Paper Title: Higher Groups and Higher Normality.


	
	Bimodal Scaling Law and Size Effect In Superelastic Nanopillars
    Xian Chen
    August 5, 2024
===============================================================

§ ABSTRACT 
		In this paper we continue Prasma's homotopical group theory program by considering homotopy normal maps in arbitrary ∞-topoi. We show that maps of group objects equipped with normality data, in Prasma's sense, are algebras for a “normal closure” monad in a way which generalizes the standard loops-suspension monad. We generalize a result of Prasma by showing that monoidal functors of ∞-topoi preserve normal maps or, equivalently, that monoidal functors of ∞-topoi preserve the property of “being a fiber” for morphisms between connected objects. We also formulate Noether's Isomorphism Theorems in this setting, prove the first of them, and provide counterexamples to the other two. 
		
		Accomplishing these goals requires us to spend substantial time synthesizing existing work of Lurie so that we may rigorously talk about group objects in ∞-topoi in the “usual way.” One nice result of this labor is the formulation and proof of an Orbit-Stabilizer Theorem for group actions in ∞-topoi.
	
	
	

§ INTRODUCTION

	
	A general theme of homotopical algebra is to take structures typically carried by sets, e.g. group, ring or module structures, and replace the sets with homotopy types, or so-called ∞-groupoids. One formalism for this approach is to work with ∞-categories as developed by Joyal, Lurie, Rezk and many others. We heavily use Lurie's work on the topic as recorded in <cit.>. For the sake of the involved reader who may be going back and forth between our document and those references, we also follow Lurie's notational conventions whenever possible.
	
	
	An orthogonal generalization of “set-based mathematics” is replace the category of sets with a topos. Recall that a (Grothendieck) topos is, roughly, a category of sheaves of sets. This latter perspective has been fruitful, for instance, in algebraic geometry, topology, and mathematical logic. The joint generalization of these two ideas is to do mathematics in ∞-topoi as defined in <cit.>, which behave something like ∞-categories of (presheaves) of ∞-groupoids. The ∞-category of ∞-groupoids is of course an ∞-topos, as it is sheaves over the point. However, one can also build ∞-topoi from Quillen model categories of simplicial (pre)sheaves of the sort that arise in, say, algebraic geometry and motivic homotopy theory.
	
	In all of these settings, one is often interested in objects that are acted on by whatever the relevant analogue of a group is. For instance, in homotopy theory we are often interested in actions of loop spaces, e.g. S^1, on other spaces. In algebraic geometry, especially in Galois theory, we can frequently encode useful mathematical structure via actions of group schemes or group stacks. Classically, understanding groups themselves aids in the understanding of their representations and the ways in which they can act on sets, spaces, or schemes. However, a general theory of higher groups has not yet been fully developed. Arguably, as this paper will indicate, this is due to the fact that much of classical group theory “breaks” in the higher setting. Nonetheless, important steps have been made in understanding higher group theory, primarily by Prasma and collaborators <cit.>. Higher group theory has also been investigated by authors interested in mathematical physics, especially from the perspective of higher principal G-bundles and topological field theories e.g. in <cit.>.
	
	The work of Prasma's which is most relevant to us is <cit.>, which introduces the idea of a normality datum for a map of loop spaces. That work is in turn based on earlier work of Farjoun and Segev <cit.> and augmented by concomitant work of Hess and Farjoun <cit.>. The crucial observation there is that in higher group theory one should replace the property of being a normal subgroup with data. This data is, essentially, the data of a group structure on the quotient by the “subgroup.” We will see that in this setting, unlike in discrete group theory, there may be more than one such group structure, necessitating the use of data instead of property. Another significant departure from the classical theory is that “sub-objects” are not particularly well behaved in homotopical mathematics and “normal subgroups” are replaced by “normal maps.” 
	
	In this paper, we take Prasma's definition of normality seriously but generalize it to an arbitrary ∞-topos and explore its ramifications in higher group theory. Some of these ramifications include: formulating and proving a higher Orbit-Stabilizer Theorem (Theorem <ref>); proving that group maps equipped with normality data are the algebras for the “normal closure” monad (Theorem <ref>); formulating and proving Noether's First Isomorphism Theorem (Theorem <ref>); and providing counterexamples to (natural generalizations of) her other two isomorphism theorems (Section <ref>). 
	
	We warn the reader that group theory in this setting should not be thought of as a “topological” generalization of group theory. For instance, even the group objects in the ∞-topos of “spaces” do not have an underlying topological space at all, but rather an underlying homotopy type. Instead, we should think of higher group theory as a kind of deformation of discrete group theory (in the same way that the sphere spectrum is a deformation of the integers). One piece of evidence for this intuition is the fact that our constructions all become the standard group-theoretic constructions after applying π_0. 
	
	
	

  §.§.§ Acknowledgements

	
	The authors thank Jake Bian, Bastian Cnossen, Aras Ergus, Sonja Farr, Kiran Luecke, Eric Peterson, Maxime Ramzi, Lorenzo Riva, and Chris Rogers for many helpful conversations regarding the material of this paper. Kiran Luecke in particular is responsible for many of the counterexamples. The first author would also like to extend a special thanks to Matan Prasma for a number of delightful conversations about higher group theory that occurred during the 2015 Young Topologists' Meeting at EPFL. Though it has been almost 10 years since then, those ideas have never fully disappeared. While working on this paper, the first author was partially supported by a Simons Foundation collaboration grant, Award ID #853272. Much of the work of this paper is based on the second author's master's thesis at the University of Nevada, Reno.
	
	
	
	
	

§ BACKGROUND ON  ∞-TOPOI

	
	We briefly remind the reader of some of the central concepts in the theory of ∞-topoi as developed in <cit.>. Throughout,  and  will always denote ∞-topoi. Recall that for any ∞-category  we can form the ∞-category of presheaves (valued in ∞-groupoids ) which we will denote (). An ∞-topos is defined to be an ∞-category  which admits a fully faithful right adjoint functor ↪() whose left adjoint L()→ is accessible and left exact. We will avoid discussing accessibility in this paper as it will not be immediately relevant but we remind the reader that being left exact means that L preserves finite limits (the reader should be reminded of the left exact functors of abstract algebra that preserve kernels, which are finite limits). 
	
	Central to the theory of ∞-categories is the notion of effective epimorphisms, which should be thought of as covers. These will occur frequently throughout this work so we recall them now.
	
	
		Let f X→ Y be a morphism in , thought of as an object of ^Δ^1. Noticing that there is an inclusion Δ^1≃{[0]→[-1]}⊂Δ_+^ of Δ^1 into the augmented simplex category, we call the right Kan extension of f along this inclusion the augmented Čech nerve of f. The (unaugmented) Čech nerve of f is then taken to be the restriction of the augmented Čech nerve to the subcategory Δ^⊂Δ^_+. We say that f is an effective epimorphism of  if the colimit of the Čech nerve of f is equivalent to f. 
	 
