal to the 2-category Cat.





 §.§ A symmetric monoidal 3-category of 2-term truncated dg-categories


Let R be a commutative, associative and unital -algebra and consider
the symmetric monoidal category _R of cochain complexes of R-modules
from  Subsection <ref>.
We denote by

_R^[-1,0] ⊆  _R

the full subcategory whose objects are all cochain complexes V∈_R
concentrated in degrees {-1,0}, i.e. V^i=0 for all i∈∖{-1,0}.
We shall often denote such objects by

V  =  (@C=2em
V^-1  [r]^-_V       V^0
)
 ∈  _R^[-1,0]   .

Note that the tensor product ⊗_R on _R does not directly restrict
to the full subcategory _R^[-1,0]⊆_R because
V⊗_R W∈_R is, in general, concentrated in degrees {-2,-1,0},
for V,W∈_R^[-1,0]. This issue can be resolved by composing ⊗_R with
the good truncation functor τ^[-1,0] : _R →_R^[-1,0] to define
the truncated tensor product


⊗_R : 
@C=3em
_R^[-1,0]×_R^[-1,0][r]^-⊆       _R×_R[r]^-⊗_R      
_R [r]^-τ^[-1,0]       _R^[-1,0]
  .

Explicitly, given two objects V,W∈_R^[-1,0], the truncated tensor product reads as

V⊗_R W =  (
@C=4em
(V^-1⊗_R W^0) ⊕(V^0⊗_R W^-1)/_V⊗_R W(V^-1⊗_R W^-1) [r]^-_V⊗_R W      
V^0⊗_R W^0

)  .


One directly checks that _R^[-1,0] forms a symmetric monoidal category
with respect to the truncated tensor product ⊗_R, the monoidal unit R∈_R^[-1,0]
and the symmetric braiding

γ_V,W^ :  V⊗_R W ⟶ W⊗_R V ,  
v⊗_R w ⟼  w⊗_R v  ,

where all Koszul signs are trivial
because there are no odd-odd degree combinations in the truncated tensor product (<ref>).
For later use, let us observe that the symmetric monoidal categories
(_R,⊗_R,R,γ) and (_R^[-1,0],⊗_R,R,γ)
are related by a lax symmetric monoidal functor.

The truncation functor τ^[-1,0] : _R →_R^[-1,0] is canonically lax symmetric monoidal.


The functor τ^[-1,0] preserves the monoidal units strictly, i.e. τ^[-1,0](R)=R,
and the lax structure τ^[-1,0](V) ⊗_R τ^[-1,0](W)→τ^[-1,0](V⊗_R W) for the tensor product of V,W∈_R 
is given by the obvious cochain map (drawn vertically)


@C=4em@R=2em
[d] (V^-1_V V^-2⊗_R 𝖹^0(W)) ⊕(𝖹^0(V)⊗_R W^-1_W W^-2)/_V⊗_R W (V^-1⊗_R W^-1) [r]^-_V⊗_R W       𝖹^0(V)⊗_R 𝖹^0(W)
[d]

(V⊗_R W)^-1/_V⊗_R W(V⊗_R W)^-2[r]_-_V⊗_R W     𝖹^0(V⊗_R W)

  ,

where 𝖹^0(U) := (_U : U^0 → U^1) denotes the submodule of 0-cocycles
associated with a cochain complex U∈_R.


The goal of the remaining part of this subsection is to provide details
for the following result.

There exists a 3-category

_R^[-1,0],ps  ∈ Cat

in the sense of Definition <ref> whose


  * objects are _R^[-1,0]-enriched categories,

  * 1-morphisms are _R^[-1,0]-enriched functors,

  * 2-morphisms are _R^[-1,0]-enriched pseudo-natural transformations, and

  * 3-morphisms are _R^[-1,0]-enriched modifications.

This 3-category admits a symmetric monoidal structure in the sense of
Definition <ref> which agrees on objects, 1-morphisms 
and strictly natural 2-morphisms with the usual one from enriched
category theory, see e.g. <cit.>.


The objects and 1-morphisms in _R^[-1,0],ps are defined 
according to the standard definitions of enriched category theory.
The precise definitions of the 2-morphisms and 3-morphisms are as follows.

Let F,G : → be two _R^[-1,0]-enriched functors between two
_R^[-1,0]-enriched categories  and . A _R^[-1,0]-enriched pseudo-natural
transformation ζ : F⇒ G consists of the following data:


  (i) For each object c∈, an element ζ_c ∈(F(c),G(c))^0 
of degree 0 in the -object (F(c),G(c))∈_R^[-1,0].


  (ii) For each pair of objects c,c^'∈, an R-linear map

ζ_c,c^' :  (c,c^')^0 ⟶ (F(c),G(c^'))^-1  .


These data have to satisfy the following properties:


  (1) For all h∈(c,c^')^0,


G(h)  ζ_c - ζ_c^' F(h) = _( ζ_c,c^'(h))  ,

and for all k∈(c,c^')^-1,

G(k)  ζ_c - ζ_c^' F(k) =  ζ_c,c^'(_(k))  .




  (2) For all c∈,

ζ_c,c(𝕀_c)  = 0  .



  (3) For all h∈(c,c^')^0 and h^'∈(c^',c^'')^0,

ζ_c,c^''(h^' h) =  ζ_c^',c^''(h^')  F(h) + G(h^') ζ_c,c^'(h)  .





Note that in the special case where ζ_c,c^'=0, for all c,c^'∈, 
Definition <ref> reduces to the usual definition of a _R^[-1,0]-enriched (strict) 
natural transformation.



Let ζ,η : F⇒ G: → be two 
_R^[-1,0]-enriched pseudo-natural transformations.
A _R^[-1,0]-enriched modification Γ : ζ⇛η
consists of the following data:


  (i) For each object c∈, an element Γ_c ∈(F(c),G(c))^-1 
of degree -1 in the -object (F(c),G(c))∈_R^[-1,0].

These data have to satisfy the following properties:


  (1) For all c∈, 

ζ_c-η_c = _(Γ_c)  .



  (2) For all h∈(c,c^')^0,

Γ_c^'  F(h) + ζ_c,c^'(h) = η_c,c^'(h)+ G(h) Γ_c  .




In the following paragraphs we describe explicitly
the symmetric monoidal 3-category structure of _R^[-1,0],ps 
in the sense of Definitions <ref> and <ref>.
To avoid overloading this appendix, we shall only spell out the relevant data 
but not the verification of their properties,  which are mostly straightforward calculations.



  
The homomorphism 2-categories: Given two _R^[-1,0]-enriched
categories  and , we denote by _R^[-1,0],ps(,)∈Cat^ps
the strict 2-category whose objects are _R^[-1,0]-enriched functors,
1-morphisms are _R^[-1,0]-enriched pseudo-natural transformations and
2-morphisms are _R^[-1,0]-enriched modifications. The composition
of two 1-morphisms


    𝐂      𝐃[""name=0, anchor=center, inner sep=0, "G"description, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "H"',curve=height=32pt, from=1-1, to=1-3]
    	["ζ"', shorten <=5pt, shorten >=5pt, Rightarrow, from=1, to=0]
    	["η"', shorten <=5pt, shorten >=5pt, Rightarrow, from=0, to=2]
      ∘⟼  𝐂      𝐃[""name=1, anchor=center, inner sep=0, "F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "H"', curve=height=32pt, from=1-1, to=1-3]
    	["η∘ζ"', shorten <=5pt, shorten >=5pt, Rightarrow, from=1, to=2]

in the 2-category _R^[-1,0],ps(,) is defined by

(η∘ζ)_c    :=  η_c ζ_c  ,

(η∘ζ)_c,c^'(h)    :=  η_c,c^'(h) ζ_c + η_c^' ζ_c,c^'(h)  ,


for all c,c^'∈ and h∈(c,c^')^0. This composition
is strictly associative and strictly unital with respect to the identity 1-morphisms
_F : F⇒ F : → defined by (_F)_c = 𝕀_F(c) and
(_F)_c,c^'=0.


The horizontal composition of two 2-morphisms


    𝐂      𝐃[""name=0, anchor=center, inner sep=0, "G"description, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "H"', curve=height=36pt, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "F", curve=height=-36pt, from=1-1, to=1-3]
    	[""name=3, anchor=center, inner sep=0, " ζ^'", curve=height=-16pt, shorten <=5pt, shorten >=5pt, Rightarrow, from=2, to=0]
    	[""name=4, anchor=center, inner sep=0, "ζ "', curve=height=16pt, shorten <=5pt, shorten >=5pt, Rightarrow, from=2, to=0]
    	[""name=5, anchor=center, inner sep=0, "η "', curve=height=16pt, shorten <=5pt, shorten >=5pt, Rightarrow, from=0, to=1]
    	[""name=6, anchor=center, inner sep=0, " η^'", curve=height=-16pt, shorten <=5pt, shorten >=5pt, Rightarrow, from=0, to=1]
    	["Γ", shorten <=5pt, shorten >=5pt, triple, from=4, to=3]
    	["Θ", shorten <=5pt, shorten >=5pt, triple, from=5, to=6]
      ∘⟼  𝐂         𝐃[""name=1, anchor=center, inner sep=0, "H"', curve=height=36pt, from=1-1, to=1-4]
    	[""name=2, anchor=center, inner sep=0, "F", curve=height=-36pt, from=1-1, to=1-4]
    	[""name=3, anchor=center, inner sep=0, " η^'∘ζ^'", curve=height=-20pt, shorten <=5pt, shorten >=5pt, Rightarrow, from=2, to=1]
    	[""name=4, anchor=center, inner sep=0, "η∘ζ "', curve=height=20pt, shorten <=5pt, shorten >=5pt, Rightarrow, from=2, to=1]
    	["Θ∘Γ", shorten <=5pt, shorten >=5pt, triple, from=4, to=3]

in the 2-category _R^[-1,0],ps(,) is defined by

(Θ∘Γ)_c :=  Θ_c ζ_c + η^'_c Γ_c  ,


for all c∈. This composition is strictly associative and strictly unital
with respect to the identity 2-morphisms
__F : _F⇛_F : F⇒ F : → defined by (__F)_c = 0.


The vertical composition of two 2-morphisms


    𝐂         𝐃[""name=0, anchor=center, inner sep=0, "G"', curve=height=36pt, from=1-1, to=1-4]
    	[""name=1, anchor=center, inner sep=0, "F", curve=height=-36pt, from=1-1, to=1-4]
    	[""name=2, anchor=center, inner sep=0, " ζ^''", curve=height=-28pt, shorten <=8pt, shorten >=8pt, Rightarrow, from=1, to=0]
    	[""name=3, anchor=center, inner sep=0, "ζ "', curve=height=28pt, shorten <=8pt, shorten >=8pt, Rightarrow, from=1, to=0]
    	[""name=4, anchor=center, inner sep=0, "ζ^'"description, shorten <=6pt, shorten >=6pt, Rightarrow, from=1, to=0]
    	["Γ", shorten <=4pt, shorten >=4pt, triple, from=3, to=4]
    	["Γ^'", shorten <=4pt, shorten >=4pt, triple, from=4, to=2]
      ⟼  𝐂         𝐃[""name=0, anchor=center, inner sep=0, "G"', curve=height=36pt, from=1-1, to=1-4]
    	[""name=1, anchor=center, inner sep=0, "F", curve=height=-36pt, from=1-1, to=1-4]
    	[""name=2, anchor=center, inner sep=0, " ζ^''", curve=height=-28pt, shorten <=8pt, shorten >=8pt, Rightarrow, from=1, to=0]
    	[""name=3, anchor=center, inner sep=0, "ζ "', curve=height=28pt, shorten <=8pt, shorten >=8pt, Rightarrow, from=1, to=0]
    	["Γ^' Γ", shorten <=4pt, shorten >=4pt, triple, from=3, to=2]

in the 2-category _R^[-1,0],ps(,) is defined by

(Γ^' Γ)_c  :=  Γ^'_c + Γ_c  ,


for all c∈. This composition is strictly associative and strictly unital
with respect to the identity 2-morphisms
_ζ : ζ⇛ζ : F⇒ F : → defined by (_ζ)_c = 0.





  
The composition pseudo-functors and identities: Given three _R^[-1,0]-enriched
categories ,  and , we denote by 

∗ : _R^[-1,0],ps(,) ×_R^[-1,0],ps(,)  ⟶ 
_R^[-1,0],ps(,)

the composition pseudo-functor between the homomorphism 2-categories
from the previous paragraph.
On objects in the homomorphism 2-categories,
this pseudo-functor acts in terms of the usual composition of _R^[-1,0]-enriched functors,
which we will denote by

    𝐂   𝐃   𝐄["F", from=1-1, to=1-2]
    	["F^'", from=1-2, to=1-3]
      ∗⟼  𝐂   𝐄["F^'∗ F", from=1-1, to=1-2]
      .

On 1-morphisms in the homomorphism 2-categories, the pseudo-functor ∗ acts as


    𝐂      𝐃      𝐄[""name=0, anchor=center, inner sep=0, "F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "G"', curve=height=32pt, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "F^'", curve=height=-32pt, from=1-3, to=1-5]
    	[""name=3, anchor=center, inner sep=0, "G^'"', curve=height=32pt, from=1-3, to=1-5]
    	["ζ "', shorten <=6pt, shorten >=6pt, Rightarrow, from=0, to=1]
    	[" ζ^'"', shorten <=6pt, shorten >=6pt, Rightarrow, from=2, to=3]
      ∗⟼  𝐂      𝐄[""name=0, anchor=center, inner sep=0, "F^'∗ F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "G^'∗ G"', curve=height=32pt, from=1-1, to=1-3]
    	["ζ^'∗ζ "', shorten <=6pt, shorten >=6pt, Rightarrow, from=0, to=1]

with

(ζ^'∗ζ)_c    := ζ^'_G(c) F^'(ζ_c)  ,

(ζ^'∗ζ)_c,c^'(h)    := ζ^'_G(c),G(c^')(G(h)) F^'(ζ_c)
+ζ^'_G(c^') F^'(ζ_c,c^'(h))  ,


for all c,c^'∈ and h∈(c,c^')^0. 
On 2-morphisms in the homomorphism 2-categories, the pseudo-functor ∗ acts as


    𝐂      𝐃      𝐄[""name=0, anchor=center, inner sep=0, "F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "G"', curve=height=32pt, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "F^'", curve=height=-32pt, from=1-3, to=1-5]
    	[""name=3, anchor=center, inner sep=0, "G^'"', curve=height=32pt, from=1-3, to=1-5]
    	[""name=4, anchor=center, inner sep=0, "ζ "', curve=height=18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	[""name=5, anchor=center, inner sep=0, " η", curve=height=-18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	[""name=6, anchor=center, inner sep=0, " ζ^'"', curve=height=18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=2, to=3]
    	[""name=7, anchor=center, inner sep=0, " η^'", curve=height=-18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=2, to=3]
    	["Γ", shorten <=5pt, shorten >=5pt, triple, from=4, to=5]
    	["Γ^'", shorten <=5pt, shorten >=5pt, triple, from=6, to=7]
      ∗⟼  𝐂         𝐄[""name=0, anchor=center, inner sep=0, "F^'∗ F", curve=height=-32pt, from=1-1, to=1-4]
    	[""name=1, anchor=center, inner sep=0, "G^'∗ G"', curve=height=32pt, from=1-1, to=1-4]
    	[""name=4, anchor=center, inner sep=0, "ζ^'∗ζ "', curve=height=18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	[""name=5, anchor=center, inner sep=0, " η^'∗η", curve=height=-18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	["Γ^'∗Γ", shorten <=5pt, shorten >=5pt, triple, from=4, to=5]

with

(Γ^'∗Γ)_c := Γ^'_G(c) F^'(ζ_c) + η^'_G(c)  F^'(Γ_c)  ,


for all c∈.


The pseudo-functor coherences for the identities
are trivial, i.e. _F^'∗ F = _F^'∗_F,
but those for the compositions are not. Given any composable diagram of the form


    𝐂      𝐃      𝐄[""name=0, anchor=center, inner sep=0, "F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "G"description, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "H"', curve=height=32pt, from=1-1, to=1-3]
    	[""name=3, anchor=center, inner sep=0, "F^'", curve=height=-32pt, from=1-3, to=1-5]
    	[""name=4, anchor=center, inner sep=0, "H^'"', curve=height=32pt, from=1-3, to=1-5]
    	[""name=5, anchor=center, inner sep=0, "G^'"description, from=1-3, to=1-5]
    	["ζ"', shorten <=5pt, shorten >=5pt, Rightarrow, from=0, to=1]
    	["η"', shorten <=5pt, shorten >=5pt, Rightarrow, from=1, to=2]
    	["ζ^'"', shorten <=5pt, shorten >=5pt, Rightarrow, from=3, to=5]
    	["η^'"', shorten <=5pt, shorten >=5pt, Rightarrow, from=5, to=4]
      ,

the corresponding pseudo-functor coherence

∗_(η^',η),(ζ^',ζ)^ : (η^'∗η)∘(ζ^'∗ζ)      (η^'∘ζ^')∗(η∘ζ)

is given by the _R^[-1,0]-enriched modification
whose components are defined by

(∗_(η^',η),(ζ^',ζ)^)_c := η^'_H(c) ζ^'_G(c),H(c)(η_c) F^'(ζ_c)  ,


for all c∈.


The composition pseudo-functors ∗ are strictly associative and strictly unital
with respect to the identities 𝕀_∈_R^[-1,0],ps(,) 
given by the identity _R^[-1,0]-enriched functors 𝕀_:→.




  
The symmetric monoidal structure: The monoidal product

⊠ :  
_R^[-1,0],ps ×_R^[-1,0],ps
 ⟶ 
_R^[-1,0],ps

is given by the Cat^ps-enriched functor which assigns to a pair
of objects ,∈_R^[-1,0],ps the 
usual _R^[-1,0]-enriched tensor product category ⊠∈_R^[-1,0],ps
and to a pair of 1-morphisms F : →^' and G : →^'
in _R^[-1,0],ps the 
usual _R^[-1,0]-enriched tensor product functor 
F⊠ G : ⊠→^'⊠^'
from <cit.>.  


The monoidal product 
of a pair of 2-morphisms


    𝐂      𝐂^'[""name=0, anchor=center, inner sep=0, "F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "F^'"', curve=height=32pt, from=1-1, to=1-3]
    	["ζ "', shorten <=6pt, shorten >=6pt, Rightarrow, from=0, to=1]
      𝐃      𝐃^'[""name=0, anchor=center, inner sep=0, "G", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "G^'"', curve=height=32pt, from=1-1, to=1-3]
    	["η "', shorten <=6pt, shorten >=6pt, Rightarrow, from=0, to=1]
      ⊠⟼  𝐂⊠𝐃      𝐂^'⊠𝐃^'[""name=0, anchor=center, inner sep=0, "F⊠ G", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "F^'⊠ G^'"', curve=height=32pt, from=1-1, to=1-3]
    	["ζ⊠η "', shorten <=6pt, shorten >=6pt, Rightarrow, from=0, to=1]

in  _R^[-1,0],ps is defined by

(ζ⊠η)_(c,d)    := ζ_c⊗_Rη_d  ,

(ζ⊠η)_(c,d),(c^',d^')(h⊗_R k)    := 
ζ_c,c^'(h)⊗_R G^'(k) η_d + ζ_c^' F(h)⊗_Rη_d,d^'(k)  ,


for all c,c^'∈, d,d^'∈, h∈(c,c^')^0
and k∈(d,d^')^0. Note that for two _R^[-1,0]-enriched 
strictly natural transformations, i.e. ζ_c,c^'=0 and η_d,d^'=0
for all objects c,c^'∈ and d,d^'∈, the tensor
product ζ⊠η agrees with the usual one from enriched category theory.


The monoidal product of a pair of 3-morphisms


    𝐂      𝐂^'[""name=0, anchor=center, inner sep=0, "F", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "F^'"', curve=height=32pt, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "ζ "', curve=height=18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	[""name=3, anchor=center, inner sep=0, " ζ^'", curve=height=-18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	["Γ", shorten <=5pt, shorten >=5pt, triple, from=2, to=3]
      𝐃      𝐃^'[""name=0, anchor=center, inner sep=0, "G", curve=height=-32pt, from=1-1, to=1-3]
    	[""name=1, anchor=center, inner sep=0, "G^'"', curve=height=32pt, from=1-1, to=1-3]
    	[""name=2, anchor=center, inner sep=0, "η "', curve=height=18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	[""name=3, anchor=center, inner sep=0, " η^'", curve=height=-18pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	["Θ", shorten <=5pt, shorten >=5pt, triple, from=2, to=3]
      ⊠⟼  𝐂⊠𝐃         𝐂^'⊠𝐃^'[""name=0, anchor=center, inner sep=0, "F⊠ G", curve=height=-32pt, from=1-1, to=1-4]
    	[""name=1, anchor=center, inner sep=0, "F^'⊠ G^'"', curve=height=32pt, from=1-1, to=1-4]
    	[""name=2, anchor=center, inner sep=0, "ζ⊠η "', curve=height=22pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	[""name=3, anchor=center, inner sep=0, " ζ^'⊠η^'", curve=height=-22pt, shorten <=7pt, shorten >=7pt, Rightarrow, from=0, to=1]
    	["Γ⊠Θ", shorten <=5pt, shorten >=5pt, triple, from=2, to=3]

in  _R^[-1,0],ps is defined by

(Γ⊠Θ)_(c,d) := Γ_c⊗_R η_d + ζ^'_c ⊗_RΘ_d  ,


for all c∈ and d∈.


The monoidal unit u :ℐ→_R^[-1,0],ps
is the _R^[-1,0]-enriched category

𝐑 ∈   _R^[-1,0],ps

consisting of a single object, say ⋆, with morphisms
𝐑(⋆,⋆):= R∈_R^[-1,0] given by
the monoidal unit of the enrichment category.


The associator and unitors for this monoidal structure 
are canonically given by the ones of the enrichment 
symmetric monoidal category _R^[-1,0].
The component at ,∈_R^[-1,0],ps of the symmetric braiding 


τ_, : ⊠ ⟶ ⊠
is the _R^[-1,0]-enriched functor which sends objects
(c,d)∈⊠ to (d,c)∈⊠
and acts on morphisms via the symmetric braiding of the enrichment 
symmetric monoidal category _R^[-1,0]


@C=5em@R=1.5em
(⊠)((c,d),(c^',d^')) @=[d]@–>[r]     @=[d]
(⊠)((d,c),(d^',c^'))

(c,c^')⊗_R(d,d^') [r]_-γ_(c,c^'),(d,d^')
     (d,d^')⊗_R(c,c^')

  .


Note that the symmetric braiding satisfies the strict hexagon identities, i.e.

τ_⊠,    = (τ_,⊠𝕀_)∗(𝕀_⊠τ_,)  ,

τ_,⊠    = (𝕀_⊠τ_,)∗(τ_,⊠𝕀_)  ,


for all ,,∈_R^[-1,0],ps.
Moreover, the symmetric braiding τ is a Cat^ps-enriched natural 
transformation, which in particular implies that it is strictly natural with respect to 
_R^[-1,0]-enriched functors F : →^', i.e.

(𝕀_⊠F)∗τ_,=τ_^',∗(F⊠𝕀_)  ,

for all ∈_R^[-1,0],ps.


To conclude this subsection, we provide a convenient
definition of braided monoidal category objects
in the symmetric monoidal 3-category _R^[-1,0],ps.
Such objects can be defined with various levels of weakness since
our ambient category _R^[-1,0],ps is a 3-category.
For the purpose of our paper, it suffices to consider 
the following semi-strict variant.

A (semi-strict) braided monoidal category object in _R^[-1,0],ps 
is a tuple (,⊗,1,α,λ,ρ,γ)
consisting of the following data:


  (i) a _R^[-1,0]-enriched category ,


  (ii) a _R^[-1,0]-enriched functor ⊗ : ⊠→ (called monoidal product),


  (iii) a _R^[-1,0]-enriched functor 1 : 𝐑→ (called monoidal unit),


  (iv) invertible _R^[-1,0]-enriched pseudo-natural transformations 
α : ⊗∗ (⊗⊠𝕀_) ⇒⊗∗ (𝕀_⊠⊗)
(called associator), λ : ⊗∗(1⊠𝕀_) ⇒𝕀_ (called left unitor)
and ρ: ⊗∗(𝕀_⊠ 1)⇒𝕀_ (called right unitor),


  (v) an invertible _R^[-1,0]-enriched pseudo-natural transformation
γ : ⊗⇒⊗∗τ_, (called braiding).

These data have to satisfy the usual coherence conditions for braided monoidal categories
internal to the 3-category _R^[-1,0],ps strictly,
i.e. the pentagon, triangle and hexagon diagrams commute strictly 
up to equality of _R^[-1,0]-enriched pseudo-natural transformations.


A (semi-strict) symmetric monoidal category object in _R^[-1,0],ps 
is a braided monoidal category object (,⊗,1,α,λ,ρ,γ)
with the property that the braiding γ squares strictly to the identity,
i.e. the ∘-composition of

@C=3.5em
⊗@=>[r]^-γ      ⊗∗τ_, @=>[r]^-γ∗_τ_,
     ⊗∗τ_,∗τ_,  =  ⊗

is equal to the identity _R^[-1,0]-enriched pseudo-natural transformation _⊗.





 §.§ Homotopy 2-categories of symmetric monoidal dg-categories


Let R be a commutative -algebra and denote by _R the 2-category
of _R-enriched categories (dg-categories), _R-enriched functors (dg-functors)
and _R-enriched strict natural transformations (dg-natural transformations).
We shall regard _R as a 3-category (with trivial 3-morphisms) 
in the sense of Definition <ref>.
The lax symmetric monoidal truncation functor  τ^[-1,0]: _R→_R^[-1,0]
from Lemma <ref> induces a change of base 3-functor

𝖧𝗈_2 :  _R ⟶ _R^[-1,0] ⊆ _R^[-1,0],ps  ,

which sends dg-natural transformations to strict
_R^[-1,0]-enriched pseudo-natural transformations.
Observe that, at the level of objects, this 3-functor assigns to a dg-category ∈_R, 
with objects _0 and morphism cochain complexes (c,c^')∈_R, 
its homotopy 2-category 𝖧𝗈_2()∈_R^[-1,0],ps, 
i.e. the _R^[-1,0]-enriched category with the same objects _0 and morphisms
given by the truncated cochain complexes τ^[-1,0]((c,c^'))∈_R^[-1,0].


Since the enrichment category _R of cochain complexes of R-modules is symmetric monoidal
with respect to the relative tensor product ⊗_R, there exists an induced
symmetric monoidal structure on _R, see e.g. <cit.>.
The 3-functor (<ref>) assigning homotopy 2-categories
is lax symmetric monoidal with the lax structure 
𝖧𝗈_2()⊠𝖧𝗈_2() →𝖧𝗈_2(⊠)
induced from the lax structure of the good truncation functor in
Lemma <ref>, for all ,∈_R.
For later use, we record the following direct consequence of this result.

Let (,⊗,1,γ) be a braided (or symmetric) monoidal dg-category.
Then the homotopy 2-category 𝖧𝗈_2()∈_R^[-1,0],ps
is canonically a braided (or symmetric) monoidal category object in
the symmetric monoidal 3-category _R^[-1,0],ps from Proposition <ref>.



Note that the braided (or symmetric) monoidal category objects in _R^[-1,0],ps 
which are obtained from this construction always have strictly natural
associators, unitors and braidings. Our symmetric monoidal 
3-category _R^[-1,0],ps from Proposition <ref>
is designed to provide more flexibility through the concept 
of enriched pseudo-natural transformations, see also Definition 
<ref>, which will be relevant 
for our deformation constructions in Section <ref>.







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 Paper Title: Galois theory of differential schemes.




[2020]12H05, 18F20, 18D40, 18E50, 14L30








     











 
Behrang Noohi

School of Mathematical Sciences

  	Queen Mary University of London

         London, E1 4NS

        United Kingdom
b.noohi@qmul.ac.uk

 
Ivan Tomašić

         School of Mathematical Sciences

  	Queen Mary University of London

         London, E1 4NS

        United Kingdom
i.tomasic@qmul.ac.uk
Supported by EPSRC Standard grant EP/V028812/1



§ ABSTRACT 
Since 1883, Picard-Vessiot theory had been developed as the Galois theory of differential field extensions associated to linear differential equations. Inspired by categorical Galois theory of Janelidze, and by using novel methods of precategorical descent applied to algebraic-geometric situations, we develop a Galois theory that applies to morphisms of differential schemes, and vastly generalises the linear Picard-Vessiot theory, as well as the strongly normal theory of Kolchin. 


Galois theory of differential schemes
    Ivan Tomašić
    August 5, 2024
=====================================























§ INTRODUCTION




 §.§ History

Clasically, a Picard-Vessiot extension (L,δ_L)/(K,δ_K) of differential fields yields a linear algebraic group

    G=^PV(L/K)

over the common field of constants k=(L,δ_L)=(K,δ_K) such that there is a Galois correspondence between the intermediate differential field extensions and Zariski closed subgroups of G, and

    G(k)≃((L,δ_L)/(K,δ_K)),

as explained in <ref>.

The Picard-Vessiot ring is a (K,δ_K)-algebra 
    (A,δ_A)
 with fraction field (L,δ_L), such that (X,δ_X)=(A,δ_A) is a G-torsor over (Y,δ_Y)=(K,δ_K) in the sense that

    (X,δ_X)×_(Y,δ_Y)(X,δ_X)≃ (X,δ_X)×_(X_0,0)(G,0),

where we consider X_0=(k) and G as differential schemes endowed with zero derivations. 
 
Janelidze realised that Picard-Vessiot theory fits into the framework of his categorical Galois theory through the adjunction

    [cross line/.style=preaction=draw=white, -,
    line width=3pt]
    (m)[matrix of math nodes, minimum size=1.7em,
    inner sep=0pt, 
    row sep=3.3em, column sep=1em, text height=1.5ex, text depth=0.25ex]
      
      |(dc)|
    
     |(c)|
    ;
    [->,font=,>=to, thin]
    
    
    
    (dc) edge [bend right=30] node (ss) [left]S (c)
    (c) edge [bend right=30] node (ps) [right]C (dc)
    
    (ss) edge[draw=none] node⊣ (ps)
    ;

between the categories of affine differential schemes and affine schemes, where 

    S((R,δ_R))=((R,δ_R)),       C((R))=(R,0).

He emphasised in <cit.> and <cit.> that the key object, the morphism of relative Galois descent, is in fact the torsor 

    f:(X,δ_X)→ (Y,δ_Y),

and not the extension of differential fields, and that the Picard-Vessiot Galois group agrees with the categorical Galois group,

    ^PV(L/K)≃[f]=S(X×_YX),

as explained in <ref>.

During the visit of the second author to Tsukuba in 2023, Akira Masuoka asked whether categorical Galois theory approach automatically recovers the classical Picard-Vessiot Galois correspondence. To our surprise, we observed that Carboni-Janelidze-Magid correspondence from <cit.> in the affine context only establishes a correspondence between split affine quotients of X over Y and effective subgroups of G=[f], i.e., those closed subgroups H such that G/H is represented by an affine scheme, see <ref>.


We realised that, contrary to the popular belief, Picard-Vessiot theory is not an entirely affine affair, and that the categorical Picard-Vessiot theory must be extended to the correspondence between the split quasi-projective fpqc quotients of (X,δ_X) over (Y,δ_Y), and closed subgroups of G, and that the intermediate differential fields from the classical correspondence appear as function fields of those quasi-projective quotients. 

On the other hand, the Galois theory of a strongly normal differential field extension (L,δ_L)/(K,δ_K) of Kolchin <cit.> is known to give rise to a general algebraic group G over k and a general differential scheme (X,δ_X) that acts as a torsor for G through a relation analogous to the above, <cit.>, <cit.>, <cit.>. 

Michael Wibmer informed us of the idea of a number of researchers in differential algebra, including Jerry Kovacic, to interpret the strongly normal theory using  categorical Galois theory, and this was independently posed as a desirable project by Janelidze in <cit.>. The difficulty lies in the fact that, with no reasonable notion of differential scheme, does the natural extension of the functor C above to a functor between schemes and differential schemes

    [cross line/.style=preaction=draw=white, -,
    line width=3pt]
    (m)[matrix of math nodes, minimum size=1.7em,
    inner sep=0pt, 
    row sep=3.3em, column sep=1em, text height=1.5ex, text depth=0.25ex]
      
      |(dc)|
    
     |(c)|
    ;
    [->,font=,>=to, thin]
    
    
    
    
    (c) edge [bend right=30] node (ps) [right]C (dc)
    
    
    ;

admit a left adjoint, so the classical categorical Galois theory cannot be invoked. 

We resolve the apparent conundrum by constructing a very partial left adjoint using Bardavid's idea of categorical scheme of leaves <cit.>, and by using an indexed version of categorical Galois theory from <cit.>,  
eventually proving Theorem <ref> that simultaneously explains the quasi-projective aspect of Picard-Vessiot theory and applies to strongly normal theory. 

We proceed much further and develop a Galois theory of arbitrary differential scheme morphisms, as explained below. 





 §.§ Differential algebraic geometry


In this paper, a differential scheme 

    (X,δ_X)=(X,(_X,δ_X))

is a scheme (X,_X) endowed with a derivation δ_X on its structure sheaf _X, i.e., with a vector field.

Given a scheme S, we define a category of S-differential schemes 

    _S

consisting of schemes over S endowed by S-derivations. 

The spectrum of a differential ring has a natural structure of a differential scheme, affording a right adjoint

    :^→

to the global sections functor. A differential scheme is affine, if it is isomorphic to a spectrum of a differential ring, and the category of affine differential schemes is anti-equivalent to the category of differential rings, i.e., 

    ≃^,

which is consistent with our discussion of affine differential schemes above. 

A natural functor

    C:→,      (X,_X)↦ (X,(_X,0))

allows us to consider a scheme as a differential scheme with a trivial derivation/zero vector field, but it does not have a left adjoint. 
Based on the ideas of Bardavid <cit.>, we define a categorical scheme of leaves 
    π_0(X)
 of a differential scheme (X,δ_X) to be the morphism 

    (X,δ_X)→ C(π_0(X))

which is universal from (X,δ_X) to C. It exists only for rare differential schemes (X,δ_X), but this construction provides a partial left adjoint π_0 to C when it does. 





 §.§ Differential schemes as precategory actions and descent


We make a key novel observation that S-differential schemes can be viewed as actions of an internal precategory 

    (S)

in _ S associated to infinitesimal augmentations of S that we can symbolically write

    (m)[matrix of math nodes, row sep=0em, column sep=1em, text height=1.5ex, text depth=0.25ex]
     
    |(2)|S[ϵ_0,ϵ_1]/(ϵ_0^2,ϵ_0ϵ_1,ϵ_1^2)    [1em] |(1)|S[ϵ]/(ϵ^2)   [1em] |(0)|S
    ; 
    [->,font=,>=to, thin]
    ([yshift=.4em]2.east) edge node[above=-2pt] ([yshift=.4em]1.west) 
    (2) edge node[above=-2pt] (1)
    ([yshift=-.4em]2.east) edge node[above=-2pt] ([yshift=-.4em]1.west) 
    ([yshift=.4em]1.east) edge node[above=-2pt] ([yshift=.4em]0.west) 
    (0)  edge node[above=-2pt] (1) 
    ([yshift=-.4em]1.east) edge node[above=-2pt] ([yshift=-.4em]0.west) 
    
     
    ;

i.e., that there is an equivalence of categories (<ref>)

    _S≃ (_ S)^(S).

It was previously known that differential schemes can be viewed as `actions of a pointed set' associated to the augmentation structure S[ϵ^2]/(ϵ^2)→ S, but the fact that we can view them as precategory actions gives us a very direct access to a theory of descent, given that precategory actions are essentially generalised descent data. More generally, we can translate problems from differential algebraic geometry into the known realm of algebraic geometry. 

In algebraic geometry, descent usually works only for very specific indexed data on schemes, so we work with a chosen pseudofunctor

    :(_ S)^→,

and extend it in a natural way, using precategory actions, to a pseudofunctor on differential schemes

    δ:_S^→.

In the special case when (V) is a certain class of scheme morphisms with target V, the category δ(X,δ_X) consists of differential scheme morphisms with target (X,δ_X) whose underlying scheme morphism belongs to the class . 

If =(X_2,X_1,X_0) is a precategory in _ S that corresponds to a differential scheme (X,δ_0) viewed as an (S)-action, then the categorical scheme of leaves of (X,δ_X) agrees with the scheme of connected components of ,

    [cramped, column sep=normal, ampersand replacement=&]
    π_0(X)≃π_0()=(X_1 .[yshift=2pt]rd_0[yshift=-2pt]r[swap]d_1&.X_0)

whenever either of them exist in _ S. 

Moreover, (X,δ_X) is simple with respect to , provided the above coequaliser exists and is universal for . 
This condition ensures that the precategory morphism

    η_X:→π_0()

to the dicrete precategory associated to the scheme π_0(X)
is of precategorical descent, i.e., the induced functor of precategory actions

    C_X:(π_0(X))≃^π_0()η_^*⟶^≃δ(X,δ_X)

is fully faithful. 

By developing the theory of descent in the category of differential schemes as precategory actions, we establish in <ref> that a morphism

    f:(X,δ_X)→ (Y,δ_Y)

is of effective descent for δ, if the associated morphism (f_0,f_1,f_2) of precategory actions satisfies that the underlying scheme morphism f_0:X→ Y is of effective descent for , and f_1 is descent for . 

Consequently, such an f is of effective descent for the fibration of polarised quasi-projective differential morphisms if f_0 is faithfully flat quasi-compact (fpqc). Moreover, if the target of f is a spectrum of a differential field, then it is of effective descent for the class of quasi-projective differential morphisms. 



 §.§ Galois theory of differential schemes


A good understanding of the two types of descent provides all the ingredients needed for a Picard-Vessiot style categorical Galois theory of differential schemes. The framework for the indexed version of categorical Galois theory involves the following choices:


  * let S be a scheme, and 
    =_ S
 be the category of schemes over S;

  * let  be the full subcategory of _S of those S-differential schemes that have a categorical scheme of leaves, hence we obtain a functor

    π_0:→;


  * a pseudo-functor 
    :^→,
 determining a pseudofunctor δ:^→;

  * let 
    C=C^:∘π_0→δ
 be the pseudo-natural transformation whose component at (X,δ)∈ is the functor C_X discussed above.


We define the category 

    _C(f)

as the full subcategory of objects P∈δ(Y) such that 
f^*P≃ C_X(Q) for some Q∈(π_0(X)).

 
 
 We define a morphism f:(X,δ_X)→ (Y,δ_Y) in  to be pre-Picard-Vessiot with respect to , if
 
 
  (i) f is a morphism of effective descent for δ;
 
  (ii) X, X×_YX, X×_YX×_YX are simple for .
 
 If we have 
 
 
  (5) a suitable (in the sense of <ref>) pseudo-natural transformation 
 
    U:⇒
 to a fibration  associated with a class of scheme morphisms,
 
 
 then we define f to be Picard-Vessiot with respect to U if, in addition to the first, it satisfies the strengthening of  the second condition, 
  
 
  (ii') X is simple for  and f is auto-split for  in the sense that f∈_C^(f). 
 
 

 
 The assumption that f is pre-Picard-Vessiot for  ensures that the kernel-pair groupoid 

    (m)[matrix of math nodes, row sep=0em, column sep=1em, text height=1.5ex, text depth=0.25ex]
     
    |(n)|_f:   [1em] |(2)|X×_Y X×_Y X    [1em] |(1)|X×_Y X   [1em] |(0)|X
    ; 
    [->,font=,>=to, thin]
    ([yshift=.4em]2.east) edge node[above=-2pt] ([yshift=.4em]1.west) 
    (2) edge node[above=-2pt] (1)
    ([yshift=-.4em]2.east) edge node[above=-2pt] ([yshift=-.4em]1.west) 
    ([yshift=.4em]1.east) edge node[above=-2pt] ([yshift=.4em]0.west) 
    (0)  edge node[above=-2pt] (1) 
    ([yshift=-.4em]1.east) edge node[above=-2pt] ([yshift=-.4em]0.west) 
    
     
    ;

is a category in , and that we can define the Galois precategory

    [f]=π_0(_f)

as a precategory in . Moreover, if f is Picard-Vessiot for U, then it is also pre-Picard-Vessiot for  and G[f] happens to be an internal groupoid in .



[Galois theorem for differential schemes, <ref>]  
A pre-Picard-Vessiot morphism f for  induces an equivalence 

    _C(f)≃^[f]

between the category of objects of δ(Y) that are C-split by f and the category of -actions of the precategory [f].

If f is Picard-Vessiot for U, 
the latter becomes the category of -actions of the groupoid [f]. 


From our perspective, 


Galois theory is all about descent.

More precisely, condition (i) from the definition of pre-Picard-Vessiot morphisms ensures effective classical descent for precategory actions, while condition (ii) ensures the new type of precategorical descent where needed.  
 
 

 §.§ Applications

 
 Using the above template Galois theorem, we prove the following result, which simultaneously refines classical Picard-Vessiot theory and strongly normal theory of differential field extensions. 
 
[Quasi-projective differential Galois correspondence <ref>]
Let (K,δ) be a differential field of characteristic 0 with the field of constants k, let S=(k) and assume

    f:(X,δ)→ (Y,δ)=(K,δ)

is a quasi-projective morphism of S-differential schemes such that X is integral and its only leaf is its generic point and f is auto-split.  Then there is a Galois correspondence between split S-differential quasi-projective fpqc quotients of f in  and closed subgroups of the S-algebraic group [f], which takes

    [cross line/.style=preaction=draw=white, -,
    line width=4pt]
    (m)[matrix of math nodes, row sep=.9em, column sep=.5em, text height=1.5ex, text depth=0.25ex]
        |(ij)| X   
       [.02em]
    	|(i)|P      
      [.02em]
    			    |(0)|Y   
    ;
    [->,font=,>=to, thin]
    (ij) edge node[left,pos=0.2]
        (i) edge node[right,pos=0.5]f (0) 
        (i) edge node[below left,pos=0.5]p (0) 
        ;
     
    to 
        [X→ P].
    
    Conversely, a closed subgroup G' of [f] corresponds to the quotient
    
        X/G'.
    
     
    As far as we are aware, we are able to formulate the first result in differential Galois theory which works over differential schemes with arbitrary categorical  scheme of constants.
    
    [Polarised quasi-projective differential Galois theory, <ref>]
    
    Let f:(X,δ_X)→ (Y,δ_Y) be a morphism of S-differential schemes such that 
    
      *  the underlying scheme morphism X→ Y is fpqc;
    
      * (X,δ_X) is simple with respect to S-scheme morphisms, with scheme of leaves G_0;
    
      *  there is an S-morphism G_1→ G_0 such that
    
        (X,δ_X)×_(Y,δ_Y)(X,δ_X)≃ (X,δ_X)×_(G_0,0)(G_1,0).
    
    Then f is Picard-Vessiot for the forgetful functor from polarised morphisms to scheme morphisms, [f] is the groupoid (G_1 G_0) and we have an equivalence between the category of quasi-projective polarised S-differential morphisms (P,_P)→ Y split by f and the category of quasi-projective polarised actions (Q,_Q)→ G_0 of [f].
    
    
     §.§ Layout of the paper
    
    
    A reader interested only in differential Galois theory can start perusing the paper from Section <ref>, and occasionally look up the prerequisites from previous sections.  There, we develop differential algebraic geometry, and discuss the numerous benefits of our original approach to differential schemes as precategory actions, including the consideration of the categorical schemes of leaves as connected components of precategories, simplicity of differential schemes through universality of connected components, polarised differential schemes. One of the most important topics we develop is the theory of descent for differential scheme morphisms.
    
    In Section <ref>, we explore the extent to which the classical categorical Galois theory of Janelidze explains the affine Picard-Vessiot Galois correspondence. 
    
    In Section <ref>, we reap the benefits of the flexibility that the indexed version of categorical Galois theory brings to differential algebraic geometry, and formulate a very general template for the Galois theory of differential schemes. We apply this theory in concrete settings of quasi-projective differential Galois correspondence that works over a base differential field, and polarised quasi-projective differential Galois theory which works over arbitrary differential schemes. 
    
    On the other hand, we must emphasise that previous sections contain a myriad of results that may be of independent interest in category theory, especially in view of connections to recent work related to descent theory 
    <cit.>
    <cit.>, 
    <cit.>
    <cit.>
    that we do not fully understand yet. 
    
    In Section <ref>, we discuss precategories and their actions, which, viewed as generalised descent data, give rise to a new form of precategorical descent, and find sufficient conditions for effective descent. We develop a whole calculus of pullbacks of descent data, and apply it to obtain a result on descent of quasi-projective morphisms. We also discuss classical descent of precategory actions that gets applied in the differential context later.
    
    In Section <ref>, we recall the foundations of Janelidze's categorical Galois theory, its indexed form from his book with Borceux, and we expand the minute details of the Carboni-Magid-Janelidze categorical Galois theory that are implicit in the original paper <cit.>. 
    
    This paper would was largely motivated by numerous discussions with George Janelidze, Andy Magid, Akira Masuoka,  Tom Scanlon and Michael Wibmer, so we thank them all for sharing their time and knowledge with us. 
    
    
    
    § DESCENT
    
    
    Throughout this appendix, let  denote a category with pullbacks, and let
    
        →
    
    be a fibred category equipped with a cleavage. Equivalently, we have an indexed category associated to a pseudofunctor
    
        P:^→
    
    where, for an object U in , the fibre of  over U is the U-indexed component of P,
    
        (U)=P(U),
    
    and, for a morphism Uf→V in , we have a pullback functor
    
        f^*=P(f):(V)→(U).
    