	
	It may be useful for the reader to have the following (standard) schematic depiction of the Čech nerve of a morphism f X→ Y:
	
	
    Y     X    X×_YX   X×_Y X×_Y X   ⋯["f"', from=1-2, to=1-1]
    		[shift right, from=1-3, to=1-2]
    		[shift left, from=1-3, to=1-2]
    		[from=1-4, to=1-3]
    		[shift left=2, from=1-4, to=1-3]
    		[shift right=2, from=1-4, to=1-3]
    		[shift right, from=1-5, to=1-4]
    		[shift left, from=1-5, to=1-4]
    		[shift right=3, from=1-5, to=1-4]
    		[shift left=3, from=1-5, to=1-4]

	
	We will later need a restricted version of the Čech nerve whose codomain will be the group objects of interest in this paper. We will rarely, in this paper, take the Čech nerve of arbitrary morphisms in . For that reason we reserve the symbol , which often denotes the general Čech nerve functor in the literature, for later use.
	
	An important characterization ∞-topoi is given in <cit.> as ∞-categories satisfying a version of Giraud's axioms. This characterization implies, among other things, that ∞-topoi admit all small limits and colimits and we will regularly use this fact without comment. We will also need the fact that every groupoid object in an ∞-topos is effective, but because this potentially requires some explanation, we will discuss it further when necessary.
	
	Just as classical topoi generalize the theory of sets, ∞-topoi generalize the the theory of ∞-groupoids, or spaces. It is not surprising then that they can be equipped with an internal homotopy theory including things like homotopy groups and Postnikov towers. Some of the work in this paper is “obviously true” in , the ∞-topos of ∞-groupoids, but requires some work to re-prove in full generality. To aid us, and the reader, in this generalization process, we recall in this section some of the ways in which an ∞-topos can be made to behave like . The material of this section can all be found in <cit.>.
	
	To begin we must get clear on what we mean by n-connected, n-connective and n-truncated, for both objects and morphisms. This terminology is not entirely standardized in the literature. Because so much of this work uses <cit.> and <cit.> as its foundational references, we follow the conventions therein. 
	
	We will say that an ∞-groupoid X is n-connective if, for every point x∈ X, the homotopy sets π_i(X,x) are trivial for 0≤ i≤ n-1. For intuition regarding this terminology, we suggest the reader keep the case of spectra or chain complexes in mind, where “connective” refers to those objects which have trivial homotopy groups in negative degrees. Therefore, assuming “0-connective” should correspond to “connective,” every ∞-groupoid should be 0-connective. This then requires that n-connective ∞-groupoids can have non-trivial homotopy in degree n.
	
	We will then say that a morphism f X→ Y of ∞-groupoids is n-connective if its fibers are n-connective. In traditional algebraic topology, a morphism of spaces is called n-connected if it has fibers which have trivial homotopy groups in degrees 0≤ i≤ n-1. Therefore an n-connective morphism in our sense is the same as an n-connected morphism in the traditional sense. It is only that the terminology differs in indexing for individual spaces. One advantage of our  terminology (which is actually Lurie's, of course) is that the connectivity of the fiber of a map, and the map itself, have the same index. 
	
	In the other direction, we say that an ∞-groupoid X is n-truncated if, for every x∈ X, the homotopy sets π_i(X,x) are trivial for all i>n. We think of an n-truncated ∞-groupoid as having its homotopy groups cut down to degree n, i.e. they still include degree n. Then, similarly to above, we say that a morphism of ∞-groupoids X→ Y is n-truncated if each of its fibers is an n-truncated ∞-groupoid. 
	
	Note that the definitions of n-connective and n-truncated morphisms can also be formulated in terms of the long exact sequence in homotopy. Suppose that a morphism f X→ Y is n-connective. Then its fiber has trivial homotopy sets up through degree n-1 and thus the long exact sequence implies that π_i(f) is an isomorphism for 0≤ i≤ n-1 and that π_n(f) is a surjection. Similarly, if f is n-truncated then π_i(f) is an isomorphism for i>n+1 and an injection for i=n+1. In other words, if f is n-truncated then X and Y are “the same” after degree n. 
	
	We now follow <cit.> in making the corresponding definitions for an arbitrary ∞-topos. They are necessarily more complicated in this setting because objects of ∞-topoi are more like (pre)sheaves of ∞-groupoids and therefore things like homotopy groups are substantially more complicated. An important difference is that connectivity and truncatedness of morphisms in an ∞-topoi are not determined, in general, by connectivity or truncatedness of the fibers of that morphism.
	
	
		Let  be an ∞-topos and let -1≤ n. Then we define the category of n-truncated objects ^≤ n to be the full subcategory of  spanned by objects with the property that (Z,X) is an n-truncated ∞-groupoid for every Z∈. We write τ_≤ n→^≤ n for the left adjoint to the inclusion ^≤ n↪, as described in <cit.>. For n=-2 we make the convention that the only object of ^≤ -2 is the terminal object and τ_≤ -2 is the terminal functor. We call a morphism f X→ Y of  n-truncated if for every Z∈ the induced morphism of ∞-groupoids (Z,X)(Z,Y) is n-truncated.
	
	
	
		Later, we will use the fact that ∞-topoi are symmetric monoidal with respect to the Cartesian product (cf. <cit.>) and that the truncation functors are symmetric monoidal with respect to this structure, by virtue of <cit.>.
	
	
	For the following definition it is necessary to recall that if  is an ∞-topos and X∈ then the slice category _/X is again an ∞-topos (cf. <cit.>). 
	
	
		Let  be an ∞-topos and X∈. Choosing a base point p∗→ S^k of the k-sphere in  induces, by cotensoring, a morphism p^∗ X^S^k→ X given by precomposition with p. We write π_k(X) for τ_≤ 0(p^∗)∈_/X. Note that if f X→ Y is a morphism of  then it is an object of _/Y which is again an ∞-topos. Therefore is makes sense to define π_k(f)∈(_/Y)_/f≃_/f≃_/X.
		
		If η∗→ X is a base point for X then pulling back induces a functor η^∗_/X→_/∗≃ and we could write η^∗π_k(X) for the resulting object of τ_≤ 0. Typically, we will have already assumed X is a pointed object and in that case, so long as it will not create confusion, we will simply write π_k(X).
	
	
	
	
		When = the straightening functor of <cit.> implies that π_k(X) determines a functor X→ which factors through . At a point x∗→ X the value of this functor is the set of homotopy classes of morphisms S^k→ X which take the chosen base point of S^k to the point x∈ X. In the case of X pointed, pulling back along the base point gives a functor ∗→ that factors through . In other words, the choice of base point for X causes π_k(X) to simply be a set. It is straightforward to verify <cit.> which asserts that π_k(X), for X∈_∗, recovers the usual k^th homotopy group of X. 
	
	
	
		Note that for f X→ Y in , the “homotopy sets of f” are not morphisms of τ_≤ 0 but, rather, objects of τ_≤ 0_/X. These can be thought of as sets “indexed by X.” In the case that =, the homotopy set π_k(f) at each x∈ X is precisely the k^th homotopy set of the fiber of f over x. This justifies the following definition.
	