    
     §.§ Precategories and their actions
    
    
    Let Δ_2 be the diagram category
    
        (m)[matrix of math nodes, row sep=0em, column sep=3em, text height=1.5ex, text depth=0.25ex]
         
        |(2)|∙    [1em] |(1)|∙   [1em] |(0)|∙
        ; 
          at (2.90) 2;
          at (1.90) 1;
          at (0.90) 0;
        [->,font=,>=to, thin]
        ([yshift=1em]2.east) edge node[above=-2pt]r_0 ([yshift=1em]1.west) 
        (2) edge node[above=-2pt]m (1)
        ([yshift=-1em]2.east) edge node[above=-2pt]r_1 ([yshift=-1em]1.west) 
        ([yshift=1em]1.east) edge node[above=-2pt]d_0 ([yshift=1em]0.west) 
        (0)  edge node[above=-2pt]n (1) 
        ([yshift=-1em]1.east) edge node[above=-2pt]d_1 ([yshift=-1em]0.west) 
        
         
        ;
    
    with
    
        d_0 r_1    =d_1 r_0, 	    d_0 n    = 𝕀_0, 
        
        d_0 m    = d_0 r_0,	    d_1 n    =𝕀_0, 
        
        d_1 m    = d_1 r_1.
    
    
    A precategory in a category  is a functor
    
        :Δ_2→.
    
    Equivalently, it is a diagram 
    
        (m)[matrix of math nodes, row sep=0em, column sep=3em, text height=1.5ex, text depth=0.25ex]
         
        |(2)|C_2    [1em] |(1)|C_1   [1em] |(0)|C_0
        ; 
        [->,font=,>=to, thin]
        ([yshift=1em]2.east) edge node[above=-2pt]r_0 ([yshift=1em]1.west) 
        (2) edge node[above=-2pt]m (1)
        ([yshift=-1em]2.east) edge node[above=-2pt]r_1 ([yshift=-1em]1.west) 
        ([yshift=1em]1.east) edge node[above=-2pt]d_0 ([yshift=1em]0.west) 
        (0)  edge node[above=-2pt]n (1) 
        ([yshift=-1em]1.east) edge node[above=-2pt]d_1 ([yshift=-1em]0.west) 
        
         
        ;
    
    in , where the morphisms satisfy the relations indicated above. 
    
    The category of precategories in  is the functor category
    
        ()=[Δ_2,].
    
    The discrete precategory associated to an object U of  is the diagram
    
        (m)[matrix of math nodes, row sep=0em, column sep=3em, text height=1.5ex, text depth=0.25ex]
         
        |(n)|U:   [1em] |(2)|U    [1em] |(1)|U   [1em] |(0)|U
        ; 
        [->,font=,>=to, thin]
        ([yshift=.4em]2.east) edge node[above=-2pt] ([yshift=.4em]1.west) 
        (2) edge node[above=-2pt] (1)
        ([yshift=-.4em]2.east) edge node[above=-2pt] ([yshift=-.4em]1.west) 
        ([yshift=.4em]1.east) edge node[above=-2pt] ([yshift=.4em]0.west) 
        (0)  edge node[above=-2pt] (1) 
        ([yshift=-.4em]1.east) edge node[above=-2pt] ([yshift=-.4em]0.west) 
        
         
        ;
    
    where all the morphisms are identities on U. 
    
    The precategory associated to the kernel pair of a morphism u:U'→ U in a category  admitting pullbacks is the diagram
    
        (m)[matrix of math nodes, row sep=0em, column sep=3em, text height=1.5ex, text depth=0.25ex]
         
        |(n)|_u:   [1em] |(2)|U'×_U U'×_U U'    [1em] |(1)|U'×_UU'   [1em] |(0)|U'
        ; 
        [->,font=,>=to, thin]
        ([yshift=1em]2.east) edge node[above=-2pt]π_01 ([yshift=1em]1.west) 
        (2) edge node[above=-2pt]π_02 (1)
        ([yshift=-1em]2.east) edge node[above=-2pt]π_12 ([yshift=-1em]1.west) 
        ([yshift=1em]1.east) edge node[above=-2pt]π_0 ([yshift=1em]0.west) 
        (0)  edge node[above=-2pt]Δ (1) 
        ([yshift=-1em]1.east) edge node[above=-2pt]π_1 ([yshift=-1em]0.west) 
        
         
        ;
    
    which happens to be a groupoid without the inversion of morphisms named. Note that, using the notation from <ref>,
    
        U=_𝕀_U.
    
    For ∈(), the category of -actions in  is the bilimit
    
        ^=lim(∘)
    
    of the diagram of categories and functors
    
        (m)[matrix of math nodes, row sep=0em, column sep=3em, text height=1.5ex, text depth=0.25ex]
         
        |(2)|(C_2)    [1em] |(1)|(C_1)   [1em] |(0)|(C_0).
        ; 
        [->,font=,>=to, thin]
        ([yshift=1em]1.west) edge node[above=-2pt]r_0^* ([yshift=1em]2.east) 
        (1) edge node[above=-2pt]m^* (2)
        ([yshift=-1em]1.west) edge node[above=-2pt]r_1^* ([yshift=-1em]2.east) 
        
         ([yshift=1em]0.west) edge node[above=-2pt]d_0^* ([yshift=1em]1.east) 
        (1)  edge node[above=-2pt]n^* (0) 
        ([yshift=-1em]0.west) edge node[above=-2pt]d_1^* ([yshift=-1em]1.east) 
        
         
        ;
    
    More explicitly, an action P∈^ consists of objects
    
        P_2   ∈(C_2),     P_1   ∈(C_1),     P_0   ∈(C_0),
    
    and isomorphisms
    
        P_2    ρ_0→ r_0^* P_1,     P_1   δ_0→d_0^*P_0,     P_0η→n^*P_1,
        
        P_2    μ→ m^* P_1,     P_1   δ_1→d_1^*P_0,    
        
        P_2    ρ_1→ r_1^* P_1,
    
    such that the diagram
    
        [cross line/.style=preaction=draw=white, -,
        line width=4pt]
        (m)[matrix of math nodes, row sep=1.5em, column sep=1.7em, text height=1.5ex, text depth=0.25ex]
        	|(i)|n^*d_0^*P_0   	|(c)|n^*P_1    |(k)|n^*d_1^*P_0
            |(j)|P_0   
        ;
        [->,font=,>=to, thin]
        (c) edge node[above,pos=0.5]n^*δ_0 (i) edge node[above,pos=0.5]n^*δ_1 (k) 
        (j) edge node[above=-2pt, sloped]∼ (i) edge node[right,pos=0.4]η (c) edge node[above=-2pt, sloped]∼ (k) 
        ;
     
    commutes in (C_0), and the diagram
    
        [cross line/.style=preaction=draw=white, -,
        line width=4pt]
        (m)[matrix of math nodes, row sep=1.2em, column sep=.8em, text height=1.5ex, text depth=0.25ex]
               |(m0)|m^*d_0^*P_0   [1em] |(mm)|m^*P_1   [1em] |(m1)|m^*d_1^*P_0      
           [2em]
        |(r0d0)|r_0^*d_0^*P_0          |(p2)|P_2         |(r1d1)|r_1^*d_1^*P_0
          [.2em]
        					   	|(r0)|r_0^*P_1             |(r1)|r_1^*P_1   
          [.02em]
        					       |(r0d1)|r_0^*d_1^*P_0        |(r1d0)|r_1^*d_0^*P_0      
        ;
        [->,font=,>=to, thin]
        (r0) edge node[below left,pos=0.5]r_0^*δ_0 (r0d0) edge node[below left,pos=0.5]r_0^*δ_1 (r0d1)
        (mm) edge node[above,pos=0.5]m^*δ_0 (m0) edge node[above, pos=0.5]m^*δ_1 (m1) 
        (r1) edge node[below right,pos=0.5]r_1^*δ_0 (r1d0) edge node[below right,pos=0.5]r_1^*δ_1 (r1d1)
        (p2) edge node[right]μ (mm) edge node[above]ρ_0 (r0) edge node[above]ρ_1 (r1) 
        (r0d0) edge node[above=-2pt, sloped]∼  (m0) 
        (m1) edge node[above=-2pt, sloped]∼  (r1d1) 
        (r0d1) edge node[above=-2pt, sloped]∼  (r1d0) 
        ;
     
    commutes in (C_2), where the unnamed arrows are coherence isomorphisms. 
    
    The category of -actions in  is  
    
        ^= Self()^,
    
    where Self()(C)=_ C is the self-indexing of  by slicing. 
    
    By considering the composite
    
        [cross line/.style=preaction=draw=white, -,
        line width=4pt]
        (m)[matrix of math nodes, row sep=1.5em, column sep=1.7em, text height=1.5ex, text depth=0.25ex]
        	|(i)|d_0^*P_0   	|(c)|P_1    |(k)|d_1^*P_0
        ;
        [->,font=,>=to, thin]
        (c) edge node[above,pos=0.3]δ_0 (i) edge node[above,pos=0.3]δ_1 (k) 
        (i) edge[bend left=30] node[above,pos=0.5]φ (k)
        ;
     
    we see that an action P is equivalently given by an object P_0∈(C_0) and an isomorphism φ:d_0^*P_0→ d_1^*P_0 in (C_1) such that the diagram
    
        [cross line/.style=preaction=draw=white, -,
        line width=4pt]
        (m)[matrix of math nodes, row sep=1.5em, column sep=1.7em, text height=1.5ex, text depth=0.25ex]
        	|(i)|n^*d_0^*P_0       |(k)|n^*d_1^*P_0
            |(j)|P_0   
        ;
        [->,font=,>=to, thin]
        (i) edge node[above,pos=0.5]n^*α (k) 
        (j) edge node[above=-2pt, sloped]∼ (i)  edge node[above=-2pt, sloped]∼ (k) 
        ;
     
    commutes in (C_0), and the cocycle condition holds, i.e., the diagram
    
        [cross line/.style=preaction=draw=white, -,
        line width=4pt]
        (m)[matrix of math nodes, row sep=1.2em, column sep=.8em, text height=1.5ex, text depth=0.25ex]
               |(m0)|m^*d_0^*P_0   [1em] 	   [1em] |(m1)|m^*d_1^*P_0      
           [2em]
        |(r0d0)|r_0^*d_0^*P_0                  |(r1d1)|r_1^*d_1^*P_0
          [.2em]
        					                  
          [.02em]
        					       |(r0d1)|r_0^*d_1^*P_0        |(r1d0)|r_1^*d_0^*P_0      
        ;
        [->,font=,>=to, thin]
         (r0d0) edge node[below left,pos=0.5]r_0^*φ (r0d1)
         (m0) edge node[above, pos=0.5]m^*φ (m1) 
        (r1d0) edge node[below right,pos=0.5]r_1^*φ (r1d1)
        (r0d0) edge node[above=-2pt, sloped]∼  (m0) 
        (m1) edge node[above=-2pt, sloped]∼  (r1d1) 
        (r0d1) edge node[above=-2pt, sloped]∼  (r1d0) 
        ;
     
    commutes in (C_2).
    
    A morphism 
        (P,φ)→ (P',φ')
    
    between two -actions considered this way is given by a morphism ψ: P_0→ P_0' in (C_0) such that the diagram
    
        [cross line/.style=preaction=draw=white, -,
        line width=4pt]
        (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
        	|(0i)|d_0^*P    |(1i)| d_0^*P'
           [.02em]
        	|(0j)|d_1^*P    |(1j)| d_1^*P'
        ;
        [->,font=,>=to, thin]
        (0i) edge node[above,pos=0.5]d_0^*ψ (1i) edge node[left,pos=0.5]φ (0j) 
        (0j) edge node[above,pos=0.5]d_1^*ψ (1j) 
        (1i) edge node[right,pos=0.5]φ' (1j)
        ;
     
    commutes in (C_1). 
    
    In terms of the fibration →, an action in ^ is given by a precategory ∈()
        (m)[matrix of math nodes, row sep=0em, column sep=3em, text height=1.5ex, text depth=0.25ex]
         
        |(2)|P_2    [1em] |(1)|P_1   [1em] |(0)|P_0
        ; 
        [->,font=,>=to, thin]
        ([yshift=1em]2.east) edge node[above=-2pt]r_0 ([yshift=1em]1.west) 
        (2) edge node[above=-2pt]m (1)
        ([yshift=-1em]2.east) edge node[above=-2pt]r_1 ([yshift=-1em]1.west) 
        ([yshift=1em]1.east) edge node[above=-2pt]d_0 ([yshift=1em]0.west) 
        (0)  edge node[above=-2pt]n (1) 
        ([yshift=-1em]1.east) edge node[above=-2pt]d_1 ([yshift=-1em]0.west) 
        
         
        ;
    
    living above , where the arrows in  are cartesian. 
    
    
    
     §.§ Connected components of precategories
    
    The object of connected components of a precategory ∈() is the reflexive coequaliser
    
        [cramped, column sep=2em, ampersand replacement=&]
        C_1 [yshift=2pt]rd_0[yshift=-2pt]r[swap]d_1& C_0 r&π_0(),
      
    provided it exists in . 
    
    If :^→ is a pseudofunctor, the object of connected components of an action P∈^ is defined as the object
    of connected components of the precategory ∈() associated to  via <ref>, 
    
        π_0(P)=π_0(),
    
    provided it exists in the fibred category  over  .
    
    With the notation from <ref>, suppose that π_0() and π_0(P) exist. Then π_0(P) projects to π_0(), i.e., a reflexive coequaliser 
    
        [cramped, column sep=2em, ampersand replacement=&]
        P_1 [yshift=2pt]rd_0[yshift=-2pt]r[swap]d_1& P_0 rρ&π_0(P)
      
    in 
    projects to a reflexive coequaliser
    
        [cramped, column sep=2em, ampersand replacement=&]
        C_1 [yshift=2pt]rd_0[yshift=-2pt]r[swap]d_1& C_0 rc&π_0()
      
    in .
    
    Writing p:→ for the given fibration, and using the fact that c is a coequaliser, there exists a unique morphism t:S→ p(π_0(P)) in  such that p(ρ)=t c. Writing θ: t^*π_0(P)→π_0(P) for a cartesian lift of t, there is a unique γ:P_0→ t^*π_0(P) such that p(γ)=c and θγ=ρ. 
    
    Since θ is cartesian, we obtain that γ coequalises d_0^P and d_1^P, so, since ρ is a coequaliser, there exists a unique morphism σ:π_0(P)→ t^*π_0(P) such that σρ=γ. 
    
    From the above, θσρ=θγ=ρ, and, since ρ is an epimorphism, we obtain that
    
        θσ=𝕀.
    
    Thus, θσθ=θ, and, since θ is cartesian, in order to show that
    
        σθ=𝕀,
    
    it suffices to verify that p(σθ)=𝕀. Indeed, p(σθ)c=p(σ)tc=p(σ)p(ρ)=p(σρ)=p(γ)=c, and the conclusion follows since c is an epimorphism. 
    
    
    With notation and assumptions of <ref>, the pullback functor
    
        η_^*:(π_0())≃^π_0()→^
    
    induced by the unique precategory morphism 
    
        η_:→π_0()
    
    determined by η_0=c has a partial left adjoint π_0 defined at P, i.e., we have a bijection
    
        (π_0())(π_0(P), Q)≃^(P,η_^*Q),
    
    natural in Q∈(π_0()), and P∈^, whenever π_0(P) exists.
    
    The assumptions yield the diagram
    
        [cross line/.style=preaction=draw=white, -,line width=4pt, proj/.style=dotted,-]
         #1#2
         ([yshift=.4em]#1.east) edge 
         ([yshift=.4em]#2.west) 
        (#1) edge 
        (#2)
        ([yshift=-.4em]#1.east) edge 
        ([yshift=-.4em]#2.west) #1#2
        ([yshift=.4em]#1.east) edge 
        ([yshift=.4em]#2.west) 
        (#2)  edge 
        (#1) 
        ([yshift=-.4em]#1.east) edge 
        ([yshift=-.4em]#2.west) #1#2
         ([yshift=.4em]#1.east) edge[cross line] 
         ([yshift=.4em]#2.west) 
        (#1) edge[cross line] 
        (#2)
        ([yshift=-.4em]#1.east) edge[cross line] 
        ([yshift=-.4em]#2.west) #1#2
        ([yshift=.4em]#1.east) edge[cross line] 
        ([yshift=.4em]#2.west) 
        (#2)  edge[cross line] 
        (#1) 
        ([yshift=-.4em]#1.east) edge[cross line] 
        ([yshift=-.4em]#2.west) (m)[matrix of math nodes, row sep=1em, column sep=1em,  
        text height=1.2ex, text depth=0.25ex]
        
        |(P2)|P_2      [0em] |(P1)|P_1   [0em]		    |(P0)|P_0   
            |(p2)|π_0(P)      |(p1)|π_0(P)       |(p0)|π_0(P)
            
        |(Q2)|η_2^*Q      [0em] |(Q1)|η_1^*Q   [0em]		    |(Q0)|η_0^*Q   
            |(q2)|Q      |(q1)|Q       |(q0)|Q
            			      
        |(C2)|C_2      |(C1)|C_1       |(C0)|C_0   
           |(c2)|π_0()      |(c1)|π_0()       |(c0)|π_0()
        ;
        [->,font=,>=to, thin,inner sep=1pt]
        P2P1P1P0Q2Q1Q1Q0C2C1C1C0
        
        
        (Q2) edge[proj] (C2)
        (Q1) edge[proj] (C1)
        (Q0) edge[proj] (C0)
        
        (P2) edge[dashed] node[pos=0.25,left]f_2(Q2)
        (P1) edge[dashed] node[pos=0.25,left]f_1(Q1)
        (P0) edge[dashed] node[pos=0.25,left]f_0(Q0)
        p2p1p1p0q2q1q1q0c2c1c1c0
        
        (Q2) edge node[pos=0.5,below left]q_2 (q2)
        (Q1) edge node[pos=0.5,below left]q_1(q1)
        (Q0) edge node[pos=0.5,below left]q_0(q0)
        
        (P2) edge node[pos=0.6,above right]η^P_2 (p2)
        (P1) edge node[pos=0.6,above right]η^P_1 (p1)
        (P0) edge node[pos=0.6,above right]η^P_0 (p0)
        
        (C2) edge node[pos=0.5,below left]η_2(c2)
        (C1) edge node[pos=0.5,below left]η_1(c1)
        (C0) edge node[pos=0.5,below left]η_0(c0)
        
        (q2) edge[cross line, proj] (c2)
        (q1) edge[cross line, proj] (c1)
        (q0) edge[cross line, proj] (c0)
        
        (p2) edge[cross line, dashed] node[pos=0.25,left]f(q2)
        (p1) edge[cross line, dashed] node[pos=0.25,left]f(q1)
        (p0) edge[cross line, dashed] node[pos=0.25,left]f(q0)
        ;
     
    without the dashed arrows, where all horizontal arrows in  except possibly η^P_0, η^P_1 and η^P_2 are cartesian. 
    
    By construction, q_0 coequalises the source and the target morphisms of η^*Q, whence, if we take a morphism (f_0,f_1,f_2)∈^(P,η^*Q), q_0f_0 coequalises d_0^P, d_1^P, so there exists a unique morphism f:π_0(P)→ Q that makes the rightmost vertical square, and ultimately the whole diagram, commutative. 
    
    Conversely, if we start with a morphism f:π_0(P)→ Q, using that q_0 is cartesian, we obtain a unique morphism f_0:P_0→η_0^*Q that makes the rightmost vertical square commutative. Since the composite η_1^*Q→η_0^*Q→ Q is cartesian, we obtain a unique morphism f_1:P_1→η_1^*Q which makes the square involving P_1,η_1^*Q, Q, π_0(P) commutative, and, given that q_0 is cartesian, f_1 and f_0 commute with source and target morphisms. Using that q_1 is cartesian, f_1 and f_0 commute with the identity sections. Continuing in the same fashion, we obtain a unique f_2:P_2→η_2^*Q that makes the whole diagram commutative. 
    
    
     §.§ Precategorical descent
    
    Let 
    
        f:→
    
    be a morphism in (). We say that
    
      * f is a descent morphism for , if the functor f^*:^→^ is fully faithful, and
    
      * f is of effective descent for , if the functor f^*:^→^ is an equivalence of categories. 
    
    Suppose that the reflexive coequaliser
    
        [cramped, column sep=2em, ampersand replacement=&]
        C_1 [yshift=2pt]rd_0[yshift=-2pt]r[swap]d_1& C_0 r&π_0()
      
    exists in . 
    
    The morphism of precategories 
    
        η:→π_0()
    
    is a descent morphism for  if and only if the above reflexive coequaliser is universal for , i.e., for every Q∈(π_0()), the diagram
    
        [cramped, column sep=2em, ampersand replacement=&]
        η_1^*Q [yshift=2pt]r[yshift=-2pt]r[swap]&η_0^*Q r& Q
      
    remains a coequaliser. 
    
    The above diagram is a coequaliser if and only if π_0(η^*Q)≃ Q, making η^* fully faithful. 
    
    
     §.§ Descent data
    
    Given a family ={U_iu_i→U : i∈ I} of morphisms in , the category of descent data
        ()
    
    consists of tuples
    
        ( (P_i)_i∈ I, (φ_ij)_(i,j)∈ I^2),
    
    where 
    
      *  for every i∈ I, P_i∈(U_i);
    
      *  for every i,j∈ I, φ_ij:π^*_ij,iP_i→π^*_ij,jP_j is  an isomorphism in (U_ij), 
    where U_ij=U_i×_UU_j and the corresponding projections are given by the pullback diagram
    
        [cross line/.style=preaction=draw=white, -,
        line width=4pt]
        (m)[matrix of math nodes, row sep=.9em, column sep=.5em, text height=1.5ex, text depth=0.25ex]
            |(ij)| U_ij   
           [.02em]
        	|(i)|U_i       |(j)|U_j
          [.02em]
        			    |(0)|U   
        ;
        [->,font=,>=to, thin]
        (ij) edge node[left,pos=0.2]π_ij,i (i) edge node[right,pos=0.2]π_ij,j (j) 
        (i) edge node[left,pos=0.6]u_i (0) 
        (j) edge node[right,pos=0.6]u_j (0)
        ;
    
    satisfying the cocycle condition: for every i,j,k∈ I, considering the triple pullback U_ijk=U_i×_UU_j×_UU_k and the corresponding projections
    
        [cross line/.style=preaction=draw=white, -,
        line width=4pt]
        (m)[matrix of math nodes, row sep=1.2em, column sep=.8em, text height=1.5ex, text depth=0.25ex]
            |(ijk)| U_ijk   
           [.02em]
        	|(ij)|U_ij    |(ik)|U_ik    |(jk)|U_jk
          [.2em]
        	|(i)|U_i    |(j)|U_j    |(k)|U_k
          [.02em]
        				    |(0)|U   
        ;
        [->,font=,>=to, thin]
        (ijk) edge node[left,pos=0.2]
            (ij) edge node[right,pos=0.2]
                (ik) edge node[right,pos=0.2]
                    (jk) 
                    (ik) edge node[left,pos=0.2]
                        (i) edge node[right,pos=0.2]
                            (k)
                            (ij) edge[cross line] node[left,pos=0.2]
                                (i) edge[cross line] node[right,pos=0.2]
                                    (j) 
                                    (jk) edge[cross line] node[left,pos=0.2]
                                        (j) edge[cross line] node[right,pos=0.2]
                                            (k)  
                                            (i) edge node[left,pos=0.6]u_i (0) 
                                            (j) edge node[right,pos=0.4]u_j (0)
                                            (k) edge node[right,pos=0.6]u_k (0)
                                            ;
                                         
                                        the diagram 
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=2em, column sep=1.2em, text height=1.5ex, text depth=0.25ex]
                                            	|(i)|π^*_ijk,iP_i       |(k)|π^*_ijk,kP_k
                                              [.02em]
                                            			    |(j)|π^*_ijk,jP_j   
                                            ;
                                            [->,font=,>=to, thin]
                                            (i) edge node[left,pos=0.6]π^*_ijk,ijφ_ij (j) edge node[above,pos=0.5]π^*_ijk,ikφ_ik (k) 
                                            (j) edge node[right,pos=0.4]π^*_ijk,jkφ_jk (k) 
                                            ;
                                         
                                        commutes in (U_ijk).
                                        
                                        A ()-morphism
                                            ((P_i)_i, (φ_ij)_ij)→((P'_i)_i, (φ'_ij)_ij)
                                        
                                        consists of a family of morphisms (P_iψ_i→P'_i)_i∈ I in (U_i) such that the diagrams
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	|(0i)|π^*_ij,iP_i    |(1i)| π^*_ij,iP'_i
                                               [.02em]
                                            	|(0j)|π^*_ij,jP_j    |(1j)| π^*_ij,iP'_j
                                            ;
                                            [->,font=,>=to, thin]
                                            (0i) edge node[above,pos=0.5]π^*_ij,iψ_i (1i) edge node[left,pos=0.5]φ_ij (0j) 
                                            (0j) edge node[above,pos=0.5]π^*_ij,jψ_j (1j) 
                                            (1i) edge node[right,pos=0.5]φ'_ij (1j)
                                            ;
                                         
                                        commute in (U_ij). 
                                        
                                        
                                        
                                         §.§ Pullback of descent data
                                        
                                        Let ={ U_iu_i→ U  :  i∈ I} and ={ V_jv_j→ V  :  j∈ J} be two families in . 
                                        
                                        A morphismγ:→ is given by a map of indices α:I→ J, a morphism g:U→ V in  and a family of commutative diagrams 
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	|(0i)|U_i    |(1i)| V_α(i)
                                               [.02em]
                                            	|(0j)|U    |(1j)| V
                                            ;
                                            [->,font=,>=to, thin]
                                            (0i) edge node[above,pos=0.5]g_i (1i) edge node[left,pos=0.5]u_i (0j) 
                                            (0j) edge node[above,pos=0.5]g (1j) 
                                            (1i) edge node[right,pos=0.5]v_α(i) (1j)
                                            ;
                                         
                                        for i∈ I.
                                        
                                        If γ':(α',g',g'_i) is another morphism → with g=g', we say that morphisms γ and γ' are homotopy equivalent and write
                                        
                                            γ∼γ'.
                                        
                                        A morphismγ:→ as above gives a functor
                                        
                                            γ^*:     ()→ DD(),  
                                                 (P_j,φ_jj')↦ (g_i^*P_α(i), (g_i× g_i')^* φ_α(i)α(i')).
                                        
                                        Moreover,  if γ∼γ', their associated pullback functors are canonically isomorphic, 
                                        
                                            γ^*≃γ'^*.
                                        
                                        With notation from <ref>, we say that 
                                        
                                          * γ:→ is a descent morphism, if the functor γ^*:()→() is fully faithful, and
                                        
                                          * γ:→ is of effective descent, if γ^*:()→() establishes an equivalence of categories. 
                                        
                                        Given a morphism f:U'→ U in , the above definition applied to the morphism of families f_ given by the diagram
                                         
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	 |(1i)| U   	|(0i)|U'
                                               [.02em]
                                            	|(1j)| U     |(0j)|U
                                            ;
                                            [->,font=,>=to, thin]
                                            (0i) edge node[above,pos=0.5]f (1i) edge node[right,pos=0.5]f (0j) 
                                            (0j) edge node[above,pos=0.5]𝕀 (1j) 
                                            (1i) edge node[left,pos=0.5]𝕀 (1j)
                                            (1i) edge[draw=none] node [above=-.3em]f_ (0j)
                                            ;
                                         
                                        tells us that f_ is a morphism of descent (resp., effective descent) if the functor
                                        
                                            f^*_ :(U)=(𝕀_U)→(f)
                                        
                                        is fully faithful (resp., an equivalence). 
                                        
                                        Hence, f is a morphism of descent/effective descent in the classical sense whenever f_ is descent/effective descent in the sense of <ref>.
                                         
                                        The horizontal composition of boxes φ and ψ in 
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	 |(1i)| U   	|(2i)|V    	|(3i)|W
                                               [.02em]
                                            	 |(1j)| U'   	|(2j)|V'    	|(3j)|W'
                                            ;
                                            [->,font=,>=to, thin]
                                            (2i) edge node[above,pos=0.5]f (1i) edge node[right,pos=0.4]v (2j) 
                                            (2j) edge node[above,pos=0.5]f' (1j) 
                                            (1i) edge node[left,pos=0.4]u (1j)
                                            (1i) edge[draw=none] node [above=-.2em]φ (2j)
                                            (3i) edge node[above,pos=0.5]g (2i) edge node[right,pos=0.4]w (3j) 
                                            (3j) edge node[above,pos=0.5]g' (2j) 
                                            
                                            (2i) edge[draw=none] node [above=-.2em]ψ (3j)
                                            ;
                                         
                                        is the box φ∘ψ given by the diagram
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	 |(1i)| U   	|(2i)|W
                                               [.02em]
                                            	 |(1j)| U'   	|(2j)|W'
                                            ;
                                            [->,font=,>=to, thin]
                                            (2i) edge node[above,pos=0.5]f∘ g (1i) edge node[right,pos=0.4]w (2j) 
                                            (2j) edge node[above,pos=0.5]f'∘ g' (1j) 
                                            (1i) edge node[left,pos=0.4]u (1j)
                                            (1i) edge[draw=none] node [above=-.2em]φ∘ϕ (2j)
                                            ;
                                        
                                        With notation of <ref>, as an immediate consequence of the definition of pullback functors from <ref>, we obtain an isomorphism of functors
                                        
                                            (φ∘ψ)^*≃ψ^*∘φ^*.
                                        
                                        The vertical composition of boxes φ and φ' in 
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	 |(1i)| U   	|(2i)|V
                                               [.02em]
                                            	 |(1j)| U'   	|(2j)|V'
                                               [.02em]
                                            	|(1k)| U”        |(2k)|V”
                                            ;
                                            [->,font=,>=to, thin]
                                            (2i) edge node[above,pos=0.5]f (1i) edge node[right,pos=0.5]v (2j) 
                                            (2j) edge node[above,pos=0.5]f' (1j) edge node[right,pos=0.4]v' (2k)
                                            (1i) edge node[left,pos=0.5]u (1j)
                                            (1j) edge node[left,pos=0.5]u' (1k)
                                            (2k) edge node[above,pos=0.5]f” (1k)
                                            (1i) edge[draw=none] node [above=-.2em]φ (2j)
                                            (1j) edge[draw=none] node [above=-.2em]φ' (2k)
                                            ;
                                         
                                        is the box φ'*φ given by the diagram
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	 |(1i)| U   	|(2i)|V
                                               [.02em]
                                            	 |(1j)| U”   	|(2j)|V”
                                            ;
                                            [->,font=,>=to, thin]
                                            (2i) edge node[above,pos=0.5]f (1i) edge node[right,pos=0.4]v'∘ v (2j) 
                                            (2j) edge node[above,pos=0.5]f” (1j) 
                                            (1i) edge node[left,pos=0.4]u'∘ u (1j)
                                            (1i) edge[draw=none] node [above=-.2em]φ'*φ (2j)
                                            ;
                                        
                                        With the notation of <ref>, if φ is a descent morphism, and 
                                            (f× f)^*:(U×_U”U)→(V×_V”V)
                                         is faithful, then φ'*φ is a descent morphism.
                                        
                                        The boxes
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	 |(1i)| U   	|(0i)|U
                                               [.02em]
                                            	|(1j)| U”     |(0j)|U'
                                            ;
                                            [->,font=,>=to, thin]
                                            (0i) edge node[above,pos=0.5]𝕀 (1i) edge node[right,pos=0.4]u (0j) 
                                            (0j) edge node[above,pos=0.5]u' (1j) 
                                            (1i) edge node[left,pos=0.5]u'∘ u (1j)
                                            (1i) edge[draw=none] node [above=-.2em]ρ_U (0j)
                                            ;
                                            
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	 |(1i)| V   	|(0i)|V
                                               [.02em]
                                            	|(1j)| V”     |(0j)|V'
                                            ;
                                            [->,font=,>=to, thin]
                                            (0i) edge node[above,pos=0.5]𝕀 (1i) edge node[right,pos=0.4]v (0j) 
                                            (0j) edge node[above,pos=0.5]v' (1j) 
                                            (1i) edge node[left,pos=0.5]v'∘ v (1j)
                                            (1i) edge[draw=none] node [above=-.2em]ρ_V (0j)
                                            ;
                                        
                                        satisfy 
                                        
                                            (φ'*φ)∘ρ_V=ρ_U∘φ,
                                        
                                        so we obtain a diagram of categories
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=2.2em, text height=1.5ex, text depth=0.25ex]
                                            	|(0i)|(u'∘ u)    |(1i)| (v'∘ v)
                                               [.02em]
                                            	|(0j)|(u)    |(1j)| (v)
                                            ;
                                            [->,font=,>=to, thin]
                                            (0i) edge node[above,pos=0.5](φ'*φ)^* (1i) edge node[left,pos=0.5]ρ^*_U (0j) 
                                            (0j) edge node[above,pos=0.5]φ^* (1j) 
                                            (1i) edge node[right,pos=0.5]ρ^*_V (1j)
                                            ;
                                         
                                        where the vertical arrows are faithful. 
                                        
                                        Indeed, ρ_U^* takes objects (P,α)∈(u'∘ u) to (P,(𝕀×𝕀)^*α)=(P,(U×_U'U→ U×_U”U)^*α), and it acts on morphisms p:(P,α)→ (P',α') as identity, hence it is faithful. A similar argument applies to ρ_V^*.
                                        
                                        By assumption, the bottom arrow φ^* is also faithful, and it follows that the top arrow (φ'*φ)^* is too. 
                                        
                                        It suffices to verify that (φ'*φ)^* is full. Let (P,α), (P',α')∈(u'∘ u), let 
                                        (Q,β)=(φ'*φ)^*(P,α), (Q',β')=(φ'*φ)^*(P',α'), and let
                                        q:(Q,β)→(Q',β') be a morphism in (v'∘ v), given by a morphism q:Q→ Q' in (V) such that the diagram
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	|(0i)|π^*_1Q    |(1i)| π^*_1Q'
                                               [.02em]
                                            	|(0j)|π^*_2Q    |(1j)| π^*_2Q'
                                            ;
                                            [->,font=,>=to, thin]
                                            (0i) edge node[above,pos=0.5]π^*_1 q (1i) edge node[left,pos=0.5]β (0j) 
                                            (0j) edge node[above,pos=0.5]π^*_2 q (1j) 
                                            (1i) edge node[right,pos=0.5]β' (1j)
                                            ;
                                         
                                        commutes in (V×_V”V).
                                        
                                        Since φ^* is fully faithful, there exists a unique morphism p:ρ_U^*(P,α)→ρ_U^*(P',α') such that 
                                        φ^*(p)=ρ_V^*(q), i.e., a morphism p:P→ P' in (U) with f^*(p)=q. 
                                        
                                        We claim that p is a morphism (P,α)→ (P',α') in (u'∘ u), i.e., that the diagram
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	|(0i)|π^*_1P    |(1i)| π^*_1P'
                                               [.02em]
                                            	|(0j)|π^*_2P    |(1j)| π^*_2P'
                                            ;
                                            [->,font=,>=to, thin]
                                            (0i) edge node[above,pos=0.5]π^*_1 p (1i) edge node[left,pos=0.5]α (0j) 
                                            (0j) edge node[above,pos=0.5]π^*_2 p (1j) 
                                            (1i) edge node[right,pos=0.5]α' (1j)
                                            ;
                                         
                                        commutes in (V×_V”V). This is indeed the case, since pulling the diagram back to (V×_V”V) via the faithful functor (f× f)^* gives the above diagram for q, which is commutative. 
                                        
                                        
                                        Suppose we have a morphism f:U'→ U admitting a section s:U→ U', so that f∘ s=𝕀_U. 
                                        Then f is of effective descent. More explicitly, the functor 
                                            f^*=f^*_:(U)→(f)
                                         has a quasi-inverse σ^* associated to the box
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	 |(1i)| U'   	|(0i)|U
                                               [.02em]
                                            	|(1j)| U     |(0j)|U
                                            ;
                                            [->,font=,>=to, thin]
                                            (0i) edge node[above,pos=0.5]s (1i) edge node[right,pos=0.4]id (0j) 
                                            (0j) edge node[above,pos=0.5]𝕀 (1j) 
                                            (1i) edge node[left,pos=0.5]f (1j)
                                            (1i) edge[draw=none] node [above=-.2em]σ (0j)
                                            ;
                                        
                                        The composite σ∘ f_ represents the outer box of the diagram
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	 |(1i)| U'   	|(0i)|U'
                                               [.02em]
                                            	|(1j)| U     |(0j)|U
                                            ;
                                            [->,font=,>=to, thin]
                                            (0i.190) edge node[below,pos=0.5]𝕀 (1i.-10) 
                                            (0i.170) edge node[above,pos=0.5]s∘ f (1i.10) 
                                            (0i) edge node[right,pos=0.4]f (0j) 
                                            (0j) edge node[above,pos=0.5]𝕀 (1j) 
                                            (1i) edge node[left,pos=0.5]f (1j)
                                            
                                            ;
                                        
                                        and let us denote the inner box by 𝕀_f. These boxes are homotopy equivalent in the sense of the `moreover' clause of <ref>, whence 
                                        
                                            f^*_σ^*≃ (σ∘ f_)^*≃𝕀_f^*=𝕀: (f)→(f).
                                        
                                        Conversely, using the fact that s is a section of f,  
                                            σ^*f^*_≃ (f_∘σ)^*=𝕀:(𝕀_U)→(𝕀_U).
                                        
                                        
                                         §.§ Classical and precategorical descent
                                        
                                        
                                        Let u:U'→ U be a morphism in , and consider the groupoid 
                                        
                                            _u
                                        
                                        associated to the kernel pair of u, as in <ref>
                                        
                                        Comparing <ref> and <ref>, we see that we have an isomorphism
                                        
                                            _(u)=^_u.
                                        
                                        Moreover, a box
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	|(0i)|U'    |(1i)| V'
                                               [.02em]
                                            	|(0j)|U    |(1j)| V
                                            ;
                                            [->,font=,>=to, thin]
                                            (0i) edge node[above,pos=0.5]f' (1i) edge node[left,pos=0.5]u (0j) 
                                            (0j) edge node[above,pos=0.5]f (1j) 
                                            (1i) edge node[right,pos=0.5]v (1j)
                                            (1i) edge[draw=none] node [above=-.2em]φ (0j)
                                            ;
                                         
                                        induces a morphism 
                                        
                                            F=(f'× f'× f', f'× f', f'):_v→_u,
                                        
                                        and the functors
                                        
                                            φ^*:(u)→(v),     and     F^*:^_u→^_v
                                        
                                        are isomorphic. 
                                        
                                        Hence, φ is a morphism of descent/effective descent in the classical sense of <ref> if and only if F is a morphism of descent/effective precategorical descent in the sense of <ref>.
                                        
                                        In particular, a morphism u:U'→ U is a morphism of descent/effective descent in the classical sense if and only if u_ is of descent/effective descent, if and only if the precategory morphism _u→U=_𝕀_U is a morphism of descent/effective precategorical descent. 
                                        
                                        
                                        
                                         §.§ Descent of precategory actions
                                        
                                        Let :^→ be a pseudofunctor, and consider the pseudofunctor
                                        
                                            :()^→,    ↦^.
                                        
                                        
                                        Let f:→∈() be a morphism of precategories in  such that
                                        
                                          * f_0 is of effective descent for ; 
                                        
                                          * f_1 is descent morphism for ;
                                        
                                          * f_2^* is faithful. 
                                        
                                        Then f is of effective descent for . 
                                        
                                        We must show that the canonical morphism 
                                            ^=()→^_f=_(f)
                                         is an equivalence of categories, where 
                                        
                                            _f∈(())
                                         
                                        is given by
                                        
                                            (m)[matrix of math nodes, row sep=0em, column sep=2em, text height=1.5ex, text depth=0.25ex]
                                             
                                            |(2)|_2=×_×_    [1em] |(1)|_1=×_   [1em] |(0)|_0=
                                            ; 
                                            [->,font=,>=to, thin]
                                            ([yshift=.4em]2.east) edge 
                                            ([yshift=.4em]1.west) 
                                            (2) edge 
                                            (1)
                                            ([yshift=-.4em]2.east) edge 
                                            ([yshift=-.4em]1.west) 
                                            ([yshift=.4em]1.east) edge 
                                             ([yshift=.4em]0.west) 
                                            (0)  edge 
                                            (1) 
                                            ([yshift=-.4em]1.east) edge 
                                            ([yshift=-.4em]0.west) 
                                            
                                             
                                            ;
                                        
                                        with _0, _1, _2∈()_. Expanding the components of these precategories as columns, we obtain a diagram
                                        
                                            [cross line/.style=preaction=draw=white, -,line width=4pt, proj/.style=dotted,-]
                                             #1#2
                                             ([yshift=.4em]#1.east) edge 
                                             ([yshift=.4em]#2.west) 
                                            (#1) edge 
                                            (#2)
                                            ([yshift=-.4em]#1.east) edge 
                                            ([yshift=-.4em]#2.west) #1#2
                                            ([yshift=.4em]#1.east) edge 
                                            ([yshift=.4em]#2.west) 
                                            (#2)  edge 
                                            (#1) 
                                            ([yshift=-.4em]#1.east) edge 
                                            ([yshift=-.4em]#2.west) #1#2
                                             ([xshift=.4em]#1.south) edge 
                                             ([xshift=.4em]#2.north) 
                                            (#1) edge 
                                            (#2)
                                            ([xshift=-.4em]#1.south) edge 
                                            ([xshift=-.4em]#2.north) #1#2
                                            ([xshift=.4em]#1.south) edge
                                            ([xshift=.4em]#2.north) 
                                            (#2)  edge 
                                            (#1) 
                                            ([xshift=-.4em]#1.south) edge[cross line] 
                                            ([xshift=-.4em]#2.north) (m)[matrix of math nodes, row sep=2em, column sep=2em,  
                                            text height=1.2ex, text depth=0.25ex]
                                            
                                            |(P22)|C_2×_D_2C_2×_D_2C_2   [0em] |(P12)|C_2×_D_2C_2   [0em]	 |(P02)|C_2
                                            
                                            |(P21)|C_1×_D_1C_1×_D_1C_1   [0em] |(P11)|C_1×_D_1C_1   [0em]	 |(P01)|C_1
                                            
                                            |(P20)|C_0×_D_0C_0×_D_0C_0   [0em] |(P10)|C_0×_D_0C_0   [0em]	 |(P00)|C_0
                                            ;
                                            [->,font=,>=to, thin,inner sep=1pt]
                                            P22P12P12P02P21P11P11P01P20P10P10P00P22P21P21P20P12P11P11P10P02P01P01P00
                                            
                                            ;
                                         
                                        where the rows are the groupoids _f_2, _f_1, _f_0 associated to kernel pairs of morphisms f_2, f_1, f_0 that constitute f. 
                                        
                                        
                                        An action P∈P̃^_f consists of a diagram 
                                        
                                            (m)[matrix of math nodes, row sep=0em, column sep=2em, text height=1.5ex, text depth=0.25ex]
                                             
                                            |(2)|P_2    [1em] |(1)|P_1   [1em] |(0)|P_0
                                            ; 
                                            [->,font=,>=to, thin]
                                            ([yshift=.4em]2.east) edge 
                                            ([yshift=.4em]1.west) 
                                            (2) edge 
                                            (1)
                                            ([yshift=-.4em]2.east) edge 
                                            ([yshift=-.4em]1.west) 
                                            ([yshift=.4em]1.east) edge 
                                             ([yshift=.4em]0.west) 
                                            (0)  edge 
                                            (1) 
                                            ([yshift=-.4em]1.east) edge 
                                            ([yshift=-.4em]0.west) 
                                            
                                             
                                            ;
                                        
                                        consisting of P_2∈(_2)=^_2,  P_1∈(_1)=^_1, P_0∈(_0)=^_0 and cartesian arrows in the fibration associated to . Expanding the components of P_2, P_1 and P_0 as columns, we obtain a diagram
                                        
                                            [cross line/.style=preaction=draw=white, -,line width=4pt, proj/.style=dotted,-]
                                             #1#2
                                             ([yshift=.4em]#1.east) edge 
                                             ([yshift=.4em]#2.west) 
                                            (#1) edge 
                                            (#2)
                                            ([yshift=-.4em]#1.east) edge 
                                            ([yshift=-.4em]#2.west) #1#2
                                            ([yshift=.4em]#1.east) edge 
                                            ([yshift=.4em]#2.west) 
                                            (#2)  edge 
                                            (#1) 
                                            ([yshift=-.4em]#1.east) edge 
                                            ([yshift=-.4em]#2.west) #1#2
                                             ([xshift=.4em]#1.south) edge 
                                             ([xshift=.4em]#2.north) 
                                            (#1) edge 
                                            (#2)
                                            ([xshift=-.4em]#1.south) edge 
                                            ([xshift=-.4em]#2.north) #1#2
                                            ([xshift=.4em]#1.south) edge
                                            ([xshift=.4em]#2.north) 
                                            (#2)  edge 
                                            (#1) 
                                            ([xshift=-.4em]#1.south) edge[cross line] 
                                            ([xshift=-.4em]#2.north) (m)[matrix of math nodes, row sep=2em, column sep=2em,  
                                            text height=1.2ex, text depth=0.25ex]
                                            
                                            |(P22)|P_2,2   [1em] |(P12)|P_1,2   [1em]	 |(P02)|P_0,2
                                            
                                            |(P21)|P_2,1   [1em] |(P11)|P_1,1   [1em]	 |(P01)|P_0,1
                                            
                                            |(P20)|P_2,0   [1em] |(P10)|P_1,0   [1em]	 |(P00)|P_0,0
                                            ;
                                            [->,font=,>=to, thin,inner sep=1pt]
                                            P22P12P12P02P21P11P11P01P20P10P10P00P22P21P21P20P12P11P11P10P02P01P01P00
                                            
                                            ;
                                         
                                        in the fibred category associated to , with all morphisms cartesian. 
                                        
                                        Hence, the rows yield actions 
                                        P̅_2∈^_f_2≃_(f_2), P̅_1∈^_f_1≃_(f_1), P̅_0∈^_f_0≃_cP(f_0). Considering descent data as a pseudofunctor on the arrow category
                                        
                                            _:()^→,
                                        
                                        the diagram
                                        
                                            (m)[matrix of math nodes, row sep=0em, column sep=2em, text height=1.5ex, text depth=0.25ex]
                                             