	
	
	
		Let f X→ Y be a morphism of an ∞-topos  and let 1≤ n. Then we say that f is n-connective if it is an effective epimorphism in  and π_i(f)≃∗ for all 0≤ i<n. A morphism is called 0-connective exactly when it is an effective epimorphism. By convention, every morphism of  is considered to be (-1)-connective. We say that an object X∈ is n-connective if the terminal morphism X→∗ is n-connective. Write ^≥ n for the full subcategory of n-connective objects of . For n=1 we will simply refer to objects of ^≥ 1 as connected rather than 1-connective. Notice that if an object X∈ is 0-connective then it must have π_0(X)≠∅, i.e. it must be “inhabited” in a certain sense. This is, of course, not an issue when we restrict ourselves to pointed objects.
	
	
	
	
		Let  be an ∞-topos. Then we define the category of pointed objects of , denoted _∗, to be the slice category ^\∗. We write _∗^≥ n for the full subcategory of _∗ spanned by objects which are n-connective after application of the forgetful functor _∗→. 
	
	
	
	
	
	
		Let  be an ∞-topos and write K for the full subcategory of the augmented simplex category Δ_+^ spanned by [-1], [0] and [1], sometimes called (Δ^_+)_≤ 1. Let iΔ^1→ K be the inclusion of the subcategory [-1]← [0] and let jΔ^1→ K be the inclusion of the subcategory [0]→[1] (as the degeneracy map). Then we define a pair of adjoint functors as follows:
		
			
  * Thinking of _∗ as a full subcategory of ^Δ^1 define the loops functor Ω__∗→_∗ to be the composite of right Kan extension along i followed by restriction along j. 
			
  * By the general yoga of Kan extensions this functor has a left adjoint Σ__∗→_∗ given by left Kan extension along i followed by restriction along j.
		
		The ambient ∞-topos will always be clear from context so we will drop the subscript from the suspension and loops functors and simply write Σ and Ω.
	

	
		If X∈_∗ then Σ X is equivalent to the pushout of the span ∗← X→∗ and Ω X is equivalent to the pullback of the cospan ∗→ X←∗, both computed in _∗. 
	
	
	
	The case of Ω is well known (to those who know it well). The pointwise formula for right Kan extensions computes the value of Ran_i(∗→ X) at [1]∈Δ_+^ to be the limit over the comma category (Δ_+^)_≤ 0^\ [1]. This last category is isomorphic to the cospan category {[0]→[-1]←[0]}. Because, by assumption, [0] is sent to ∗, this gives the result.
		
		
	Now we prove the suspension case. Let S={a→ b← c} denote the cospan category. In <cit.>, Lurie proves that the functor σ S→Δ_≤ 1, given by a,c↦ [0] and b↦[1], is “right cofinal” in the sense of the definition given on <cit.>. This is equivalent to saying that σ^ S^→Δ_≤ 1^ is cofinal in the sense of <cit.>. We warn the reader that this terminology is the opposite of the terminology used in <cit.>.
	
	Now consider an object X∈_∗ thought of as a functor {[0]→[1]}→_∗ with [0]↦∗ and [1]↦ X. Left Kan extending along the inclusion {[0]→[1]}↪Δ^_≤ 1 necessarily gives a diagram in which both of the induced morphisms X→∗ are the trivial ones (there is nothing else they can be).  Because Kan extensions paste, the left Kan extension of the diagram ∗→ X along the inclusion into (Δ_+^)_≤ 1 is equivalent to first left Kan extending to Δ_≤ 1^ and then left Kan extending again along the inclusion Δ^_≤ 1↪ (Δ_+^)_≤ 1. But (Δ_+^)_≤ 1 is precisely the cocone (Δ_≤ 1^)^▹. Therefore the desired left Kan extension can be computed by simply taking the colimit of the previously obtained diagram Δ_≤ 1^→_∗. We now use that S^→Δ_≤ 1^ is cofinal, along with <cit.>, to deduce that this colimit is equivalent to the desired pushout. 
	
	
	
	We will repeatedly use the following result of <cit.> so we reproduce it here.
	
	
	
		Let  be an ∞-topos and assume n≥ 0. Let f X→ Y be a morphism of  which admits a section s Y→ X:
		
    Y X Y

		Then f is n-connective if and only if s is (n-1)-connective. 
	
	
	
		If  is an ∞-topos and X∈_∗^≥ n then Σ X∈_∗^≥ n+1. In particular, Σ X is connective for every X∈_∗.
	
	
	
		Suppose that X is n-connective, i.e. the terminal morphism X→∗ is n-connective. By <cit.> we have that n-connective morphisms are stable under pushout, so the right vertical morphism in the following commutative diagram is also n-connective.
		
		
    X    ∗
    ∗   Σ X[from=1-1, to=1-2]
    			[from=1-1, to=2-1]
    			[from=1-2, to=2-2]
    			[from=2-1, to=2-2]
    			["⌟"anchor=center, pos=0.125, rotate=180, draw=none, from=2-2, to=1-1]

		That morphism fits into a retract ∗→Σ X→∗ and therefore, by Lemma <ref>, the morphism Σ X→∗ is (n+1)-connective.
	
	
	
	
		Let Ω^≥ 1 denote the restriction of Ω to _∗^≥ 1. The reduced suspension functor Σ→ is left adjoint to Ω and corestricts to a left adjoint of  Ω^≥ 1.
	
	
	
		The fact that Σ is left adjoint to Ω follows from definitions. Because Σ factors through the full subcategory ^≥ 1_∗, the unit morphism id_^∗⇒ΩΣ defining the Ω⊢Σ adjunction restricts to a unit morphism id_^≥ 1_∗⇒Ω^≥ 1Σ.
	
	
	
	

§ GROUP OBJECTS IN ∞-TOPOI

	
	The concept of “group object in an ∞-category” is defined in <cit.>. These are the objects that we will study for the remainder of the paper. This goal of this section is to carefully lay out the necessary definitions and to prove that things like delooping and taking quotients behave “as expected.”
	
	It is important to note that in <cit.> Luried introduces “grouplike associative monoids” and “grouplike associative algebras” as well. All of these deserve to be called “group objects,” and Lurie shows them to be equivalent for a Cartesian monoidal ∞-category, but we will primarily work with the first notion in this paper. Nonetheless, in Section <ref> we review the equivalences between each of these models and prove that they are compatible with quotients. 
	
	

 §.§ Simplicial Models for ∞-Groups and Actions

	
	
		Let  be an ∞-topos. We say that a simplicial object Z_∙Δ^→ is a category object if the canonical Segal maps Z_n→ Z_1×_Z_0⋯×_Z_0Z_1 are equivalences for all n (cf. <cit.>). We say that a category object Z_∙ is a monoid object if Z_0 is terminal in  (cf. <cit.>). We will write () for the full subcategory of ^Δ^ spanned by category objects and () for the full subcategory of ^Δ^ spanned by monoid objects.
	