                                            |(2)|P̅_2    [1em] |(1)|P̅_1   [1em] |(0)|P̅_0
                                            ; 
                                            [->,font=,>=to, thin]
                                            ([yshift=.4em]2.east) edge 
                                            ([yshift=.4em]1.west) 
                                            (2) edge 
                                            (1)
                                            ([yshift=-.4em]2.east) edge 
                                            ([yshift=-.4em]1.west) 
                                            ([yshift=.4em]1.east) edge 
                                             ([yshift=.4em]0.west) 
                                            (0)  edge 
                                            (1) 
                                            ([yshift=-.4em]1.east) edge 
                                            ([yshift=-.4em]0.west) 
                                            
                                             
                                            ;
                                        
                                        determines an action 
                                        
                                            P̅∈_^f
                                        
                                        where f=(f_2,f_1,f_0)∈(()). 
                                        
                                        Hence, we have shown that
                                        
                                            _(f)≃_^f.
                                        
                                        Since f_0 is effective descent, there is an object Q_0∈(D_0) such that, writing Q̅_0=(Q_0,𝕀) for the trivial descent datum, we have P̅_0≃ f_0^*Q̅_0. The action isomorphism d_0,f^*P̅_0α→ d_1,f^*P̅_0 yields an isomorphism
                                        
                                            f_1^*d̅_0^*Q̅_0≃ d_0,f^*f_0^*Q̅_0≃ d_0,f^*P̅_0α→ d_1,f^*P̅_0≃ d_1,f^*f_0^*Q̅_0≃ f_1^*d̅_1^*Q̅_0,
                                        
                                        where we wrote d̅_0 and d̅_1 for the obvious boxes/morphisms 𝕀_D_1→𝕀_D_0 in ().  
                                        Given that f_1 is descent, we obtain a unique action morphism 
                                        
                                            d̅_0^*Q̅_0β̅→d̅_1^*Q̅_0
                                        
                                        such that f_1^*β̅≃α. Note that β̅ is uniquely determined by an isomorphism
                                        
                                            d_0^*Q_0β→d_1^*Q_0,
                                        
                                        and it remains to verify that β satisfies the cocycle condition
                                        
                                            r_1^*β∘ r_0^*β≃ m^*β
                                        
                                        in (D_2), or, equivalently, that r̅_1^*β̅∘r̅_0^*β̅ and m̅^*β̅ agree up to coherence. Applying the functor f_2^* to both yields
                                        
                                            f_2^*r̅_1^*β̅∘ f_2^*r̅_0^*β̅≃ r_1,f^*f_1^*β̅∘ r_0,f^*f_1^*β̅≃ r_1,f^*α∘∘ r_0,f^*α
                                        
                                        and
                                        
                                            f_2^*m̅^*β̅≃ m_f^*f_1^*β̅≃ m_f^*α,
                                        
                                        which agree up to coherence by the cocycle condition for α. By faithfulness of f_2^*, we obtain the cocycle condition for β, and we have constructed a unique action (Q_0,β)∈^ (up to isomorphism) that lifts to P, as desired. 
                                        
                                        
                                         §.§ Descent for quasi-projective morphisms
                                        
                                        A scheme morphism f:X→ Y whose codomain Y is the spectrum of a field k is of effective descent for quasi-projective morphisms. 
                                        
                                        Let k' be a finite extension of k with X(k')≠∅. Writing Y'=(k'), we have a pullback diagram
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	 |(1i)| X   	|(0i)|X'
                                               [.02em]
                                            	|(1j)| Y     |(0j)|Y'
                                            ;
                                            [->,font=,>=to, thin]
                                            (0i) edge node[above,pos=0.5]g' (1i) edge node[right,pos=0.4]f' (0j) 
                                            (0j) edge node[above,pos=0.5]g (1j) 
                                            (1i) edge node[left,pos=0.5]f (1j)
                                            
                                            ;
                                        
                                        where f' admits a section s afforded by a k'-point of X'. The morphism g is finite locally free surjective, hence of effective descent for quasi-projective morphisms by <cit.>, and so is g' as a base-change of g. 
                                        Consider the diagram 
                                         
                                         
                                         
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=2em, column sep=2em, text height=1.5ex, text depth=0.25ex]
                                            	 |(1i)| X   	|(2i)|X'    	|(3i)|Y'    |(4i)|X'
                                               [.02em]
                                            	 |(1j)| Y   	|(2j)|Y    	|(3j)|Y    |(4j)|Y
                                            ;
                                            [->,font=,>=to, thin]
                                            (2i) edge node[above,pos=0.5]g' (1i) edge node[right=-1.8em,pos=0.4]fg'=gf' (2j) 
                                            (2j) edge node[above,pos=0.5]𝕀 (1j) 
                                            (1i) edge node[left,pos=0.4]f (1j)
                                            (1i) edge[draw=none] node [above=-.2em]α (2j)
                                            (3i) edge node[above,pos=0.5]s (2i) edge node[right,pos=0.4]g (3j) 
                                            (3j) edge node[above,pos=0.5]𝕀 (2j) 
                                            (2i) edge[draw=none] node [above=-.2em]β (3j)
                                            (4i) edge node[above,pos=0.5]f' (3i) edge node[right,pos=0.4]gf' (4j) 
                                            (4j) edge node[above,pos=0.5]𝕀 (3j) 
                                            (3i) edge[draw=none] node [above=-.2em]γ (4j)
                                            ;
                                         
                                        defining boxes α, β and γ in . 
                                        
                                        Writing σ for the box
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                            	 |(1i)| X   	|(0i)|Y'
                                               [.02em]
                                            	|(1j)| Y'     |(0j)|Y'
                                            ;
                                            [->,font=,>=to, thin]
                                            (0i) edge node[above,pos=0.5]s (1i) edge node[right,pos=0.4]id (0j) 
                                            (0j) edge node[above,pos=0.5]𝕀 (1j) 
                                            (1i) edge node[left,pos=0.5]f' (1j)
                                            (1i) edge[draw=none] node [above=-.2em]σ (0j)
                                            ;
                                        
                                        we directly verify that β=𝕀_g*σ and γ=𝕀_g*f'_. 
                                        
                                        As in the proof of <ref>, we have that σ∘ f'_∼𝕀_f', so 
                                        
                                            β∘γ=(𝕀_g*σ)∘(𝕀_g*f'_)=𝕀_g*(σ∘ f'_)∼𝕀_g*𝕀_f'=𝕀_g∘ f',
                                         whence
                                        
                                            γ^*β^*≃𝕀.
                                         Conversely, γ∘β yields identity on the nose, so we conclude that γ^* is an equivalence of categories. 
                                        
                                        Using the fact that g' is of effective descent and that g'× g' is finite faithfully flat, Lemma <ref> gives that the functor α^* associated to α=𝕀_f*g'_ is fully faithful. 
                                        
                                        We directly verify that g_∘γ=f_∘α, whence we obtain a diagram of categories
                                        
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=.9em, column sep=.5em, text height=1.5ex, text depth=0.25ex]
                                                |(ij)| (f)   
                                               [.02em]
                                            	|(i)|(Y)       |(j)|(fg')
                                              [.02em]
                                            			    |(0)|(g)   
                                            ;
                                            [->,font=,>=to, thin]
                                            (i) edge node[above left,pos=0.5]f^*_ (ij) edge node[below left,pos=0.5]g^*_ (0) 
                                            (0) edge node[below right,pos=0.5]γ^* (j) 
                                            (ij) edge node[above right,pos=0.5]α^* (j)
                                            ;
                                         
                                        where we know that α^* f^*_=γ^* g^*_ is an equivalence of categories, and α^* is fully faithful, so we deduce that f^*_ is also an equivalence. 
                                        
                                        
                                        § CATEGORICAL GALOIS THEORY
                                        
                                        
                                         §.§ Classical Janelidze's categorical Galois theory
                                        
                                        
                                        Consider an adjoint pair of functors
                                        
                                         
                                         [cross line/.style=preaction=draw=white, -,
                                        line width=3pt]
                                        (m)[matrix of math nodes, minimum size=1.7em,
                                        inner sep=0pt, 
                                        row sep=3.3em, column sep=1em, text height=1.5ex, text depth=0.25ex]
                                          
                                          |(dc)|
                                        
                                         |(c)|
                                        ;
                                        [->,font=,>=to, thin]
                                        
                                        
                                        
                                        (dc) edge [bend right=30] node (ss) [left]S (c)
                                        (c) edge [bend right=30] node (ps) [right]C (dc)
                                        
                                        (ss) edge[draw=none] node⊣ (ps)
                                        ;
                                        
                                        
                                        with unit η:𝕀→ CS and counit ϵ:SC→𝕀.
                                        If  admits pullbacks, for any X∈ we obtain an adjunction
                                        
                                         
                                         [cross line/.style=preaction=draw=white, -,
                                        line width=3pt]
                                        (m)[matrix of math nodes, minimum size=1.7em,
                                        inner sep=0pt, 
                                        row sep=3.3em, column sep=1em, text height=1.5ex, text depth=0.25ex]
                                          
                                          |(dc)|_ X
                                        
                                         |(c)|_ S(X)
                                        ;
                                        [->,font=,>=to, thin]
                                        
                                        
                                        
                                        (dc) edge [bend right=30] node (ss) [left]S_X (c)
                                        (c) edge [bend right=30] node (ps) [right]C_X (dc)
                                        
                                        (ss) edge[draw=none] node⊣ (ps)
                                        ;
                                        
                                        
                                        where  
                                        
                                            S_X(AX)=S(A)S(X),
                                        
                                        and  C_X(ES(X)) is obtained by forming the pullback
                                        
                                        (m)[matrix of math nodes, row sep=2em, column sep=2em, text height=1.9ex, text depth=0.25ex]
                                         
                                         |(1)|C_X(e)    |(2)|C(E)
                                        
                                         |(l1)|X    |(l2)|CS(X)
                                        ; 
                                        [->,font=,>=to, thin]
                                        (1) edge  (2) edge   (l1)
                                        (2) edge node[right]C(e) (l2) 
                                        (l1) edge node[above]η_X  (l2);
                                        
                                        
                                        in .
                                        
                                        A morphism XYin  gives rise to the pullback/base change functor
                                        
                                            f^*:_ Y→_ X,
                                        
                                        which admits a left adjoint
                                        
                                            f_!:_ X→_ Y,      a↦ f∘ a.
                                        Following <cit.> an object AY∈_ Y is split by XY∈ when the unit η^X:𝕀→ C_XS_X of adjunction S_X⊣ C_X gives an isomorphism
                                        
                                            η^X_f^*a:f^*a→ C_XS_X(f^*a).
                                        
                                        If C_X is fully faithful, a is split by f(<cit.>), if and only if there exists an object ES(X) such that 
                                        
                                            f^*a≃ C_X(e).
                                        
                                        We write
                                        
                                            _Y(f)
                                        
                                        for the full subcategory of _ Y of objects split by f.
                                        
                                        The morphism f is of relative Galois descent if
                                        
                                          * f^* is monadic;
                                        
                                          *  the counit ϵ^X:S_XC_X→𝕀 of adjunction S_X⊣ C_X is an isomorphism;
                                        
                                          *  for every ES(X) in _ S(X), the object f_!  C_X(e)∈_ Y is split by f.
                                         
                                         If XY is of relative Galois descent, the Galois precategory
                                        
                                            [f]=S(_f)
                                        
                                        is actually an internal groupoid in  given by the data
                                        
                                        
                                        (m)[matrix of math nodes, row sep=0em, column sep=3em, text height=1.5ex, text depth=0.25ex]
                                         
                                        |(0)|S(X×_YX)×_S(X)S(X×_YX)   [3em]  |(1)|S(X×_YX)   [1em] |(2)|S(X)
                                        ; 
                                        [->,font=,>=to, thin]
                                        (0) edge node[above](S(π_1),S(π_4)) (1)
                                        ([yshift=1em]1.east) edge node[above=-2pt]S(π_1) ([yshift=1em]2.west) 
                                        (2)  edge node[above=-2pt]S(Δ) (1) 
                                        ([yshift=-1em]1.east) edge node[above=-2pt]S(π_2) ([yshift=-1em]2.west) 
                                        
                                         (1) edge [loop below] node S(τ) (1)
                                        ;
                                        
                                        
                                        where τ is the morphism interchanging the copies of X, and Δ is the diagonal.
                                        
                                        Janelidze's Galois theorem(<cit.>) gives  an equivalence of categories
                                        
                                            _Y(f)≃^[f].
                                        
                                        of f-split objects and the actions of the internal groupoid [f] in , as in <ref>.
                                        
                                        The proof consists in verifying that the monad 𝕋 associated to the adjunction 
                                        
                                         
                                         [cross line/.style=preaction=draw=white, -,
                                        line width=3pt]
                                        (m)[matrix of math nodes, minimum size=1.7em,
                                        inner sep=0pt, 
                                        row sep=3.3em, column sep=1em, text height=1.5ex, text depth=0.25ex]
                                          
                                          |(dc)|_Y(f)
                                        
                                         |(c)|_ S(X)
                                        ;
                                        [->,font=,>=to, thin]
                                        
                                        
                                        
                                        (dc) edge [bend left=30] node (ss) [right]U=S_X f^* (c)
                                        (c) edge [bend left=30] node (ps) [left]F=f_! C_X (dc)
                                        
                                        (ss) edge[draw=none] node⊣ (ps)
                                        ;
                                        
                                        
                                        of the monadic functor U(<cit.>), with functorial part T=UF, is isomorphic to the monad 𝕋' on _ S(X) associated to the adjunction 
                                        
                                         
                                         [cross line/.style=preaction=draw=white, -,
                                        line width=3pt]
                                        (m)[matrix of math nodes, minimum size=1.7em,
                                        inner sep=0pt, 
                                        row sep=3.3em, column sep=1em, text height=1.5ex, text depth=0.25ex]
                                          
                                          |(dc)|^[f]
                                        
                                         |(c)|_ S(X)
                                        ;
                                        [->,font=,>=to, thin]
                                        
                                        
                                        
                                        (dc) edge [bend left=30] node (ss) [right]U' (c)
                                        (c) edge [bend left=30] node (ps) [left]F' (dc)
                                        
                                        (ss) edge[draw=none] node⊣ (ps)
                                        ;
                                        
                                        
                                        where U' is the forgetful functor omitting the groupoid action, and F' is the `representable internal diagram' functor, whose functorial part is T'=d_1!d_0^* and the category of algebras is the category ^[f]. Hence, we obtain equivalences
                                        
                                            _Y[f]K^𝕋⟶ (_ S(X))^𝕋≃ (_ S(X))^𝕋'K^𝕋'⟵^[f],
                                        
                                        where we wrote K^𝕋 and K^𝕋' for the comparison functors of the respective monads. 
                                        
                                        In this case, modulo the identification of the category of 𝕋-algebras (_ S(X))^𝕋 with  ^[f], 
                                        the functors realising the sought-after equivalence are the comparison functor 
                                        
                                            Φ=K^𝕋:_Y(f) (_ S(X))^𝕋,     p↦ (U(p),U(ε_p)),
                                        
                                        and its left adjoint
                                        
                                            Ψ: (q,ν)↦[cramped, column sep=normal, ampersand replacement=&]
                                            (FUF(q) .[yshift=2pt]rF(ν)[yshift=-2pt]r[swap]ε_Fq&.F(q)),
                                        
                                        where (Qq→X_0,ν) is a 𝕋-algebra and ε:FU→𝕀 is the counit of the adjunction F⊣ U, and the coequaliser exists by the proof of Beck's monadicity criterion as in <cit.>.
                                        
                                        By identifying 𝕋 and 𝕋', and writing [f]=(S(X×_YX),S(X))=(G_1,X_0), the top arrow appearing in the coequaliser is obtained by applying F to the action G_1×_X_0 Qν→ Q, which, modulo the identification X×_X_0(G_1×_X_0Q)≃ X×_X_0G_1×_X_0Q gives the morphism
                                        
                                            𝕀×ν: X×_X_0G_1×_X_0Q→ X×_X_0Q.
                                        
                                        If p∈_Y[f] then f^*(p)≃ C_X(q) for some q, whence the counit ε_p is
                                        
                                            FUp=f_!C_XS_Xf^*p≃ f_!C_XS_XC_Xq≃ f_!C_Xq≃ f_!f^*p→ p,
                                        
                                        so the bottom arrow ε_Fq identifies with f_!f^*Fq→ Fq, which, modulo the isomorphism
                                        
                                            X×_Y(X×_X_0Q)≃ (X×_YX)×_X(X×_X_0Q)≃ (X×_X_0G_1)×_X(X×_X_0Q)≃ X×_X_0G_1×_X_0Q,
                                        
                                        identifies with 
                                        
                                            μ×𝕀: X×_X_0G_1×_X_0Q→ X×_X_0Q,
                                        
                                        where μ denotes the [f]-action on X. Thus, we may symbolically write the above coequaliser as the quotient
                                        
                                            Ψ(q,ν)=(𝕀×ν,μ×𝕀)=F(q)/[f]=(X×_X_0Q)/[f],
                                        
                                        by the twisted-diagonal action of the Galois groupoid on X×_X_0Q. 
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                        
                                         §.§ Carboni-Magid-Janelidze Galois correspondence
                                        [<cit.>]
                                        Let  be a category with pullbacks and coequalisers of equivalence relations. In the presence of pullbacks, regular epimorphisms coincide with effective epimorphisms, and we call them quotients for short. 
                                        
                                        Let G=(G_1, G_0,d_0,d_1,e,m) be an internal groupoid in . Let us write G̃∈^G for the canonical action of G on itself. There is a bijection
                                        
                                            (G)≃(G̃)
                                        
                                        between the set of subgroupoids of G with the same object of objects and the set of equivalence relations on G̃ in ^G as follows.
                                        
                                        Given a subgroupoid G'=(G_1',G_0) with ι:G_1'↪ G_1, the corresponding equivalence relation on G̃ is
                                        
                                            [cramped, column sep=4em, ampersand replacement=&]
                                            R_G'=G_1×_G_0G_1'  [yshift=2pt]rπ_1[yshift=-2pt]r[swap]m(𝕀_G_1×ι)&G_1
                                          
                                        
                                        Conversely, if R↪G̃×G̃ is an equivalence relation on G̃ in ^G, we define the corresponding subgroupoid by the pullback
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=1.8em, column sep=3.2em, text height=1.5ex, text depth=0.25ex]
                                            	|(0i)|G_1'    |(1i)| R
                                               [.02em]
                                            	|(0j)|G_1    |(1j)| G_1×_G_0G_1
                                            ;
                                            [->,font=,>=to, thin]
                                            (0i) edge node[above,pos=0.5] (1i) edge[right hook->] node[left,pos=0.5] (0j) 
                                            (0j) edge node[above,pos=0.5](𝕀_G_1, ed_1) (1j) 
                                            (1i) edge[right hook->]  node[right,pos=0.5] (1j)
                                            ;
                                         
                                        where G_1×_G_0G_1 is the kernel pair of d_1.
                                        
                                        A subgroupoid is called effective is the associated equivalence relation is effective in the sense that it is the kernel pair of its coequaliser. 
                                        
                                        With notation from <ref>, suppose that f:X→ Y is of relative Galois descent with Galois groupoid G=[f] internal in . There is an anti-isomorphism
                                        
                                            SplitQuo[f]≃ EffSub([f])
                                        
                                        that assigns
                                        
                                            [cross line/.style=preaction=draw=white, -,
                                            line width=4pt]
                                            (m)[matrix of math nodes, row sep=.9em, column sep=.5em, text height=1.5ex, text depth=0.25ex]
                                                |(ij)| X      
                                               [.02em]
                                            	|(i)|P      ↦   [X→ P]	
                                              [.02em]
                                            			    |(0)|Y      
                                             [1em]
                                            			    X/G'				      	G'		
                                            ;			
                                            [->,font=,>=to, thin]
                                            (ij) edge node[left,pos=0.2]
                                                (i) edge node[right,pos=0.5]f (0) 
                                                (i) edge node[below left,pos=0.5]p (0) 
                                                ;
                                             
                                            between the ordered set of quotients of X over Y in [f] and the ordered set of effective subgroupoids of [f].
                                            
                                            We use the equivalence established by functors Φ and Ψ from <ref> and the fact that, in the presence of pullbacks, quotients (regular epimorphisms) agree with effective epimorphisms.
                                            
                                            If the quotient p of X from the above diagram is f-split by Qq→X_0=S(X), i.e., f^*p≃ C_X(q), applying the comparison functor Φ
                                            gives an effective quotient
                                             
                                                [cross line/.style=preaction=draw=white, -,
                                                line width=4pt]
                                                (m)[matrix of math nodes, row sep=.9em, column sep=.5em, text height=1.5ex, text depth=0.25ex]
                                                    |(ij)| G̃
                                                   [.02em]
                                                	|(i)|Q   
                                                  [.02em]
                                                			    |(0)|X_0
                                                ;
                                                [->,font=,>=to, thin]
                                                (ij) edge node[left,pos=0.2]
                                                    (i) edge node[right,pos=0.5]
                                                        (0) 
                                                        (i) edge node[below left,pos=0.5]q (0) 
                                                        ;
                                                     
                                                    and an effective equivalence relation G̃×_QG̃↪G̃×_X_0G̃ on G̃. The corresponding  effective subgroupoid G_P of G is given by
                                                    
                                                        [cross line/.style=preaction=draw=white, -,line width=4pt]
                                                        (m)[matrix of math nodes, row sep=1.8em, column sep=3.2em, text height=1.5ex, text depth=0.25ex]
                                                        	|(0i)|G_P,1    |(1i)| G_1×_Q G_1
                                                           [.02em]
                                                        	|(0j)|G_1    |(1j)| G_1×_G_0G_1
                                                        ;
                                                        [->,font=,>=to, thin]
                                                        (0i) edge node[above,pos=0.5] (1i) edge[right hook->] node[left,pos=0.5] (0j) 
                                                        (0j) edge node[above,pos=0.5](𝕀_G_1, ed_1) (1j) 
                                                        (1i) edge[right hook->]  node[right,pos=0.5] (1j)
                                                        ;
                                                    
                                                    
                                                    
                                                    
                                                    
                                                    
                                                    
                                                    
                                                    
                                                    Conversely,  an effective subgroupoid G' of G is associated with the action 
                                                        G̃/G'=(R_G'G̃)
                                                     
                                                    in ^G given as the coequaliser of its associated effective equivalence relation, and then taken to the split quotient
                                                    
                                                        Ψ(G̃/G')≃ (X×G̃/G')/G≃ X/G'=f/G'
                                                        
                                                        
                                                        =
                                                        [cramped, column sep=normal, ampersand replacement=&]
                                                        (F(G_1'→ X_0)→ F(G_1→ X_0) .[yshift=2pt]rμ[yshift=-2pt]r[swap] proj&.f),
                                                    
                                                    where μ:X×_X_0G_1→ X denotes the action of G on X.  
                                                    
                                                    The assignments given above clearly establish an equivalence because they are constructed as restrictions of Φ and Ψ to the appropriate full subcategories, but we find it useful to provide a direct proof of the correspondence.
                                                    