	

	
	
		Let C_∙Δ^→ be a category object of . We say that C_∙ is a groupoid object of  if it satisfies the following condition (cf. <cit.>):
		
			
  * Let k,m,n≥ 0 with m≤ n and k≤ n. Suppose additionally that there is a pushout diagram in Δ
			
			
    [0]   [m]
    [k]   [n][from=1-1, to=1-2]
    				[from=1-1, to=2-1]
    				[from=1-2, to=2-2]
    				[from=2-1, to=2-2]
    				["⌟"anchor=center, pos=0.125, rotate=180, draw=none, from=2-2, to=1-1]
 i.e. that the union of the images of [m] and [k] is all of [n] and that their intersection is only one element. Then the image of this diagram under Z:
			
    C_n[r][d]     C_m[d]
    
    				C_k[r]     C_0
 is a pullback diagram in . We write () for the full subcategory of ^Δ^ spanned by groupoid objects.
		
	
	
	
		Note that asking for C_∙ to be a category object in Definition <ref> is superfluous. Indeed, the condition given there implies that C_n≃ C_n-1×_C_0C_1 for every n from which the “Segal condition” inductively follows. 
	
	
	
		The reader is encouraged to check that a groupoid object in , in terms of Definition <ref>, is precisely a (discrete) category in which every morphism is invertible. This follows from the fact that we have required the Segal condition for all partitions of [n] which overlap at one element, as opposed to ordered partitions alone.
	
	
	
		Given a simplicial object Z_∙Δ^→, we say that Z_∙ is a group object if it is both a groupoid object and a monoid object (cf. <cit.>). We write () for the full subcategory of ^Δ^ spanned by group objects. 
	
	
	
		The above definitions are homotopically coherent ways of saying “a monoid is a category with one object,” “a groupoid is a category in which every morphism is invertible,” and “a group is a groupoid with one object.”
	
	
	
		Define the (pointed) forgetful functor s_0^∗()→_∗ to be restriction along the zeroth degeneracy {[0][1]}↪Δ^. Define the (unpointed) forgetful functor U_()→ as the further restriction along [1]↪Δ^. If G_∙∈(), we will sometimes write G_1 for U_(G). 
	
	

	
	
		Let  be an ∞-topos. Following <cit.> we define a right action object of  to be a functor AΔ^×Δ^1→ satisfying the following two properties:
		
			
  * The restriction of A along Δ^×{1}↪Δ^×Δ^1 is a monoid object of .
			
  * The iterated face map A([n],0)→ A({0},0)≃ A([0],0) and the map A([n],0)→ A([n],1) exhibit A([n],0) as the product A([0],0)× A([n],1). 
		
		
		We will write () for the full subcategory of (Δ^×Δ^1,) spanned by right action objects. If A is a right action object with A([1],1)=M and A([0],0)=X, we will sometimes write (X,M) instead of A to indicate that A is encoding an action of a monoid M on an object X.
	
	
	
		A right action of a monoid M_∙≃ A(Δ^op,1) on X≃ A([0],0) can be depicted schematically as
		
		
    ⋮   ⋮
    X× M× M   M× M
    X× M    M 
    
    			X    ∗[from=4-1, to=4-2]
    			[shift right, from=3-1, to=4-1]
    			[shift left, from=3-2, to=4-2]
    			[shift left, from=3-1, to=4-1]
    			[shift right, from=3-2, to=4-2]
    			[from=2-1, to=3-1]
    			[from=2-2, to=3-2]
    			[shift left=2, from=2-1, to=3-1]
    			[shift right=2, from=2-1, to=3-1]
    			[shift left=2, from=2-2, to=3-2]
    			[shift right=2, from=2-2, to=3-2]
    			[from=3-1, to=3-2]
    			[from=2-1, to=2-2]
    			[shift right, from=1-1, to=2-1]
    			[shift left, from=1-1, to=2-1]
    			[shift right=3, from=1-1, to=2-1]
    			[shift left=3, from=1-1, to=2-1]
    			[shift right, from=1-2, to=2-2]
    			[shift left, from=1-2, to=2-2]
    			[shift left=3, from=1-2, to=2-2]
    			[shift right=3, from=1-2, to=2-2]

	
	
	
		Restricting along the inclusion Δ^×{1}↪Δ^×Δ^1 induces a forgetful functor ()→(). We write _M() for the pullback of this functor along {M}↪():
		
		
    _M()   ()
    {M}   ()[from=1-1, to=1-2]
    			[from=1-1, to=2-1]
    			["⌟"anchor=center, pos=0.125, draw=none, from=1-1, to=2-2]
    			[from=1-2, to=2-2]
    			[from=2-1, to=2-2]

	
	
	
	
	
		Let M_∙∈() and let (X,M)∈_M(). Then we define the bar construction for the action of M on X to be the simplicial object Δ^op→ obtained from X by precomposing with the inclusion Δ^×{0}↪Δ^×Δ^1. When the action object (X,M) is specified by context we write (X,M) for this simplicial object.
	
	
	
		Let M_∙∈() be a monoid object, hence an object of the functor category ^Δ^. By considering the identity morphism of M_∙ in (^Δ^)^Δ^1 as an object of ^Δ^×Δ^1, we obtain a right action object displaying an action of M on ∗. Note that (∗,M)≃ M_∙. In other words, this method of thinking about monoids in an ∞-category has the bar construction built in.
	
	
	

 §.§ Augmented Simplicial Objects and Quotients

	
	
	
		Let  be an ∞-topos with M_∙∈() and (X,M)∈_M(). We define several functors relating (), () and , as well as extensions of these categories to “augmented” versions. Write i_+Δ^→Δ_+^ for the inclusion of the unaugmented simplex category into the augmented simplex category and i_+×Δ^1Δ^×Δ^1↪Δ_+^×Δ^1 for its obvious extension to Δ^×Δ^1.
		
			
  * Write ()→ for the geometric realization functor. Given M∈() we will write M for its image in .
			
			
  * Write _∙^+()→^Δ_+^ for the functor which left Kan extends M_∙ along i_+. Note that _∙^+M recovers M when restricted to [-1]∈Δ_+^.
			
			
  * Write _∗()→_∗ for the functor which restricts _∙^+ to the terminal morphism {[0]→[-1]}∈Δ^_+.
			
			
  * Write -/-()→ for the functor that takes a right action object (X,M) to the geometric realization of (X,M). Given (X,M) we will write X/M for its image under this functor. 
			
			
  * Write ()→^Δ_+^×Δ^1 for the left Kan extension of a right action object (X,M) along i_+×Δ^1. Write (X,M) for the image of (X,M) under this functor. 
			
  * Write (-,-)()→^Δ^_+ for the restriction of  along Δ_+^×{0}↪Δ^_+×Δ^1. We will write (X,M) for the image of (X,M)∈() under this functor. 
		
		We will write ^+() and ^+() for the essential images of ^+_∙ and Act^+_∙ respectively, and ^+_M() for obvious restriction of ^+() to a fixed monoid. 
	
	
	
		The values of (X,M) at ([-1],0) and ([-1],1) are respectively the geometric realizations of (X,M) and (∗,M)≃ M_∙. This can be visualized as the following diagram:
		
		
    ⋮   ⋮
    X× M× M   M× M
    X× M    M 
    
    			X    ∗
    X/M   ∗/M[shift right, from=1-1, to=2-1]
    			[shift left, from=1-1, to=2-1]
    			[shift right=3, from=1-1, to=2-1]
    			[shift left=3, from=1-1, to=2-1]
    			[shift right, from=1-2, to=2-2]
    			[shift left, from=1-2, to=2-2]
    			[shift left=3, from=1-2, to=2-2]
    			[shift right=3, from=1-2, to=2-2]
    			[from=2-1, to=2-2]
    			[from=2-1, to=3-1]
    			[shift left=2, from=2-1, to=3-1]
    			[shift right=2, from=2-1, to=3-1]
    			[from=2-2, to=3-2]
    			[shift left=2, from=2-2, to=3-2]
    			[shift right=2, from=2-2, to=3-2]
    			[from=3-1, to=3-2]
    			[shift right, from=3-1, to=4-1]
    			[shift left, from=3-1, to=4-1]
    			[shift left, from=3-2, to=4-2]
    			[shift right, from=3-2, to=4-2]
    			[from=4-1, to=4-2]
    			[from=4-1, to=5-1]
    			[from=4-2, to=5-2]
    			[from=5-1, to=5-2]

		In the case that M_∙ is a group object, we will see that the bottom square will be a pullback, recovering the idea of a principal G-bundle as described, for instance, in <cit.>.
	
	
	
	
	
	
	
	
	
	
	

 §.§ Action Groupoids

	
	We now specialize to the case of group actions, rather than monoid actions. This section is primarily an exposition of <cit.>. First we need the following definitions. 
	
	
		Let F,G→ be functors of ∞-categories. We say that a natural transformation η F⇒ G is Cartesian if, for every morphism ϕ c→ d in , the following is a pullback diagram in :
		
		
    F(c)   G(c)
    F(d)   G(d)["η_c", from=1-1, to=1-2]
    			["F(ϕ)"', from=1-1, to=2-1]
    			["⌟"anchor=center, pos=0.125, draw=none, from=1-1, to=2-2]
    			["G(ϕ)", from=1-2, to=2-2]
    			["η_d"', from=2-1, to=2-2]

	
	
	
	 Write _gp() for the pullback of the restriction functor ()→() along the inclusion ()↪(). 
		All of the categories and functors of Definition <ref> can be restricted along the inclusions _gp()↪() and ()↪(), and they continue to have the same interpretations as Kan extensions and restrictions. Write ^+() and _gp^+() for the essential images of these restrictions. 
	
	
	
	
		Let  denote the full subcategory of Δ_+^×Δ^1 spanned by the objects ([-1],0), ([-1], 1) and ([0],1). Let ℰ denote the full subcategory of  ^Δ^op_+×Δ^1 spanned by functors F with the following properties:
		
			
  * The functor F is a right Kan extension of its restriction to 𝒦⊂Δ_+^×Δ^1. 
			
  * The object F([0],1) is terminal.
			
  * The object F([-1],1) is connected (equivalently, F([0],1)→ F([-1],1) is an effective epimorphism). 
		
		Then restriction along the inclusion of 𝒦 induces an equivalence ℰ→^Δ^1×_^{1}^≥ 1_∗. 
	
	
	
		This proof is simply an elaboration of Lurie's argument in the cited remark. Note that the second and third conditions above imply that all of the functors in E restrict along 𝒦 to objects of ^Δ^1×_^{1}_∗^≥ 1. It remains to show that the resulting functor is an equivalence.  This follows by applying (the opposite of) <cit.>. In the statement of that proposition, we should set =Δ_+^×Δ^1, =, ^0=𝒦 and '=∗. The proposition may be interpreted as saying that functors out of 𝒦 which admit right Kan extensions to all of Δ_+^×Δ^1 are equivalent to functors which are right Kan extensions. 
	
	
	
		The above lemma shows that functors in ℰ can be thought of as cospans X→ Y←∗ in  with Y connected. 
	
	
	
		The ∞-category ℰ of Lemma <ref> is equivalent to _gp^+(). Moreover, it fits into the following commutative diagram in which the horizontal morphisms are equivalences:
		
		
    _gp()      ^+_gp()   ^Δ^1×_^{1}^≥ 1_∗
    ()      ^+()   _∗^≥ 1["Act^+_∙", from=1-1, to=1-3]
    			[from=1-1, to=2-1]
    			["res_𝒦", from=1-3, to=1-4]
    			[from=1-3, to=2-3]
    			[from=1-4, to=2-4]
    			["^+_∙"', from=2-1, to=2-3]
    			["res_[0]→[-1]"', from=2-3, to=2-4]

	
	
	
		Recall that left Kan extending against a fully faithful functor (in this case i_+Δ^×Δ^1↪Δ^_+×Δ^1) is fully faithful. Hence the functor Act_∙^+ is an equivalence onto its essential image, which is defined to be _gp^+(). The fact that _gp^+()≃ℰ follows from the penultimate sentence of <cit.> which implies that ℰ is precisely the full subcategory of ^Δ_+^×Δ^1 spanned by functors which are left Kan extensions of their restrictions to Δ^×Δ^1 and whose restriction to Δ^×{1} is a group object. It is straightforward to check that the restriction of Act_∙^+ to (), i.e. the functor _∙^+, is an equivalence and that both squares above commute. The fact that the two right horizontal morphisms are equivalences follows from Lemma <ref>.
	
	
	
		By taking fibers over a fixed G∈() in the diagram of Lemma <ref>, we obtain equivalences 
    _G()≃_G^+()≃_/ G

	
	
	Corollary <ref> allows us to produce right action objects without having to explicitly construct simplicial objects, as in the following examples.
	
	
		If G_∙∈() then applying _∗ gives an object ∗→ G of _/ G whose fiber is G_1. Passing to _G() gives a right action object of G_∙ on G_1 that we refer to as the translation action of G_∙ on itself.
	
	
	
		The identity morphism G→ G has trivial fiber and therefore the associated right action object is the trivial action of G_∙ on the terminal object of . Note that because G is always pointed there is a morphism X→∗→ G for any X∈. The associated right action object is the trivial action of G_∙ on X.
	
	
	
		Choosing a base point p∗→ S^1, we obtain an evaluation morphism G^S^1→ G whose fiber is equivalent to Ω G≃ G_1. We refer to the resulting right action object as the conjugation action of G_∙ on itself. 
	
	
	
		Because (-,-)()→^+() is defined as a left Kan extension, it has a restriction as its right adjoint. Similarly, res_𝒦 has a right Kan extension as its right adjoint. This gives explicit formulae for the inverse equivalences of the horizontal arrows in Lemma <ref>.
	
	
	
	
	
		We will write β_gp()→^Δ^1×_^{1}_∗^≥ 1 for the equivalence obtained as the composite of res_𝒦 with . For fixed G∈(), write β_G for the restriction of β to the fiber of _gp() over G. Write ω and ω_G for inverses of β and β_G. 
		
		Note that the composite res_[0]→[-1]∘_∙^+ is precisely the functor _∗ of Definition <ref> restricted to (). Write  for an inverse of res_[0]→[-1]∘_∙. Going forward we will use _∗ exclusively as a functor with domain () (rather than ()). 
	