                                                    By construction, the groupoid associated to X/G' is
                                                    
                                                        G_X/G'=G×_G×_X_0G(G×_G/G'G)≃ G_G×_X_0G(G×_X_0G')≃ G',
                                                    
                                                    where the isomorphism holds by effectivity. 
                                                    
                                                    Conversely, if P is f-split by Q, then 
                                                    
                                                    
                                                    
                                                    
                                                    
                                                    
                                                    
                                                    
                                                    
                                                    
                                                    
                                                        P≃Ψ(Q)=Ψ(G/G_P)= (X×_X_0 G/G_P)/G≃ X/G_P.
                                                    
                                                    
                                                    It remains to show that G_P,1≃ S(X×_PX), i.e., 
                                                    
                                                        G_P≃[X→ P].
                                                    
                                                    By applying C_X to the pullback diagram defining G_P,1 above, and using the fact that the right adjoint C_X commutes with pullbacks, as well as the relations witnessing splitting of the objects involved, we obtain a pullback diagram
                                                    
                                                        [cross line/.style=preaction=draw=white, -,line width=4pt]
                                                        (m)[matrix of math nodes, row sep=1.8em, column sep=3.2em, text height=1.5ex, text depth=0.25ex]
                                                        	|(0i)|C_X(G_P,1)    |(1i)| f^*(X)×_f^*(P) f^*(X)
                                                           [.02em]
                                                        	|(0j)|f^*(X)    |(1j)| f^*(X)×_Xf^*(X)
                                                        ;
                                                        [->,font=,>=to, thin]
                                                        (0i) edge node[above,pos=0.5] (1i) edge[right hook->] node[left,pos=0.5] (0j) 
                                                        (0j) edge node[above,pos=0.5](𝕀, Δ∘proj) (1j) 
                                                        (1i) edge  node[right,pos=0.5] (1j)
                                                        ;
                                                     
                                                    which, using f^*X≃ X×_YX and the groupoid structure of  _f, simplifies to C_X(G_P,1)≃ X×_PX, whence G_P,1≃ S_X(X×_PX). 
                                                    
                                                    
                                                     §.§ Indexed categorical Galois theory
                                                    
                                                    
                                                    Suppose we are given
                                                    
                                                      *  a functor S:→;
                                                    
                                                      *  pseudo-functors K:^→ and L:^→;
                                                    
                                                      *  a pseudo-natural transformation α: K∘ S⇒ L; 
                                                    
                                                      *  a `precategorical decomposition' of a morphism Xf→Y in , i.e., a commutative diagram of morphisms of precategories
                                                    
                                                        [cross line/.style=preaction=draw=white, -,
                                                        line width=4pt]
                                                        (m)[matrix of math nodes, row sep=1.2em, column sep=1.2em, text height=1.5ex, text depth=0.25ex]
                                                        	|(i)|X       |(k)|Y
                                                          [.02em]
                                                        			    |(j)|   
                                                        ;
                                                        [->,font=,>=to, thin]
                                                        (i) edge node[left,pos=0.6]i (j) edge node[above,pos=0.5]f (k) 
                                                        (j) edge node[right,pos=0.4]π (k) 
                                                        ;
                                                     
                                                    with i_0=𝕀_X, and
                                                    such that the components α_C_0, α_C_1 and α_C_2 are full and faithful.
                                                    
                                                    
                                                    
                                                    If (f,(i,,π)) is of effective descent with respect to L, we have an equivalence of categories
                                                    
                                                        _α(f)≃ K^S∘.
                                                    
                                                    
                                                    § DIFFERENTIAL ALGEBRAIC GEOMETRY
                                                    
                                                    
                                                     §.§ Differential schemes
                                                    
                                                    
                                                    A differential scheme
                                                        (X, (_X,δ_X))
                                                     is a differentially ringed space where (X,_X) is a scheme, and δ_X∈(_X,_X).
                                                    
                                                    A morphism of differential schemes
                                                        (f,φ):(X,(_X,δ_X))→ (Y,(_Y,δ_Y))
                                                    
                                                    is a morphism of differentially ringed spaces which is also a scheme morphism, i.e., it is a scheme morphism (f,φ):(X,_X)→ (Y,_Y) whose structure homomorphism is a morphism of differential rings φ:(_Y,δ_Y)→ f_*(_X,δ_X), or, equivalently, its mate is a morphism of differential rings φ^♯:f^*(_Y,δ_Y)→ (_X,δ_X). 
                                                    
                                                    They constitute the category of differential schemes denoted
                                                    
                                                        .
                                                    
                                                    We have an obvious functor 
                                                    
                                                        C:→,    (X,_X)↦ (X,(_X,0))
                                                    
                                                    that turns a scheme into a differential scheme with the trivial derivation 0. 
                                                    
                                                    Given a scheme morphism (f,φ):(X,_X)→ (S,_S) and an _X-module , we say that an additive morphism D:_X→ is an S-derivation of _X to  if it is an f^*_S-derivation via φ^♯:f^*_S→_X, or, equivalently, if D_x:_X,x→_x is an _S,f(x)-derivation via φ^♯_x:_S,f(x)→_X,x for every x∈ X. The collection of all S-derivations of _X to  is denoted
                                                    
                                                        _S(_X,).
                                                    
                                                    
                                                    A differential scheme (X,(_X,δ_X)) equipped with a scheme morphism (f,φ):(X,_X)→ (S,_S) is called an S-differential scheme provided δ_X∈_S(_X,_X). 
                                                    
                                                    Clearly, an S-differential scheme (X,δ_X) is a morphism of differential schemes (X,δ_X)→ C(S)=(S,0). Thus, the category of S-differential schemes is the slice category
                                                    
                                                        _S=_ C(S).
                                                    
                                                    
                                                     §.§ Differential schemes and vector fields
                                                    
                                                    
                                                    Let (f,φ):X→ S be a scheme morphism. By <cit.>, the universal differential
                                                    
                                                        d_X/S:_X→Ω^1_X/S
                                                    
                                                    is an S-derivation, and composing with d_X/S induces an isomorphism of Γ(X,_X)-modules
                                                    
                                                        __X(Ω^1_X/S,)∼→_S(_X,)
                                                    
                                                    for any _X-module . 
                                                    
                                                    Let 
                                                        T_X/S=𝐕(Ω^1_X/S)
                                                     be the tangent bundle of X relative to S, defined as the vector bundle associated to the quasi-coherent _X-module Ω^1_X/S (i.e., the spectrum of the quasi-coherent _X-algebra Sym(Ω^1_X/S)) see <cit.>. 
                                                    
                                                    For a point x∈ X, the tangent space of X at x relative to S is defined <cit.> as
                                                    
                                                        T_X/S(x)=(T_X/S×_X(κ(x)))(κ(x))≃_κ(x)(Ω^1_X/S⊗__Xκ(x),κ(x)).
                                                    
                                                    
                                                    Choosing an S-derivation δ_X∈_S(_X,_X), we therefore obtain a morphism of _X-modules
                                                    
                                                        Ω^1_X/S→_X,
                                                    
                                                    which yields a section X→ T_X/S of the relative tangent bundle, that we think of as a vector field on X relative to S.
                                                    
                                                    Pointwise, for every x∈ X, pulling back the _X-modules to κ(x)-modules via the morphism (κ(x))→ X yields a morphism
                                                    
                                                        Ω^1_X/S⊗__Xκ(x)→κ(x),
                                                    
                                                    an element of T_X/S(x). This construction is in line with the classical notion of vector field in differential geometry as a map that sends a point to a vector in the corresponding tangent space. 
                                                    
                                                    A point x∈ X is a leaf for the vector field given by δ_X if the corresponding tangent vector at x is 0. 
                                                    
                                                    
                                                    
                                                    
                                                    
                                                     §.§ Differential spectra and affine differential schemes
                                                    
                                                    
                                                    The spectrum 
                                                    of the underlying ring of a differential ring (A,δ_A) carries a natural structure of a differential scheme
                                                    
                                                        (A,δ_A)=((A), (_(A),δ_(A))).
                                                    
                                                    Indeed, 
                                                        δ_(A):_(A)→_(A)
                                                     is determined on basic opens D(f) in (A), for f∈ A by setting
                                                    
                                                        δ_(A), D(f): _(A)(D(f))=A_f→ A_f=_(A)(D(f)),    a/f↦δ_A(a)f-aδ_A(f)/f^2.
                                                    
                                                    This construction extends to a functor 
                                                    
                                                        :^→,
                                                    
                                                    right adjoint to the global sections functor 
                                                    
                                                        Γ:(X,(_X,δ_X))↦ (_X(X),δ_X,X)
                                                    
                                                    as the diagram
                                                    
                                                        [cross line/.style=preaction=draw=white, -,
                                                        line width=3pt]
                                                        (m)[matrix of math nodes, minimum size=1.7em,
                                                        inner sep=0pt, 
                                                        row sep=3.3em, column sep=1em, text height=1.5ex, text depth=0.25ex]
                                                          
                                                          |(dc)|
                                                        
                                                         |(c)|^
                                                        ;
                                                        [->,font=,>=to, thin]
                                                        
                                                        
                                                        
                                                        (dc) edge [bend right=30] node (ss) [left]Γ (c)
                                                        (c) edge [bend right=30] node (ps) [right] (dc)
                                                        
                                                        (ss) edge[draw=none] node⊣ (ps)
                                                        ;
                                                    
                                                    depicts. 
                                                    
                                                    A differential scheme is affine, if it is isomorphic to a spectrum of a differential ring. Clearly, the category of affine differential schemes is anti-equivalent to the category of differential rings, i.e., 
                                                    
                                                        ≃^.
                                                    
                                                    
                                                     §.§ Differential schemes as precategory actions
                                                    
                                                    
                                                    Given a scheme S, let us consider the diagram of quasi-coherent _S-algebras
                                                    
                                                        (m)[matrix of math nodes, row sep=0em, column sep=3em, text height=1.5ex, text depth=0.25ex]
                                                         
                                                        |(2)|_S[ϵ_0,ϵ_1]/(ϵ_0^2,ϵ_0ϵ_1,ϵ_1^2)    [1em] |(1)|_S[ϵ]/(ϵ^2)   [1em] |(0)|_S
                                                        ; 
                                                        [->,font=,>=to, thin]
                                                        ([yshift=1em]1.west) edge node[above=-2pt]ϵ_0ϵ ([yshift=1em]2.east) 
                                                        (1) edge node[above=-2pt]ϵ_0+ϵ_1ϵ (2)
                                                        ([yshift=-1em]1.west) edge node[above=-2pt]ϵ_1ϵ ([yshift=-1em]2.east) 
                                                        ([yshift=1em]0.west) edge node[above=-2pt]𝕀+0 ([yshift=1em]1.east) 
                                                        (1)  edge node[above=-2pt]ϵ↦ 0 (0) 
                                                        ([yshift=-1em]0.west) edge node[above=-2pt]𝕀+0 ([yshift=-1em]1.east) 
                                                        
                                                         
                                                        ;
                                                    
                                                    Applying the spectrum of quasi-coherent _S-algebras functor <cit.>, we obtain a precategory (S)
                                                        (m)[matrix of math nodes, row sep=0em, column sep=3em, text height=1.5ex, text depth=0.25ex]
                                                         
                                                        |(2)|S_2    [1em] |(1)|S_1   [1em] |(0)|S_0
                                                        ; 
                                                        [->,font=,>=to, thin]
                                                        ([yshift=1em]2.east) edge node[above=-2pt]r_0 ([yshift=1em]1.west) 
                                                        (2) edge node[above=-2pt]m (1)
                                                        ([yshift=-1em]2.east) edge node[above=-2pt]r_1 ([yshift=-1em]1.west) 
                                                        ([yshift=1em]1.east) edge node[above=-2pt]d_0 ([yshift=1em]0.west) 
                                                        (0)  edge node[above=-2pt]n (1) 
                                                        ([yshift=-1em]1.east) edge node[above=-2pt]d_1 ([yshift=-1em]0.west) 
                                                        
                                                         
                                                        ;
                                                    
                                                    in _ S consisting of schemes affine over S_0=S, and the underlying morphisms of topological spaces are all identities. 
                                                    
                                                    Note, if we write 
                                                    
                                                        ()=(()),
                                                    
                                                    then 
                                                    
                                                        (S)=()×S=(S×([ϵ_0,ϵ_1]/(ϵ_0^2,ϵ_0ϵ_1,ϵ_1^2)),S×([ϵ]/(ϵ^2)), S).
                                                    
                                                    The category of S-differential schemes is equivalent to the category of (S)-actions in _ S (cf. <ref>), 
                                                    
                                                        _S≃ (_ S)^(S).
                                                    
                                                    Using <ref>, an action is determined by a scheme morphism X_0=X→ S=S_0 and an S_1-automorphism α:X_1→ X_1, where X_1=X×_S_0S_1 satisfying n^* α=𝕀 and the cocycle condition. Equivalently, it is given by an _S[ϵ]/(ϵ^2)-automorphism of _X_1=_X[ϵ]/(ϵ^2) which, tensored by the augmentation morphism η_S:_S[ϵ]/(ϵ^2)→_S gives 𝕀__X, and it follows that it must be of the form 
                                                    
                                                        𝕀__X[ϵ]/(ϵ^2)+ϵδ_α∘η_X,
                                                    
                                                    where δ_α∈_S(_X,_X) and η_X:_X[ϵ]/(ϵ^2)→_X is the augmentation homomorphism. Note that such a morphism always has an inverse id__X[ϵ]/(ϵ^2)-ϵδ_α∘η_X.
                                                    
                                                    The cocycle condition is trivially satisfied/superfluous, since r_i^*α is determined by the automorphism of  
                                                    _X[ϵ_0,ϵ_1]/(ϵ_0^2,ϵ_0ϵ_1,ϵ_1^2) given by 𝕀+ϵ_i δ_α∘η_X,2 for i=0,1, and m^*α is given by 𝕀+(ϵ_0+ϵ_1)δ_α∘η_X,2, where we wrote η_X,2 for the augmentation homomorphism _X[ϵ_0,ϵ_1]/(ϵ_0^2,ϵ_0ϵ_1,ϵ_1^2)→_X, and these morphisms compose in the way prescribed by the cocycle condition. 
                                                    
                                                    Thus, an action uniquely determines a derivation, and we obtain an S-differential scheme (X,(_X,δ_α)). 
                                                    
                                                    Conversely, given an S-differential scheme (X,(_X,δ_X)), we can create a (S)-action from the diagram
                                                    
                                                        (m)[matrix of math nodes, row sep=0em, column sep=3em, text height=1.5ex, text depth=0.25ex]
                                                         
                                                        |(2)|_X[ϵ_0,ϵ_1]/(ϵ_0^2,ϵ_0ϵ_1,ϵ_1^2)    [1em] |(1)|_X[ϵ]/(ϵ^2)   [1em] |(0)|_X
                                                        ; 
                                                        [->,font=,>=to, thin]
                                                        ([yshift=1em]1.west) edge node[above=-2pt]ϵ_0ϵ ([yshift=1em]2.east) 
                                                        (1) edge node[above=-2pt]ϵ_0+ϵ_1ϵ (2)
                                                        ([yshift=-1em]1.west) edge node[above=-2pt]ϵ_1ϵ ([yshift=-1em]2.east) 
                                                        ([yshift=1em]0.west) edge node[above=-2pt]𝕀+0 ([yshift=1em]1.east) 
                                                        (1)  edge node[above=-2pt]ϵ↦ 0 (0) 
                                                        ([yshift=-1em]0.west) edge node[above=-2pt]𝕀+ϵδ_X ([yshift=-1em]1.east) 
                                                        
                                                         
                                                        ;
                                                    
                                                    by applying the spectrum of quasi-coherent _P-algebras functor. All the resulting morphisms are cartesian, so we obtain an action by <ref>.
                                                    
                                                    The scheme of connected components is
                                                    
                                                        π_0((S))=S
                                                    
                                                    because it is calculated as the coequaliser of morphisms d_0, d_1 in (S), which agree. Hence, writing 
                                                    
                                                        η:(S)→S
                                                    
                                                    for the associated morphism of precategories,  the functor 
                                                    
                                                        C:_ S→_S,     T↦ (T,0)
                                                    
                                                    corresponds, through the equivalence <ref>, to the functor
                                                    
                                                        η^*: _ S→ (_ S)^(S).
                                                    
                                                    Let 
                                                    
                                                        :_ S^→
                                                    
                                                    be a pseudofunctor on the category of S-schemes. 
                                                    
                                                    We define a pseudofunctor
                                                    
                                                        δ:_S^→,    δ(X,δ_X)=^,
                                                    
                                                    where ∈(_ S) is the (S)-action corresponding to X, considered as a precategory by <ref>.
                                                    
                                                    Assume that the indexed data :_ S^→ is a full sub-pseudofunctor of the self-indexing of _ S, 
                                                    i.e., for Y∈_ S, (Y) is a full subcategory of _ Y. We can think of  as being associated to a class of morphisms in _ S that is stable under pullback, 
                                                    Then, the indexed data δ is a natural differential analogue of that class in the sense that, for (X,δ_X)∈_S,
                                                    
                                                        δ(X,δ_X)={(P,δ_P)→ (X,δ_X)∈_S: P→ X ∈(X)},
                                                    
                                                    i.e., it consists of those morphisms of differential schemes with target (X,δ_X) whose underlying morphism of schemes belongs to the class . 
                                                    
                                                    If (Y) is the category of quasi-projective morphisms Q→ Y, then, given an S-differential scheme (X,δ_S), the fibre δ(X,δ_X) is the category of S-differential scheme morphisms (P,δ_P)→ (X,δ_X) such that P→ X is quasi-projective.
                                                    
                                                    
                                                     §.§ Categorical scheme of leaves
                                                    
                                                    
                                                    The following definition is inspired by the notion of `espace des feuilles grossier' that appears in Bardavid's thesis <cit.>.
                                                    
                                                    
                                                    Let (X,δ_X) be an S-differential scheme. A scheme T∈_ S, together with a morphism of S-differential schemes η:(X,δ_X)→ C(T)=(T,0) is a categorical S-scheme of leaves, if it is a universal morphism from (X,δ_X) to the functor C in the sense that any other morphism of S-differential schemes η':(X,δ_X)→ C(T')=(T',0) factors through η,  i.e., there is a unique S-morphism f:T→ T' such that η'=C(f)∘η. In other words, the solid part of the diagram
                                                    
                                                        [cross line/.style=preaction=draw=white, -,
                                                        line width=4pt]
                                                        (m)[matrix of math nodes, row sep=1.2em, column sep=1.2em, text height=1.5ex, text depth=0.25ex]
                                                        	|(i)|(X,δ_X)       |(k)|C(T)
                                                          [.02em]
                                                        			    |(j)|C(T')   
                                                        ;
                                                        [->,font=,>=to, thin]
                                                        (i) edge node[below left,pos=0.5]η' (j) edge node[above,pos=0.5]η (k) 
                                                        (k) edge[style=dashed] node[below right,pos=0.5]∃! C(f) (j) 
                                                        ;
                                                     
                                                    can be uniquely completed by a dashed arrow to a commutative diagram.
                                                    
                                                    The categorical scheme of leaves need not exist for an arbitrary differential scheme (X,δ_X), but when it does, it is unique up to unique isomorphism, and we denote it
                                                    
                                                        π_0(X).
                                                    
                                                    Given a differential ring (A,δ_A), it is difficult to speculate whether
                                                    π_0((A,δ_A))
                                                    exists. Note that the scheme ((A,δ_A)) only satisfies the universal property required of a categorical scheme of leaves in the category of affine differential schemes, but not necessarily in the category of all differential schemes. 
                                                    
                                                    Let ∈(_ S) be the (S)-action associated to an S-differential scheme (X,δ), considered as a precategory. Then, the categorical scheme of leaves of (X,δ_X) is isomorphic to the scheme of connected components
                                                    
                                                        [cramped, column sep=normal, ampersand replacement=&]
                                                        π_0()=(X_1 .[yshift=2pt]rd_0[yshift=-2pt]r[swap]d_1&.X_0)
                                                    
                                                    of  in the category _ S, 
                                                    whenever either of the objects exist. In other words, 
                                                    
                                                        π_0()≃π_0(X).
                                                    
                                                    Using <ref>, we obtain that
                                                    
                                                        _S((X,δ_X),C(T'))≃ (_ S)^(S)(,η^*T')≃_ S(π_0(), T'),
                                                    
                                                    whenever π_0() exists, hence π_0() satisfies the universal property of the categorical scheme of leaves. 
                                                    
                                                    
                                                     §.§ Simple differential schemes
                                                    
                                                    An S-differential scheme (X,δ_X) is simple with respect to the pseudofunctor :_ S→, if its categorical scheme of leaves π_0(X) exists and the coequaliser
                                                    
                                                        [cramped, column sep=normal, ampersand replacement=&]
                                                        X_1[yshift=2pt]rd_0[yshift=-2pt]r[swap]d_1&X_0rη&π_0(X)
                                                    
                                                    is universal for  in the sense of <ref>, i.e., if  has a -universal scheme of connected components. 
                                                    
                                                    If an S-differential scheme (X,δ_X) is simple for  with categorical scheme of leaves η_X:X→π_0(X), the canonical functor
                                                    
                                                        C_X:(π_0(X))≃^π_0()η_^*⟶^≃δ(X,δ_X)
                                                    
                                                    is fully faithful, where we wrote η_:→π_0() for the associated morphism of precategories inducing the pullback of precategory actions η_^* as in <ref>.
                                                    
                                                    Using <ref>, we have that the arrow in the above diagram
                                                    
                                                    
                                                    
                                                    is fully faithful, so the composite is too.
                                                    
                                                    When  is a sub-pseudocategory of the self-indexing of _ S as in <ref>, we obtain that
                                                    
                                                        C_X(Q→π_0(X))=(X,δ_X)×_C(π_0(X))C(Q),
                                                    
                                                    so C_X agrees with the functor introduced in the classical Galois context <ref>.
                                                    
                                                    Let (A,δ) be a differential ring with the ring of constants k such that k→ A is universally injective<cit.>. For any k-algebra R, we have that 
                                                    
                                                        ((A,δ)⊗_(k,0)(R,0))≃ R.
                                                    
                                                    Since k→ A is universally injective, the short exact sequence of k-modules
                                                    
                                                        0→ k→ Aδ→(δ)→ 0
                                                    
                                                    is universally exact, so the base change via k→ R 
                                                    yields an exact sequence of R-modules
                                                    
                                                        0→ R→ A⊗_kRδ⊗𝕀_E→(δ)⊗_kR,
                                                    
                                                    whence R=(δ⊗𝕀)=((A,δ)⊗_(k,0)(R,0)). 
                                                    
                                                    In the situation of <ref>, we have the following.
                                                    
                                                      *  If k→ A is faithfully flat, then it is universally injective <cit.>.
                                                    
                                                      *  If (δ) is a flat k-module, then 0→ k→ Aδ→(δ)→ 0 is universally flat <cit.>.
                                                    
                                                      *  The above short exact sequence is universally exact if and only if it is a directed colimit of split short exact sequences
                                                    
                                                        0→ k→ A_i→ D_i→ 0
                                                    
                                                    with D_i finitely presented <cit.>.  
                                                    
                                                    If a differential scheme (X,δ_X) is simple with respect to  Zariski open immersions, with categorical scheme of leaves given as coequaliser from <ref>, then
                                                    