	
	
	
		The arguments of <cit.> imply that every functor in _gp^+() is a Cartesian natural transformation. In other words, if we think of (X,G) as a natural transformation Δ^1→^Δ_+^ then it has the property that every naturality square is a pullback. In particular there is always a fiber sequence X→ X/G→ G coming from (X,G) restricted to (Δ_+^≤ 0)^×Δ^1.	
	
	
	
		Given f X→ Y∈_/Y for Y a pointed and connected object of  the action object ω(f) displays a group structure on Ω Y and an action of it on (f).  Moreover, Y≃Ω Y. 
	
	
	
		Remark <ref> implies that ω(f) describes an action of a group object on (f). Note that ω factors through _gp^+() by first taking the right Kan extension along the inclusion ↪Δ_+^×Δ^1. It then takes the resulting augmented object to its restriction to Δ^×{1}. From <cit.> (specifically item i_0) we have that ω(f)(Δ^op,{1}) is the right Kan extension of ∗→ Y along the inclusion {[0]→[-1]}↪Δ_+^ followed by its restriction to Δ^. In other words, ω(f)(Δ^,{1}) is the Čech nerve of the pointing ∗→ Y. It follows from Proposition <ref> (and the fact that Kan extensions paste along full subcategory inclusions) that ω(f)([1],{1})≃Ω Y. The last statement follows from the bottom row of the diagram of Lemma <ref> being an equivalence. 
	
	
	
	
		Let G_∙∈() and (X,G)∈_G(). Then (X,G) is a groupoid object of  in the sense of <cit.>.
	
	
	
		For clarity in this proof we simply write G^n instead of G_1^n or, equivalently, G_n. Suppose we have a commutative square in which each morphism is a face map of (X,G): 
		
    G^n× X   G^m× X
    G^r× X    X
    			["d_j", from=1-1, to=1-2]
    			["d_i"', from=1-1, to=2-1]
    			["d_k", from=1-2, to=2-2]
    			["d_l"', from=2-1, to=2-2]

		This diagram projects onto the corresponding diagram of G_∙
		
    G^n   G^m
    G^r   ∗["d_j", from=1-1, to=1-2]
    			["d_i"', from=1-1, to=2-1]
    			["⌟"anchor=center, pos=0.125, draw=none, from=1-1, to=2-2]
    			["d_k", from=1-2, to=2-2]
    			["d_l"', from=2-1, to=2-2]

		which is a pullback, since G_∙ is a group(oid) object as in Definition <ref>. The first diagram forms the “top” and the second diagram forms the “bottom” of the following commutative cube:
		
		
    G^n× X      G^m× X
       G^r× X       X 
    G^n      G^m
       G^r      ∗[from=1-1, to=1-3]
    			[from=1-1, to=2-2]
    			[from=1-1, to=3-1]
    			[from=1-3, to=2-4]
    			[" "description, from=1-3, to=3-3]
    			[" "description, from=3-1, to=3-3]
    			[from=2-2, to=2-4]
    			[from=2-2, to=4-2]
    			[from=2-4, to=4-4]
    			[from=3-1, to=4-2]
    			[from=3-3, to=4-4]
    			[from=4-2, to=4-4]

		Because (X,G) is a Cartesian natural transformation, as in Remark <ref>, and G_∙ is a group object, every face in the above cube is a pullback by assumption except for the top one. However this implies that the top face must also be a pullback.  
	
	
	
		Let (X,G) be a right action object in _gp(). Then the restriction of (X,G) to Δ^×{0} is a groupoid resolution of X→ X/G in the sense of <cit.>.
	
	
	
		By Lemma <ref>, we have that (X,G) is a groupoid object of . Moreover, by Lemma <ref> we have that X/G is the colimit of (X,G). 
	
	
	
		The morphism X→ X/G in ^+(X,G) is an effective epimorphism.
	 
	
	
		By <cit.> we know that every groupoid object of  is effective, as in <cit.>, which means every groupoid object is equivalent to the Čech nerve of the augmentation map to its colimit. Hence _∙(X,G) is equivalent to the Čech nerve of X→ X/G. Therefore X→ X/G satisfies the conditions of Definition <ref>.
	
	
	

 §.§ Free Group Objects

	
	
	
		The pointed forgetful functor s_0^∗()→_∗ of Definition <ref> admits a left adjoint given by the composite Σ. 
	
	
	
		By construction there is an equivalence of functors s_0^∗≃Ω^≥ 1^≥ 1_∗→_∗. By Lemma <ref> we have that s_0^∗ is right adjoint to Σ_∗→_∗^≥ 1. Because _∗ and  are equivalences, they are both left adjoints and both right adjoints. Therefore we may form the composite adjunction:


    _∗         _∗^≥ 1      ()
    	[""name=0, anchor=center, inner sep=0, "Σ"', curve=height=12pt, from=1-1, to=1-4]
    	[""name=1, anchor=center, inner sep=0, "Ω^≥ 1"', curve=height=12pt, from=1-4, to=1-1]
    	[""name=2, anchor=center, inner sep=0, ""', curve=height=12pt, from=1-4, to=1-6]
    	[""name=3, anchor=center, inner sep=0, "β"', curve=height=12pt, from=1-6, to=1-4]
    	["⊣"anchor=center, rotate=90, draw=none, from=0, to=1]
    	["⊣"anchor=center, rotate=90, draw=none, from=2, to=3]

		However, Ω^≥ 1∘_∗≃ s_0^∗∘∘_∗≃ s_0^∗. Therefore Σ is left adjoint to the forgetful functor ()→_∗.
	
	
	
		For any G_∙∈() there is an isomorphism π_i(s_0^∗ G)≅π_i+1( G). 
	
	
	
		From the preceding lemmas, we have a string of adjunction equivalences:
		
    _∗(S^i,s_0^∗ G)≃()(Σ S^i, G)≃_∗^≥ 1( S^i+1,_∗ G)≃_∗(S^i+1,_∗ G)

		Applying the truncation functor gives the result.
	
	
	
		Write _+_∗→() for the functor obtained by right Kan extending along the inclusion {[0]→[-1]}↪Δ_+^ and then restricting along Δ^↪Δ_+^. Then the composite _∗_+_∗→_∗^≥ 1 is a right adjoint to the inclusion _∗^≥ 1↪_∗. 
	
	
	
		Let ϵ_∗_+⇒ id__∗ denote the counit transformation of _∗⊣_+. By <cit.> this natural transformation is an equivalence when restricted to connected objects. It follows that the conditions of (the opposite of) <cit.> are satisfied and therefore _∗_+ is a colocalization functor. 
	
	
	The following is a standard fact for colocalizations.
	
	
		Limits in _∗^≥ 1 are computed by applying _∗_+ to the limit computed in _∗.
	
	
	
		Let F K→_∗^≥ 1 be a diagram. Then we compute lim(F)≃lim(_∗_+iF)≃_∗lim(iF) where i denotes the inclusion functor and in the last step we use that _∗_+ is a right adjoint.
	
	
	
	
	
	
	
	

 §.§ Base Change of Action Objects

	
	In a Cartesian monoidal ∞-category, monoids can be identified with -algebras, heretofore referred to as associative algebras, and right action objects can be identified with right modules. We review these constructions now for the reader's convenience.
	
	Recall from <cit.> that a symmetric monoidal ∞-category ^⊗ whose tensor product is given by the Cartesian product has, essentially by definition, a “Cartesian structure” π^⊗→. Recall that if 𝒪^⊗ is an ∞-operad then 𝒪-algebras in any symmetric monoidal ∞-category are given by certain functors 𝒪^⊗→^⊗. The category of 𝒪-algebras is denoted _ O(). If  is Cartesian monoidal then we can also define 𝒪-monoids in  to be certain functors 𝒪^⊗→. The category of O-monoids in  is denoted _ O() (not to be confused with () from the preceding section).
	
	When  is Cartesian monoidal <cit.> gives that composing with π^⊗→ induces an equivalence π_∗_𝒪()→_𝒪(). In the case that 𝒪^⊗ is the associative operad 𝒜ssoc^⊗ or the operad for left modules ℛℳ^⊗, we can make a further simplification.
	
	There are two functors Δ^→𝒜ssoc^⊗ and γΔ^×Δ^1→ℛℳ^⊗ which are “approximations” in the sense of <cit.> (where ℛℳ^⊗ is the ∞-operad for right modules over an associative algebra). We then have from <cit.> and <cit.> that precomposing with  and γ respectively give equivalences ^∗_𝒜ssoc()→() and γ^∗_ℛℳ()→(). Composing these with the preceding equivalences gives equivalences
	
    ^∗π_∗_𝒜ssoc()()
 and 
    γ^∗π_∗()()

	
	
		Let  be a Cartesian monoidal ∞-category. Write _𝒜ssoc()→() and ()→() for the above equivalences. 
	
	
	
		Recall that the objects of the ∞-operad  are the pointed sets ⟨ n⟩={∗,1,2,…,n} for all n∈ℕ and that the maps are pointed functions f⟨ n⟩→⟨ m⟩ with the data of a linear order on the fiber f^-1(i) for each 1≤ i≤ m. Write μ⟨ 2⟩→⟨ 1⟩ for the morphism that takes both 1 and 2 to 1 with μ^-1(1) having 1≺ 2. One can check that if A^⊗→ is an associative monoid object of an ∞-category  then A(μ) is “the multiplication map” of A. Pulling A back along  to (), one can further check that A(μ) is the image under ^∗ A of the opposite of the inner coface map [1]→[2] in Δ which takes 0 to 0 and 1 to 2. In other words, the “multiplication map” A(μ) of A, as a monoid object, is the red arrow in the below standard simplicial diagram:
		
		
    ∗   A   A× A   ⋯[shift right, from=1-2, to=1-1]
    			[shift left, from=1-2, to=1-1]
    			[color=rgb,255:red,255;green,51;blue,78, from=1-3, to=1-2]
    			[shift left=2, from=1-3, to=1-2]
    			[shift right=2, from=1-3, to=1-2]
    			[shift right, from=1-4, to=1-3]
    			[shift left, from=1-4, to=1-3]
    			[shift right=3, from=1-4, to=1-3]
    			[shift left=3, from=1-4, to=1-3]
 
	
	
	
		By restricting to objects in () with a fixed algebra A, we obtain an equivalence _A_A()→_ (A)().  We will abuse notation by writing _A() instead of _ (A)() when we refer to the category of right action objects over the monoid associated to an associative algebra.
	
	
	Recall from <cit.> that if  is a monoidal ∞-category and A∈() then there is a Cartesian fibration p()→() whose Cartesian morphisms are those which become equivalences when projecting to the underlying module in . In general, the morphisms of () are pairs (ϕ,θ) (A,P)→ (B,Q) such that ϕ A→ B is an algebra morphism and θ P→ϕ^∗(Q) is a morphism of right A-modules where ϕ^∗(Q) denotes Q equipped with the A-module structure obtained by restriction of scalars along ϕ. Therefore the straightening p()^→ takes a morphism of algebras ϕ A→ B to the restriction of scalars functor ϕ^∗_B()→_A(). The fibration p is also coCartesian by <cit.> and, as a result, each restriction of scalars functor has a left adjoint ϕ_∗_A()→_B() given by P↦ P⊗_A B. 
	
	When  is Cartesian monoidal and A and B are monoids we can restrict and extend scalars in an a priori different way. For simplicity we only consider the case that  is an ∞-topos.
	
	
		Let F→ be a monoidal (i.e. product preserving) functor of ∞-topoi  and write F_∗()→^Δ^×Δ^1 for postcomposition with F. Then F_∗ factors through ().  Moreover, F_∗ restricts to a functor _gp()→_gp(). 
	
	
	
		Note that both monoids and right action objects are defined via products and a terminal object, so it is immediate that F_∗ factors through (). For the case of groups it suffices to show that F_∗ preserves group objects when restricted to (). By <cit.> it suffices to show that (^∗)^-1(FG_∙) is a grouplike -monoid in  for each group object G_∙∈(). If we write μ_G G× G→ G for the inner face map giving “the” group structure on G (see Remark <ref>) then G_∙ is grouplike if and only if (μ_G,π_1) G× G→ G× G and (π_2,μ_G) G× G→ G× G are both equivalences, as in <cit.>. Since F preserves products (hence projection maps) and F(μ_G)=μ_FG by definition, the corresponding morphisms for FG_∙ must still be equivalences, so FG_∙ is a group object. 
	
	
	
		Let  be an ∞-topos and let ϕ H_∙→ G_∙ be a morphism of group objects in . Define the functor _H^G  to be the composite
		
    _G()_/ G_/ H_H()
 where (ϕ)^∗ denotes pullback along ϕ H→ G. On the other hand, define _H^G to the composite functor
		
    _H()_/ H_/ G_G()
 where (ϕ)_∗ denotes postcomposition with ϕ. It is immediate that _H^G is right adjoint to _H^G.  
	
	
	
		When ϕ H_∙→ G_∙ is a morphism of group objects, we will be particularly interested in the H-action on ^G_H((G,G)), where G_∙ is acting on itself by translation (cf. Example <ref>).  Note that β_G((G,G)) is the inclusion of the base point ∗→ G. Applying (ϕ)^∗ and ω_H gives a right action object that looks like the following:
		
		
    ⋮   ⋮
    G× H× H   H× H
    G× H    H 
    
    			G    ∗[shift right, from=1-1, to=2-1]
    			[shift left, from=1-1, to=2-1]
    			[shift right=3, from=1-1, to=2-1]
    			[shift left=3, from=1-1, to=2-1]
    			[shift right, from=1-2, to=2-2]
    			[shift left, from=1-2, to=2-2]
    			[shift left=3, from=1-2, to=2-2]
    			[shift right=3, from=1-2, to=2-2]
    			[from=2-1, to=2-2]
    			[from=2-1, to=3-1]
    			[shift left=2, from=2-1, to=3-1]
    			[shift right=2, from=2-1, to=3-1]
    			[from=2-2, to=3-2]
    			[shift left=2, from=2-2, to=3-2]
    			[shift right=2, from=2-2, to=3-2]
    			[from=3-1, to=3-2]
    			[shift right, from=3-1, to=4-1]
    			[shift left, from=3-1, to=4-1]
    			[shift left, from=3-2, to=4-2]
    			[shift right, from=3-2, to=4-2]
    			[from=4-1, to=4-2]