                                                        η_*(_X,δ_X)=_π_0(X).
                                                    
                                                    From the coequaliser diagram of schemes, we obtain a diagram of structure sheaves
                                                    
                                                        [cramped, column sep=normal, ampersand replacement=&]
                                                        η_*_X_1&η_*_X_0[yshift=2pt]l[swap]η_*δ_0[yshift=-2pt]lη_*δ_1&_π_0(X)l
                                                    
                                                    which yields a unique morphism 
                                                    
                                                        _π_0(X)→(η_*δ_0,η_*δ_1)≃η_*(δ_0,δ_1)=η_*(_X,δ_X).
                                                    
                                                    By assumption, for every Zariski open V↪π_0(X), the diagram
                                                    
                                                        [cramped, column sep=normal, ampersand replacement=&]
                                                        U_1[yshift=2pt]r[yshift=-2pt]r&Ur& V
                                                    
                                                    with U=η^-1V and U_1 the pullback of V to X_1, remains a coequaliser. Substituting V=(A) in the above equaliser of sheaves, we obtain a ring homomorphism
                                                    
                                                        A→(_U(U)_U_1(U)).
                                                    
                                                    The universal property of V being coequaliser, applied to varying affine schemes, yields that in fact 
                                                        _π_0(X)(V)=A≃(_U(U)_U_1(U)=η_*(_X)(V),
                                                    
                                                    for an arbitrary affine open V,  whence we obtain the desired conclusion.  
                                                    
                                                    
                                                    Let (X,δ_X) be an S-differential scheme with categorical scheme of leaves S such that the canonical morphism
                                                    
                                                        η:X→π_0(X)=S
                                                    
                                                    is fpqc and universally open (for example, fppf), and, for every Zariski open U in X, we have
                                                    
                                                        π_0(U)=η(U).
                                                    
                                                    Then (X,δ_X) is simple with respect to S-scheme morphisms. 
                                                    
                                                    Writing  for the precategory in _ S associated to (X,δ_X), our first assumption means that the diagram 
                                                    
                                                        [cramped, column sep=normal, ampersand replacement=&]
                                                        X_1[yshift=2pt]rd_0[yshift=-2pt]r[swap]d_1&X_0r& S
                                                    
                                                    is a coequaliser. We need to show that it is universal for S-scheme morphisms, i.e., that the base change 
                                                    
                                                        (m)[matrix of math nodes, row sep=2em, column sep=2em, text height=1.5ex, text depth=0.25ex]
                                                         
                                                        |(u2)|X_1×_S Q    [1em] |(u1)|X_0×_S Q   [1em] |(u0)|Q
                                                        
                                                        |(2)|X_1    [1em] |(1)|X_0   [1em] |(0)|S
                                                        ; 
                                                        [->,font=,>=to, thin]
                                                        ([yshift=.2em]2.east) edge node[above=-2pt]d_0 ([yshift=.2em]1.west) 
                                                        ([yshift=-.2em]2.east) edge node[below=-2pt]d_1 ([yshift=-.2em]1.west) 
                                                        (1) edge node[above=-2pt]η (0) 
                                                        ([yshift=.2em]u2.east) edge node[above=-2pt] ([yshift=.2em]u1.west) 
                                                        ([yshift=-.2em]u2.east) edge node[below=-2pt] ([yshift=-.2em]u1.west) 
                                                        (u1) edge node[above=-2pt] (u0) 
                                                        (u2) edge (2)
                                                        (u1) edge (1)
                                                        (u0) edge node[right]q (0)
                                                        ;
                                                    
                                                    along any S-scheme morphism Q→ S remains a coequaliser. Indeed, let f:X_0×_S Q→ Y be a morphism of S-schemes that coequalises the two arrows from X_1×_SQ. Every point p∈ X_0×_S Q has an open affine neighbourhood U_p×_S_pQ_p, where U_p, S_p and Q_p are open affine with η(U_p)⊆ S_p and q(Q_p)⊆ S_p such that  f_p=f_U_p×_S_pQ_p factors through an affine open subset of Y. 
                                                    Since η is open, we may assume that η(U_p)= S_p. 
                                                    
                                                    As a base-change of a surjective morphism X_0→ S, the morphism X_0×_SQ→ Q is surjective, so, since  
                                                    U_p×_S_pQ_p cover X_0×_SQ, we obtain that Q_p cover Q. The morphisms d_0 and d_1 are identites on the underlying topological spaces of X_1 and X_0, and let us write  U_p,1=U_p×_S S_1 for the preimages of U_p in X_1.
                                                    
                                                    By assumption,
                                                    
                                                        [cramped, column sep=normal, ampersand replacement=&]
                                                        U_p,1[yshift=2pt]r[yshift=-2pt]r[swap]&U_pr& S_p
                                                    
                                                    is a coequaliser, and by <ref>, we obtain that
                                                    
                                                        [cramped, column sep=normal, ampersand replacement=&]
                                                        U_p,1×_SQ_i[yshift=2pt]r[yshift=-2pt]r[swap]&U_p×_S_p Q_prη_p& Q_p
                                                    
                                                    is a coequaliser in _ S_p. 
                                                    
                                                    Since f_p factors through an affine open in Y, there exists a unique h_p:Q_p→ Y such that 
                                                    
                                                        f_p=h_p∘η_p.
                                                    
                                                    Every point in Q_p∩ Q_p' has an affine open neighbourhood W which is standard open in Q_p and Q_p', so by the same affine argument applied to U_p×_S W and U_p'×_S W, we obtain that h_p_W=h_p'_W. 
                                                    
                                                    
                                                    Hence 
                                                        h_p_Q_p∩ Q_p'=h_p'_Q_p∩ Q_p',
                                                    
                                                    so the h_p can be glued into a morphism 
                                                    
                                                        h:Q→ Y
                                                    
                                                    verifying the universal property of coequaliser for Q. 
                                                    
                                                    Let k be a field of characteristic 0, let S=(k), and let (X,δ_X) be an integral S-differential scheme whose only leaf is the generic point (i.e., it is simple in the sense of <cit.>). Then (X,δ_X) is simple with respect to S-scheme morphisms. 
                                                    
                                                    Bardavid has shown <cit.> that S is the categorical scheme of leaves of (X,δ_X) and that S remains the categorical scheme of leaves for any open subscheme of X, so we may apply <ref>.
                                                    
                                                    
                                                     §.§ Polarised differential scheme morphisms
                                                    
                                                    The category of polarised quasi-projective morphisms has objects 
                                                    
                                                        (P,_P)p→U,
                                                    
                                                    consisting of a scheme morphism p:P→ U and a p-relatively ample invertible _P-module _P. 
                                                    
                                                    A morphism (f,α) between polarised quasi-projective morphisms (P,_P)p→U and (Q,_Q)q→V is a commutative diagram
                                                    
                                                        (m)[matrix of math nodes, row sep=2em, column sep=2em, text height=1.9ex, text depth=0.25ex]
                                                         
                                                         |(1)|(P,_P)    |(2)|(Q,_Q)
                                                        
                                                         |(l1)|U    |(l2)|V
                                                        ; 
                                                        [->,font=,>=to, thin]
                                                        (1) edge node[above](f,α) (2) edge node[left]p  (l1)
                                                        (2) edge node[right]q (l2) 
                                                        (l1) edge node[above]  (l2);
                                                    
                                                    consisting of a scheme morphism f:P→ Q that makes the underlying scheme diagram commutative, together with an _P-module isomorphismα:f^*_Q→_P.
                                                    
                                                    If (g,β) is another morphism from (Q,_Q)q→V to (R,_R)r→W, the composite is computed as
                                                    
                                                        (g,β)∘(f,α)=(g∘ f, α∘ f^*β∘coherence),
                                                     
                                                    where, more precisely, the _P-module isomorphism is given as the composite
                                                    
                                                        (g∘ f)^*_R≃ f^*g^*_Rf^*β→f^*_Qα→_P.
                                                    
                                                    The category of polarised quasi-projective morphisms has a natural codomain fibration over the category of schemes.
                                                    In this subsection, we write
                                                     
                                                        :_ S^→
                                                     
                                                     for the associated pseudofunctor, i.e., (V) is the category consisting of pairs
                                                    
                                                        (Qq→V,),
                                                    
                                                    where q is a morphism of finite type and  is an invertible q-ample _Q-sheaf.
                                                    
                                                    We will write 
                                                        =Self(_ S)
                                                     for the self-indexing of the category of S-schemes over itself, and 
                                                    
                                                        U:→
                                                    
                                                    for the natural forgetful functor. 
                                                    
                                                    The category of polarised quasi-projective differential scheme morphisms is
                                                    the fibered category 
                                                    
                                                        δ
                                                    
                                                    obtained by construction <ref>.
                                                    
                                                    For an S-differential scheme (X,δ_X), the category of polarised quasi-projective differential scheme morphisms with codomain (X,δ) is equivalent to the category of pairs 
                                                        ((P,δ_P)p→ (X,δ_X),(_P,δ__P)),
                                                    
                                                    where p:(P,δ_P)→ (X,δ_X) is a morphism of S-differential schemes, and (_P,δ__P) is an invertible (_P,δ_P)-module with _P relatively ample with respect to the underlying scheme morphism P→ X. 
                                                    
                                                    An object P∈δ(X,δ)=^ gives rise to a diagram 
                                                    
                                                        [cross line/.style=preaction=draw=white, -,line width=4pt, proj/.style=dotted,-]
                                                         #1#2
                                                         ([yshift=.4em]#1.east) edge 
                                                         ([yshift=.4em]#2.west) 
                                                        (#1) edge 
                                                        (#2)
                                                        ([yshift=-.4em]#1.east) edge 
                                                        ([yshift=-.4em]#2.west) #1#2
                                                        ([yshift=.4em]#1.east) edge 
                                                        ([yshift=.4em]#2.west) 
                                                        (#2)  edge 
                                                        (#1) 
                                                        ([yshift=-.4em]#1.east) edge 
                                                        ([yshift=-.4em]#2.west) #1#2
                                                         ([xshift=.4em]#1.south) edge 
                                                         ([xshift=.4em]#2.north) 
                                                        (#1) edge 
                                                        (#2)
                                                        ([xshift=-.4em]#1.south) edge 
                                                        ([xshift=-.4em]#2.north) #1#2
                                                        ([xshift=.4em]#1.south) edge
                                                        ([xshift=.4em]#2.north) 
                                                        (#2)  edge 
                                                        (#1) 
                                                        ([xshift=-.4em]#1.south) edge[cross line] 
                                                        ([xshift=-.4em]#2.north) (m)[matrix of math nodes, row sep=2em, column sep=1em,  
                                                        text height=1.2ex, text depth=0.25ex]
                                                        
                                                        |(P22)|(P_2,_P_2)   [1em] |(P12)|(P_1,_P_1)   [1em]	 |(P02)|(P_0,_P_0)
                                                        
                                                        |(P21)|X_2   [1em] |(P11)|X_1   [1em]	 |(P01)|X_0
                                                        
                                                        |(P20)|S_2   [1em] |(P10)|S_1   [1em]	 |(P00)|S_0
                                                        ;
                                                        [->,font=,>=to, thin,inner sep=1pt]
                                                        P22P12P12P02P21P11P11P01P20P10P10P00
                                                        
                                                        (P22) edge (P21)
                                                        (P21) edge (P20)
                                                        
                                                        (P12) edge (P11)
                                                        (P11) edge (P10)
                                                        
                                                        (P02) edge (P01)
                                                        (P01) edge (P00)
                                                        
                                                        ;
                                                     
                                                    where the bottom level is the precategory (S) as in <ref>, and all the arrows in the top and middle levels are cartesian. Hence, forgetting the polarisations, we see that P is also an S-differential scheme, whose differential structure is given by an S_1-automorphism P_1f→ P_1, which is identity on the underlying space, and it is given by an automorphism φ:_P_0[ϵ]/(ϵ^2)→_P_0[ϵ]/(ϵ^2) associated to an S-derivation δ_P on _P as in <ref>, and the action on invertible sheaves is given by an isomorphism (f,α):(p_1,d_P,0^*_0)→ (p_1,d_P,0^*_0), i.e., by an isomorphism
                                                    
                                                        α:_P_0[ϵ]/(ϵ^2) _φ⊗__P_0[ϵ]/(ϵ^2)_0[ϵ]/(ϵ^2)→_0[ϵ]/(ϵ^2).
                                                    
                                                    The identity section requirement yields that 
                                                        α=𝕀+ϵδ_α∘η_L,
                                                     where η_L: _0[ϵ]/(ϵ^2)→_0 is the augmentation morphism, and δ_α:_0→_0 makes _0 into an (_P,δ_P)-module.
                                                    
                                                    Let (X,δ_X) be a differential scheme which is simple with respect to Zariski open immersions, with the categorical scheme of leaves given by the coequaliser
                                                    
                                                        [cramped, column sep=normal, ampersand replacement=&]
                                                        X_1[yshift=2pt]rd_0[yshift=-2pt]r[swap]d_1&X_0rη&π_0(X).
                                                    
                                                    Consider invertible sheaves   and ' on π_0(X) and  an _X_0-module isomorphism 
                                                    
                                                        α:η^*→η^*'
                                                    
                                                    making the diagram
                                                    
                                                        [cross line/.style=preaction=draw=white, -,
                                                        line width=4pt]
                                                        (m)[matrix of math nodes, row sep=1.8em, column sep=1.8em, text height=1.5ex, text depth=0.25ex]
                                                        	|(0i)|d_0^*η^*    |(1i)| d_0^*η^*'
                                                           [.02em]
                                                        	|(0j)|d_1^*η^*    |(1j)| d_1^*η^*'
                                                        ;
                                                        [->,font=,>=to, thin]
                                                        (0i) edge node[above,pos=0.5]d_0^*α (1i) edge node[below=-2pt,pos=0.5,sloped]∼ (0j) 
                                                        (0j) edge node[above,pos=0.5]d_1^*α (1j) 
                                                        (1i) edge node[above=-2pt,pos=0.5,sloped]∼ (1j)
                                                        ;
                                                     
                                                    commutative. 
                                                    
                                                    Then, there exists a unique isomorphism σ:→' such that 
                                                        α=η^*σ.
                                                    
                                                    Let V_i constitute an open cover of π_0(X) such that, for every i, _V_i and '_V_i are free of rank 1. Then U_i=η^-1(V_i) cover X_0 and η^*_U_i≃_U_ie_i, η^*'_U_i≃_U_ie'_i are free rank 1, for some e_i and e_i', so α_U_i is given as multiplication by some a_i∈_X(U_i)^×. 
                                                    
                                                    By identifying 
                                                    
                                                        d_0^*η^*_U_i=_U_i[ϵ]/(ϵ^2)⊗__U_i_U_ie_i ≃_U_i(1⊗ e_i)⊕_U_i(ϵ⊗ e_i),
                                                    
                                                    and similarly for d_0^*η^*'_U_i, the matrix of d_0^*α_U_i in the pair of bases
                                                    ((1⊗ e_i),(ϵ⊗ e_i)), ((1⊗ e'_i),(ϵ⊗ e'_i)) becomes
                                                    
                                                        [ a_i   0;   0 a_i ].
                                                    
                                                    In the same pair of bases, the matrix of d_1^*α_U_i is
                                                    
                                                        [      a_i        0; δ_X(a_i)      a_i ].
                                                    
                                                    By the assumption that d_0^*α and d_1^*α agree up to coherences, we conclude that δ_X(a_i)=0, i.e., by <ref>, 
                                                    
                                                        a_i∈(_X)(U_i)=η_*(_X)(V_i)≃_π_0(X)(V_i),
                                                    
                                                    whence it gives an isomorphism _V_i→'_V_i. 
                                                    
                                                    By the same argument, these isomorphisms agree on the intersections V_i∩ V_j and hence glue uniquely to an isomorphism →' with the desired property.
                                                    
                                                    The forgetful functor 
                                                    
                                                        δ→δ
                                                    
                                                    reflects coequalisers associated with connected components of differential schemes which are simple with respect to Zariski open immersions. 
                                                    
                                                    Let (X,δ_X)∈δ be simple for Zariski open immersions, and let (p_2,p_1,p_0)∈δ(X,δ_X) be such that, in the diagram
                                                    
                                                        (m)[matrix of math nodes, row sep=2em, column sep=2em, text height=1.5ex, text depth=0.25ex]
                                                         
                                                        |(u2)|(P_1,_P_1)    [1em] |(u1)|(P_0,_P_0)   [1em] |(u0)|(Q,_Q)
                                                        
                                                        |(2)|X_1    [1em] |(1)|X_0   [1em] |(0)|π_0(X)
                                                        ; 
                                                        [->,font=,>=to, thin]
                                                        ([yshift=.2em]2.east) edge node[above=-2pt] ([yshift=.2em]1.west) 
                                                        ([yshift=-.2em]2.east) edge node[below=-2pt] ([yshift=-.2em]1.west) 
                                                        (1) edge node[above=-2pt] (0) 
                                                        ([yshift=.2em]u2.east) edge node[above=-2pt](d_0,δ_0) ([yshift=.2em]u1.west) 
                                                        ([yshift=-.2em]u2.east) edge node[below=-2pt](d_1,δ_1) ([yshift=-.2em]u1.west) 
                                                        (u1) edge node[above=-2pt](r,ρ) (u0) 
                                                        (u2) edge node[left]p_1 (2)
                                                        (u1) edge node[left]p_0 (1)
                                                        (u0) edge (0)
                                                        ;
                                                    
                                                    we have that (r,ρ) coequalises (d_0,δ_0) and (d_1,δ_1), and the underlying diagram of scheme morphisms is a coequaliser. 
                                                    
                                                    We need to show that the diagram is a coequaliser of polarised quasi-projective morphisms. Let (h,χ):(P_0,_P_0)→ (T,_T) be an arbitrary morphism that coequalises (d_0,δ_0) and (d_1,δ_1). Since the underlying diagram of scheme morphisms is a coequaliser, there is a unique morphism s:Q→ T such that s∘ r=h. 
                                                    
                                                    Let =s^*_T, '=_Q, and consider the isomorphism α:r^*→ r^*' given as the composite
                                                    
                                                        r^*s^*_T≃ h^*_Tχ→_P_0ρ^-1→r^*_Q.
                                                    
                                                    Conditions (h,χ)∘(d_0,δ_1)=(h,χ)∘(d_0,δ_1) and (r,ρ)∘(d_0,δ_1)=(r,ρ)∘(d_0,δ_1) show that the diagram
                                                    
                                                        (m)[matrix of math nodes, row sep=1.5em, column sep=2em, text height=1.5ex, text depth=0.25ex]
                                                         
                                                        |(u2)|d_0^*h^*_T    [1em] |(u1)|d_0^*_P_0   [1em] |(u0)|d_0^*r^*_Q
                                                             |(mid)|_P_1   
                                                        
                                                        |(2)|d_1^*h^*_T    [1em] |(1)|d_0^*_P_0   [1em] |(0)|d_1^*r^*_Q
                                                        ; 
                                                        [->,font=,>=to, thin]
                                                        (2) edge node[above=-2pt]d_1^*χ (1) 
                                                        (1) edge node[above=-2pt]d_1^*ρ^-1 (0) 
                                                        (u2) edge node[above=-2pt]d_0^*χ (u1) 
                                                        (u1) edge node[above=-2pt]d_0^*ρ^-1 (u0) 
                                                        (u2) edge node[below=-2pt,pos=0.5,sloped]∼(2)
                                                        (u1) edge node[left=-2pt]δ_0 (mid)
                                                        (1) edge node[left=-2pt]δ_1 (mid) 
                                                        (u0) edge node[above=-2pt,pos=0.5,sloped]∼(0)
                                                        ;
                                                    
                                                    commutes, so d_0^*α and d_1^*α agree up to coherences. By <ref>, we obtain an isomorphism 
                                                    
                                                        σ:s^*_T=→'=_Q
                                                    
                                                    such that α=r^*σ, and it follows that
                                                    
                                                        (s,σ)∘(r,ρ)=(h,χ),
                                                    
                                                    so (Q,_Q) has the universal property of a coequaliser, as required. 
                                                    
                                                    
                                                     §.§ Differential descent
                                                    
                                                    Let :_ S^→ be a pseudofunctor, and let 
                                                        f:(X,δ_X)→ (Y,δ_Y)
                                                     be a morphism of S-differential schemes such that
                                                    
                                                      *  the underlying morphism f_0:X→ Y is a morphism of effective descent for ;
                                                    
                                                      * f_1=d_0^*f_0: X_1→ Y_1 is a descent morphism for , where d_0:S_1→ S_0=S is the source morphism of precategory (S). 
                                                     
                                                    Then f is a morphism of effective descent for δ. 
                                                    
                                                    We consider the morphism of S-differential schemes as a morphism f:→𝕐 of (S)-actions, and apply <ref>, noting that the cocycle condition is superfluous for differential schemes so we may omit the condition on f_2. 
                                                    
                                                    Let f:(X,δ_X)→ (Y,δ_Y) be a morphism of differential schemes with 
                                                    
                                                    codomain the spectrum of a differential field and X quasi-compact.  Then f is a morphism of effective descent for the class of differential quasi-projective morphisms. 
                                                    
                                                    The underlying morphism f_0:X→ Y is (trivially) fpqc because the target is the spectrum of a field, and <ref> show that it is effective descent for quasi-projective scheme morphisms. Its base change f_1=d_0^*f is again fpqc, hence a morphism of descent for all scheme morphisms by <cit.>.
                                                    
                                                    Let f:(X,δ_X)→ (Y,δ_Y) be a morphism of differential schemes whose underlying scheme morphism is fpqc. 
                                                    
                                                    Then f is a morphism of effective descent for the class of differential polarised quasi-projective morphisms. 
                                                    
                                                    The underlying scheme morphism f_0:X→ Y is fpqc, so of effective descent for polarised quasi-projective morphisms by <cit.>, and the same holds for f_1 as the base change of f_0. 
                                                    
                                                    
                                                    § AFFINE PICARD-VESSIOT THEORY
                                                    
                                                    
                                                     §.§ Picard-Vessiot Galois theory for differential field extensions
                                                    
                                                    
                                                    In this section, we follow the Hopf-theoretic approach to Picard-Vessiot theory explained in <cit.>. 
                                                    
                                                    
                                                    An extension L/K of differential fields is Picard-Vessiot, if
                                                    
                                                      *  the extension L contains no new constants, i.e., L_0=K_0, and we write k for the common field of constants;
                                                    
                                                      *  there exists a differential K-subalgebra A of L such that Frac(A)=L and 
                                                    
                                                        H=(A⊗_KA)_0
                                                    
                                                    generates the left A-module A⊗_KA in the sense that
                                                    
                                                        (A⊗_KK)H=A⊗_KA.
                                                    
                                                    Such an A is called a Picard-Vessiot ring for the extension L/K.
                                                    
                                                    In the situation from the above definition, we have the following.
                                                    
                                                      * A is unique;
                                                    
                                                      * H is a Hopf algebra;
                                                    
                                                      *  there is a comodule structure θ:A→ A⊗_kH such that 
                                                    
                                                        A^ coH={a∈ A: θ(a)=a⊗ 1}= K,
                                                    
                                                    and
                                                    
                                                        A⊗_KA≃ A⊗_(k,0)(H,0).†
                                                    
                                                      *  The linear algebraic group G=^ PV(L/K)=(H) over k is called the Picard-Vessiot Galois group of L/K and we have that
                                                    
                                                        G(k)≃_(L/K).
                                                    [Classical Picard-Vessiot Galois correspondence]
                                                    Let L/K be a Picard-Vessiot extension.
                                                    
                                                    There is a one-to-one correspondence between intermediate differential field extensions and closed subgroups of the linear algebraic group ^ PV(L/K) given by
                                                    
                                                        M↦^ PV(L/M).
                                                    
                                                    Moreover, an intermediate field M in L/K is Picard-Vessiot over K if and only if ^ PV(L/M) is normal in ^ PV(L/K). In this case, we have that
                                                    
                                                        ^ PV(M/K)≃^ PV(L/K)/ ^ PV(L/M).
                                                    
                                                    
                                                     §.§ Janelidze's categorical framework for Picard-Vessiot theory
                                                    
                                                    In <cit.>, Janelidze makes the following choices for the classical categorical setup as in <ref>.
                                                    
                                                    
                                                      * ==^, the category of affine differential schemes;
                                                    
                                                      * ==^, the category of affine schemes;
                                                    
                                                      * S=^ is the functor of constants, i.e., S((A,δ))=((A,δ)), often written as S(X)=X_0;
                                                    
                                                      * C(Spec(R))=Spec(R,0) transforms a ring into a differential ring with a trivial derivation 0. 
                                                    
                                                      *  Given X∈, the functor C_X:_ X_0→_ X is given by
                                                    
                                                        C_X(Q→ X_0)=X×_C(X_0)C(Q)=(X,δ)×_(X_0,0)(Q,0).
                                                    
                                                      *  An object Pp→ Y in _ Y is split by a morphism Xf→Y if the natural morphism 
                                                    f^*p→ C_XS_X(f^*p) is an isomorphism, i.e., if
                                                    
                                                        X×_YP≃ X×_(X_0,0)((X×_YP)_0,0).
                                                    
                                                    Let A be the Picard-Vessiot ring for a differential field extension L/K, and let 
                                                        f=(K→ A):X=(A)→ Y=(K)
                                                     be the associated morphism in . 
                                                    
                                                    Then f is a morphism of Galois descent, the categorical Galois groupoid agrees with the Picard-Vessiot Galois group,
                                                    
                                                        G=[f]=^ PV(L/K),
                                                    
                                                    and X is a G-torsor over Y in the sense that
                                                    
                                                        X×_YX ≃ X×_(X_0,0)(G,0).
                                                    
                                                    There is an equivalence of categories 
                                                    
                                                        _Y[f]≃ [G,]
                                                    
                                                    between the category of objects Pp→ Y in _ Y split by f, in the sense that 
                                                        X×_YP=f^*(p)≃ C_X(q)=X×_X_0Q
                                                     for some Qq→ X_0 in _ X_0, and the category of G-actions in .
                                                    
                                                      *  The morphism f is faithfully flat (given as a spectrum of an algebra over a field) and hence it is a morphism of effective descent for affine morphisms in the sense of algebraic geometry. Using Benabou-Roubaud <cit.>, we obtain that the pullback functor f^* is monadic.
                                                    
                                                      *  Using <ref>, the counit S_XC_X→𝕀 is an isomorphism, or, equivalently, C_X is fully faithful.
                                                    
                                                      *  The morphism f is self-split by the property <ref> from <ref>.
                                                    
                                                    Categorical Galois theory stipulates that the object of morphisms of the groupoid [f] is
                                                    
                                                        G=[f]_1=S(f^*f)=(X×_YX)_0=((A⊗_KA)_0)=(H),
                                                    
                                                    while the object of objects 
                                                    
                                                        [f]_0=S(X)=X_0=(k)
                                                    
                                                    is a point, so we obtain a linear algebraic group G over k, exactly as in the Picard-Vessiot case. The torsor equation is precisely the self-splitting of f, written in terms of G, and the equivalence of categories follows from categorical Galois theory <ref> specialised to the framework <ref>.
                                                    [Algebraic group quotients and effective subgroups]
                                                    Let H→ G be a monomorphism/closed immersion of algebraic groups over a field k. Combining <cit.>, or, by using <cit.>, we obtain:
                                                    
                                                      * G is quasi-projective; 
                                                    
                                                      *  the quotient G/H is representable by a quasi-projective scheme over k; 
                                                    
                                                      *  the quotient morphism G→ G/H is faithfully flat; 
                                                    
                                                      *  we have 
                                                    
                                                        G× H≃ G×_G/HG.
                                                    
                                                    We deduce that all closed subgroups of an algebraic group over a field k are effective in the category of schemes with quasi-projective morphisms. 
                                                    
                                                    In the category of affine schemes, a closed subgroup H of an affine algebraic group G is effective if and only if G/H is affine. If H is a normal closed subgroup, then it is effective (<cit.>).
                                                    
                                                    With assumptions of <ref>, there is a one-to-one correspondence between split affine quotients of f:X→ Y
                                                    and effective subgroups of the linear algebraic group G=[f] which  takes
                                                    
                                                        [cross line/.style=preaction=draw=white, -,
                                                        line width=4pt]
                                                        (m)[matrix of math nodes, row sep=.9em, column sep=.5em, text height=1.5ex, text depth=0.25ex]
                                                            |(ij)| X   
                                                           [.02em]
                                                        	|(i)|P      
                                                          [.02em]
                                                        			    |(0)|Y   
                                                        ;
                                                        [->,font=,>=to, thin]
                                                        (ij) edge node[left,pos=0.2]
                                                            (i) edge node[right,pos=0.5]f (0) 
                                                            (i) edge node[below left,pos=0.5]p (0) 
                                                            ;
                                                         
                                                        to 
                                                        
                                                            [X→ P].
                                                        
                                                        Conversely, if G' is an effective subgroup in the sense that it is a closed subgroup such that the coset space G/G' has a structure of an affine scheme (<ref>), it corresponds to the quotient 
                                                        
                                                            X/G',
                                                        
                                                        which is f-split by the scheme G/G'. 
                                                        
                                                        Moreover, this correspondence restricts to a one-to-one correspondence between split quotients P such that P→ Y is Picard-Vessiot, and closed normal subgroups of G. In this case, 
                                                        
                                                            [P→ Y]≃[X→ Y]/[X→ P].
                                                        
                                                        The statement is a direct consequence of <ref> and <ref>.
                                                        
                                                          *  The equivalence of categories form of Picard-Vessiot theory from <ref> is new and as of yet unexplored in differential algebra. 
                                                        
                                                          *  The affine Galois correspondence from <ref> does not fully recover the classical Picard-Vessiot Galois correspondence <ref> because it only refers to effective subgroups of the Galois group, while the classical correspondence is for all closed subgroups. The `moreover' clause does recover the correspondence for Picard-Vessiot quotients and normal groups from <cit.>.
                                                        
                                                        
                                                        § CATEGORICAL GALOIS THEORY FOR DIFFERENTIAL SCHEMES
                                                        
                                                        
                                                         §.§ Indexed framework for scheme-theoretic Picard-Vessiot theory
                                                        
                                                        
                                                        
                                                        
                                                        
                                                        
                                                        
                                                        
                                                        
                                                        The indexed framework for pre-Picard-Vessiot differential Galois theory consists of the following choices for objects needed to apply <ref>. 
                                                        
                                                        
                                                          *  Let S be a base scheme, and let =_ S.
                                                        
                                                        
                                                          *  Let  be the category of S-differential schemes that have a categorical scheme of leaves. By this choice, we have a `categorical scheme of leaves' functor 
                                                        
                                                            π_0:→.
                                                        
                                                          *  Let :^→ be a pseudofunctor, which yields a pseudofunctor δ:^→ by <ref>.
                                                        
                                                        
                                                          *  Let C=C^: ∘π_0⇒δ be the pseudo-natural transformation whose (Z,δ_Z)-component is the canonical functor
                                                        
                                                            C_Z:(π_0(Z,δ_Z))→δ(Z,δ_Z)
                                                        
                                                        from <ref>.
                                                        nameOfYourChoice
                                                        
                                                        
                                                        
                                                        
                                                        
                                                        For Picard-Vessiot differential Galois theory, we choose the following additional structure.
                                                        
                                                          *  Let :^→ be a full sub-pseudofunctor of the self-indexing Self(_ S) of the category of S-schemes over itself, so that, for an S-scheme Z,  (Z) is a full subcategory of _ Z. It gives rise to  pseudofunctor  δ:^→. 
                                                        
                                                          *  Let C^:∘π_0⇒δ be the pseudo-natural transformation corresponding to .
                                                        
                                                          *  Let U:⇒ be a faithful pseudo-natural transformation. It gives rise to a morphism of fibrations, taking cartesian morphisms to cartesian, hence we obtain a pseudo-natural transformation δ U:δ⇒δ such that the diagram
                                                        
                                                            (m)[matrix of math nodes, row sep=2em, column sep=2em, text height=1.9ex, text depth=0.25ex]
                                                             
                                                             |(1)|∘π_0    |(2)|δ
                                                            
                                                             |(l1)|∘π_0    |(l2)|δ
                                                            ; 
                                                            [->,font=,>=to, thin]
                                                            (1) edge node[above]C^ (2) edge node[left]U  (l1)
                                                            (2) edge node[right]δ U (l2) 
                                                            (l1) edge node[above]C^  (l2);
                                                        
                                                        commutes.
                                                        
                                                          *  We require that δ U reflects (coequalisers associated with) -universal connected components, i.e., 
                                                        if P∈δ(X,δ_X) for some (X,δ_X)∈, and we have a diagram 
                                                        
                                                            [cramped, column sep=normal, ampersand replacement=&]
                                                            P_1[yshift=2pt]rd_0[yshift=-2pt]r[swap]d_1&P_0rr& Q.
                                                        
                                                        with r∘ d_0=r∘ d_1, which U maps onto an -universal coequaliser
                                                        
                                                        
                                                        
                                                        
                                                        
                                                        
                                                            (m)[matrix of math nodes, row sep=2em, column sep=2em, text height=1.5ex, text depth=0.25ex]
                                                             
                                                            |(u2)|UP_1    [1em] |(u1)|UP_0   [1em] |(u0)|π_0(δ U(P))
                                                            
                                                            |(2)|X_1    [1em] |(1)|X_0   [1em] |(0)|π_0(X)
                                                            ; 
                                                            [->,font=,>=to, thin]
                                                            ([yshift=.2em]2.east) edge node[above=-2pt] ([yshift=.2em]1.west) 
                                                            ([yshift=-.2em]2.east) edge node[below=-2pt] ([yshift=-.2em]1.west) 
                                                            (1) edge node[above=-2pt] (0) 
                                                            ([yshift=.2em]u2.east) edge node[above=-2pt] ([yshift=.2em]u1.west) 
                                                            ([yshift=-.2em]u2.east) edge node[below=-2pt] ([yshift=-.2em]u1.west) 
                                                            (u1) edge node[above=-2pt] (u0) 
                                                            (u2) edge node[left] (2)
                                                            (u1) edge node[left] (1)
                                                            (u0) edge (0)
                                                            ;
                                                        
                                                        associated to the -simple differential scheme δ U(P),
                                                        
                                                        then the original diagram was already a coequaliser.
                                                        
                                                        In view of <ref>, the above setup with the functor π_0 and the pseudofunctor C is not an extension of the adjunction ()_0⊣ C we used in <ref>, given that the functor ()_0 on affine differential schemes does not extend to a functor on differential schemes. 
                                                        
                                                        The category 
                                                        
                                                            _C(f)
                                                        
                                                        consists of objects P∈δ(Y) such that, for some Q∈(S(X)),
                                                        
                                                            f^*P≃ C_X(Q).
                                                        
                                                        A morphism f:(X,δ_X)→ (Y,δ_Y) of S-differential schemes is pre-Picard-Vessiot with respect to  provided:
                                                        
                                                        
                                                          (0)f is a morphism in , i.e., (X,δ_X) and (Y,δ_Y) have categorical schemes of leaves over S;
                                                        
                                                          * f is a morphism of effective descent for δ;
                                                        
                                                          * X, X×_YX and X×_YX×_YX are simple for .
                                                        
                                                        A morphism f:(X,δ_X)→ (Y,δ_Y) of S-differential schemes is Picard-Vessiot with respect to U:⇒ provided:
                                                        
                                                          (0)f is a morphism in , i.e., (X,δ_X) and (Y,δ_Y) have categorical schemes of leaves over S;
                                                        
                                                          * f is a morphism of effective descent for δ;
                                                        
                                                          * X is simple for ;
                                                        
                                                        
                                                          * f is auto-split with respect to , i.e., f∈_C^(f).
                                                        
                                                        
                                                        If  is already a sub-pseudofunctor of the self-indexing of _ S, we say that f is Picard-Vessiot with respect to , if it is such with respect to 𝕀:⇒. 
                                                        
                                                        
                                                        
                                                        A morphism f:(X,δ_X)→ (Y,δ_Y) in  is Picard-Vessiot with respect to  provided:
                                                        
                                                        
                                                          * f is a morphism of effective descent for δ;
                                                        
                                                          * X is simple for ;
                                                        
                                                          * f is auto-split, i.e., f∈_C(f).
                                                        
                                                          *  If (X,δ_X) is simple for , then it is simple for .
                                                        
                                                          *  Given an S-differential scheme (Y,δ_Y), the functor δ U_Y restricts to 
                                                            _C^(f)→_C^(f).
                                                        
                                                        
                                                        
                                                        
                                                        
                                                        For the first claim, suppose (X,δ_X) is simple with respect to , and let Q∈(π_0(X)). Let P=C^_X(Q) and consider the associated diagram 
                                                        
                                                            [cramped, column sep=normal, ampersand replacement=&]
                                                            P_1[yshift=2pt]rd_0[yshift=-2pt]r[swap]d_1&P_0rr& Q.
                                                        
                                                        Since (X,δ_X) is simple with respect to , the analogous diagram for δ U (C^_X(Q))=C^_X(UQ) is an -universal coequaliser and π_0(C^_X(UQ)=UQ. Using the fact that δ U reflects connected components, we deduce that the original diagram is a coequaliser and π_0(P)=Q.
                                                        
                                                        The second claim follows directly from the fact that U preserves cartesian morphisms.
                                                        
                                                        If f is a Picard-Vessiot morphism of differential schemes with respect to U:⇒, then f is pre-Picard-Vessiot with respect to .  
                                                        
                                                        Writing G_0=π_0(X), since 
                                                         f is auto-split with respect to C^,  for some G_1∈(G_0), 
                                                        
                                                            X×_YX≃ C^_X(G_1)=X×_C(G_0)C(G_1).
                                                        
                                                        By the definition of simplicity using universal coequalisers, any object in the essential image of C^_X is automatically simple for , whence X×_YX is simple for  and we have π_0(X×_YX)≃ G_1. Moreover, 
                                                        
                                                            X×_YX×_YX≃ (X×_YX)×_X(X×_YX)
                                                            ≃ (C(G_1)×_C(G_0)X)×_X(X×_C(G_0)C(G_1))
                                                            ≃ X×_C(X_0)C(G_1×_G_0 G_1)
                                                            ≃ C^_X(G_1×_G_0G_1),
                                                        
                                                        whence X×_YX×_YX is also simple for  with π_0(X×_YX×_YX)≃ G_1×_G_0G_1. Using <ref> again, we obtain that X, X×_YX and X×_YX×_YX are -simple, as required. 
                                                        
                                                        The assumption that f is pre-Picard-Vessiot for  ensures that the kernel-pair groupoid 
                                                        
                                                            (m)[matrix of math nodes, row sep=0em, column sep=3em, text height=1.5ex, text depth=0.25ex]
                                                             
                                                            |(n)|_f:   [1em] |(2)|X×_Y X×_Y X    [1em] |(1)|X×_Y X   [1em] |(0)|X
                                                            ; 
                                                            [->,font=,>=to, thin]
                                                            ([yshift=1em]2.east) edge node[above=-2pt]π_01 ([yshift=1em]1.west) 
                                                            (2) edge node[above=-2pt]π_02 (1)
                                                            ([yshift=-1em]2.east) edge node[above=-2pt]π_12 ([yshift=-1em]1.west) 
                                                            ([yshift=1em]1.east) edge node[above=-2pt]π_0 ([yshift=1em]0.west) 
                                                            (0)  edge node[above=-2pt]Δ (1) 
                                                            ([yshift=-1em]1.east) edge node[above=-2pt]π_1 ([yshift=-1em]0.west) 
                                                            
                                                             
                                                            ;
                                                        
                                                        is a category in , and that the Galois precategory
                                                            [f]=π_0(_f)
                                                        
                                                        exists as a precategory in .
                                                        
                                                        If f is Picard-Vessiot, then G[f] is an internal category (actually a groupoid without the inversion of arrows named) in , by the argument in <ref>.
                                                         
                                                        A pre-Picard-Vessiot morphism f for  induces an equivalence of categories
                                                        
                                                            _C(f)≃^[f]
                                                        
                                                        between the category of objects of δ(Y)C-split by f and the category of -actions of the precategory [f].
                                                        
                                                        Moreover, if f is Picard-Vessiot for U:⇒, the latter becomes the category of -actions of the groupoid [f]. 
                                                        
                                                        When f is pre-Picard-Vessiot, <ref> ensures that _f gives a precategorical decomposition of f in , and that the Galois precategory [f] is well-defined in . By assumption, f is of effective descent for _f. 
                                                        
                                                        By the assumption on simplicity of X, X×_YX, X×_YX×_YX and <ref>, we get that the functors C_X, C_X×_YX, C_X×_YX×_YX are fully faithful.  
                                                        
                                                        Hence, all the assumptions of <ref> are satisfied. When f is Picard-Vessiot, it is also pre-Picard-Vessiot by <ref>.
                                                        
                                                        Therefore, in both cases we get the desired equivalence involving actions of the precategory [f], which happens to be a groupoid in the Picard-Vessiot case, as observed in <ref>.
                                                        
                                                        
                                                         §.§ Quasi-projective Picard-Vessiot theory
                                                        
                                                        Let (K,δ) be a differential field of characteristic 0 with constants k, and write S=(k). Let f:X→ Y=(K,δ) be a morphism of S-differential schemes such that
                                                        
                                                          * X is integral quasi-projective over Y and its only leaf is the generic point;
                                                        
                                                          * f is auto-split, witnessed by a quasi-projective k-scheme G,
                                                        
                                                            X×_YX≃ X×_C(S)C(G).
                                                        
                                                        Then f is Picard-Vessiot, [f] is an S-algebraic group isomorphic to G, and there is an equivalence of categories
                                                        
                                                            {quasi-projective S-differential morphisms to Y split by f}
                                                            ≃{quasi-projective S-scheme actions of G}
                                                        
                                                        We follow the template of <ref> for  the fibred category of quasi-projective morphisms. By assumption (1) and <ref>, X is simple with categorical scheme of leaves S. By <ref>, f is of effective descent for δ. Thus, X is Picard-Vessiot and <ref> gives the desired equivalence. 
                                                        
                                                        With notation of <ref>, there is an order-reversing one-to-one correspondence between split S-differential quasi-projective fpqc quotients of f:X→ Y in 
                                                        and closed subgroups of the algebraic group G=[f] which  takes
                                                        
                                                            [cross line/.style=preaction=draw=white, -,
                                                            line width=4pt]
                                                            (m)[matrix of math nodes, row sep=.9em, column sep=.5em, text height=1.5ex, text depth=0.25ex]
                                                                |(ij)| X   
                                                               [.02em]
                                                            	|(i)|P      
                                                              [.02em]
                                                            			    |(0)|Y   
                                                            ;
                                                            [->,font=,>=to, thin]
                                                            (ij) edge node[left,pos=0.2]
                                                                (i) edge node[right,pos=0.5]f (0) 
                                                                (i) edge node[below left,pos=0.5]p (0) 
                                                                ;
                                                             
                                                            to 
                                                            
                                                                [X→ P].
                                                            
                                                            Conversely, a closed subgroup G' corresponds to the quotient 
                                                            
                                                                X/G',
                                                            
                                                            which is f-split by the quasi-projective scheme G/G'. 
                                                            
                                                            Moreover, this correspondence restricts to a one-to-one correspondence between split fpqc quotients P such that P→ Y is Picard-Vessiot, and closed normal subgroups of G. In this case, 
                                                            
                                                                [P→ Y]≃[X→ Y]/[X→ P].
                                                            
                                                            The claimed correspondence is more specific than a claim that could be extracted from <ref>, so we provide an explicit proof following an analogous strategy. 
                                                            
                                                            If h:X→ P is f-split by g:G→ Q, and h is fpqc, then f^*(h) is fpqc, hence an universal effective epimorphism by <cit.>. Since S_X preserves colimits as a left adjoint, it follows that g=U(h)=S_Xf^* (h) is a regular epimorphism, hence again an effective epimorphism in the presence of pullbacks. Since g corresponds to a quotient of an algebraic group, it is fpqc again by <ref>.
                                                            
                                                            Conversely, if G'≤ G is a closed subgroup, then, by <ref>, g:G→ G/G'=Q is fpqc. The quotient h:X→ P corresponding to it satisfies f^*(h)≃ C_X(g), whose underlying scheme morphis is fpqc since C_X acts as base change on g. Since f is fpqc, using the fact <cit.> that the properties of being `faithfully flat and quasi-compact' are local in the fpqc topology, it follows that h is fpqc itself. 
                                                            
                                                            The theory above shows that even in linear Picard-Vessiot theory, with X=(A,δ_A) the spectrum of a Picard-Vessiot ring over a differential field (K,δ), in order to extend the correspondence <ref> to all closed subgroups od the Galois group, we are forced to consider split quasi-projective quotients of X. 
                                                            
                                                            The above theory applies to strongly normal differential Galois theory of <cit.>.
                                                            
                                                            
                                                             §.§ Polarised quasi-projective differential Galois theory
                                                            
                                                            
                                                            Let f:(X,δ_X)→ (Y,δ_Y) be a  morphism of S-differential schemes such that, 
                                                            
                                                            with notation <ref>, 
                                                            
                                                              *  the underlying S-scheme morphism X→ Y is fpqc;
                                                            
                                                              * (X,δ_X) is simple for  with scheme of leaves G_0;
                                                            
                                                              *  there is  
                                                            
                                                            an S-morphism G_1→ G_0 such that
                                                            
                                                                (X,δ_X)×_(Y,δ_Y)(X,δ_X)≃ (X,δ_X)×_(G_0,0)(G_1,0).
                                                            
                                                            Then f is Picard-Vessiot for U:⇒, [f] is the groupoid (G_1 G_0) and we have an equivalence between the category of quasi-projective polarised S-differential morphisms (P,_P)→ Y split by f and the category of quasi-projective polarised actions (Q,_Q)→ G_0 of [f].
                                                            
                                                            Using <ref>, we obtain that f is of effective descent for δ. By <ref>, the forgetful functor δ U reflects -universal categorical schemes of leaves. Hence, f is indeed Picard-Vessiot for U and we can apply the template Theorem <ref>.
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