		Recall however that ω_H factors through taking the right Kan extension along the inclusion ↪Δ_+^×Δ^1 which is a diagram of the following form:
		
		
    ⋮   ⋮
    G× H× H   H× H
    G× H    H 
    
    			G    ∗
    G/H    H[shift right, from=1-1, to=2-1]
    			[shift left, from=1-1, to=2-1]
    			[shift right=3, from=1-1, to=2-1]
    			[shift left=3, from=1-1, to=2-1]
    			[shift right, from=1-2, to=2-2]
    			[shift left, from=1-2, to=2-2]
    			[shift left=3, from=1-2, to=2-2]
    			[shift right=3, from=1-2, to=2-2]
    			[from=2-1, to=2-2]
    			[from=2-1, to=3-1]
    			[shift left=2, from=2-1, to=3-1]
    			[shift right=2, from=2-1, to=3-1]
    			[from=2-2, to=3-2]
    			[shift left=2, from=2-2, to=3-2]
    			[shift right=2, from=2-2, to=3-2]
    			[from=3-1, to=3-2]
    			[shift right, from=3-1, to=4-1]
    			[shift left, from=3-1, to=4-1]
    			[shift left, from=3-2, to=4-2]
    			[shift right, from=3-2, to=4-2]
    			[from=4-1, to=4-2]
    			[from=4-1, to=5-1]
    			["⌟"anchor=center, pos=0.125, draw=none, from=4-1, to=5-2]
    			[from=4-2, to=5-2]
    			[from=5-1, to=5-2]

		But the morphism G/H→ H, by construction, is precisely the fiber of ϕ H→ G. Therefore by composing fiber sequences and extending we obtain a kind of Puppe sequence
		
    HG→ G/H→ H G

	
	
	
		When ϕ H_∙→ G_∙ is a morphism of group objects in an ∞-topos, we will call the composite fiber sequence of Example <ref> the Puppe sequence associated to ϕ. We will call the object G/H the quotient of G by H with respect to ϕ. 
	
	
	
		Let ϕ H_∙→ G_∙ be a morphism of group objects in an ∞-topos . Then the composite 
    _H()_H()_G()_G()

		is equivalent to the extension of scalars functor ϕ_∗_H()→_G(). Consequently, the composite 
		
    _G()_G()_H()_H()

		is equivalent to the restriction of scalars functor ϕ^∗_G()→_H().
	
	
		By <cit.> we have that colimit preserving functors _H()→_G() are equivalent to tensoring with (H,R)-bimodules. This equivalence implies that a specific colimit preserving functor F_H()→_G() is equivalent to -⊗_HF(H).  The first composite above evidently takes H with the translation action on itself to G with the translation action on itself and so is equivalent to the standard base change functor. The second statement follows from uniqueness of adjoints. 
	
	
	
		Let (X,G)∈_G() be a right action of a group G∈() on X∈ and let  t G→∗ be the terminal group morphism. Then there is an equivalence X/G≃ t_∗(X)=X⊗_G∗, i.e. the quotient of X by G is equivalent to base changing from G to the trivial group.
	
	
	Before stating our next main result, we need that base change is compatible with monoidal functors.
	
	
		Let  and  be monoidal categories, F→ a monoidal functor and ϕ A→ B a morphism of associative algebras in . Then the following diagram commutes:
		
		
    _B()   _A()
    _FB()   _FA()["ϕ^∗", from=1-1, to=1-2]
    			["(F)_B"', from=1-1, to=2-1]
    			["(F)_A", from=1-2, to=2-2]
    			["(Fϕ)^∗"', from=2-1, to=2-2]

		where (F)_B and (F)_A refer to the restriction of  (F) (i.e. postcomposition with F as a functor ℛℳ^⊗→^⊗) to _B() and _A() respectively.
	
	
	
		From <cit.>, along with the definition of monoidal functor, we have a commutative square whose vertical legs, which project away the module coordinate, are Cartesian fibrations and whose upper horizontal morphism preserves Cartesian morphisms:
		
		
    ()   ()
    ()   ()[from=1-1, to=1-2]
    			[from=1-1, to=2-1, "p"]
    			[from=1-2, to=2-2, "q"]
    			[from=2-1, to=2-2]

		Moreover, the morphisms in () which are p-Cartesian are precisely those which induce an equivalence on the underlying module (in particular, restriction of scalar morphisms). 
		By taking the pullback of the lower right cospan in the above square, we obtain a commutative triangle
		
		
    ()   ℰ
    ()   [from=1-1, to=1-2]
    			[from=1-1, to=2-1, swap, "p"]
    			[from=1-2, to=2-1,"r"]

		By <cit.> and <cit.> we have that r is still a Cartesian fibration and that a morphism of ℰ is r-Cartesian if and only if its image in () is q-Cartesian. Applying the straightening functor to this triangle (which is a morphism in the ∞-category of Cartesian fibrations over ()), we see that for any ϕ∈() the resulting naturality square is exactly the desired diagram. Here we are using the fact that fibers are preserved by pullbacks so that r^-1(A)≃ q^-1(FA)≃_FA() for every A∈(). 
	
	
	
		Let F→ be a monoidal (i.e. product preserving) functor of ∞-topoi. Then the following diagram commutes:
		
		
    _G()   _H()
    _FG()   _FH()["_H^G", from=1-1, to=1-2]
    			["F_∗"', from=1-1, to=2-1]
    			["F_∗", from=1-2, to=2-2]
    			["_FH^FG"', from=2-1, to=2-2]

		where F_∗ denotes postcomposition with F.
	
	
	
		First note that the vertical morphisms in the diagram are well-defined by Lemma <ref>. Now consider the extended diagram:
		
		
    _G()   _G()   _H()   _H()
    _FG()   _FG()   _FH()   _FH()["_G", from=1-1, to=1-2]
    			["(F)_G"', from=1-1, to=2-1]
    			["_H^G", from=1-2, to=1-3]
    			["F_∗"', from=1-2, to=2-2]
    			["(_H)^-1", from=1-3, to=1-4]
    			["F_∗", from=1-3, to=2-3]
    			["(F)_H", from=1-4, to=2-4]
    			["_FG", from=2-1, to=2-2]
    			["_FH^FG", from=2-2, to=2-3]
    			["(_FH)^-1", from=2-3, to=2-4]

		The leftmost and rightmost squares commute because both vertical morphisms are simply postcomposition with F. From Proposition <ref>, the upper and lower “squares” of the following diagram also commute:
		
		
    _G()   _G()   _H()   _H()
    _FG()   _FG()   _FH()   _FH()["_G", from=1-1, to=1-2]
    			["ϕ^∗", curve=height=-45pt, from=1-1, to=1-4]
    			["(F)"', from=1-1, to=2-1]
    			["_H^G", from=1-2, to=1-3]
    			["F_∗"', from=1-2, to=2-2]
    			["(_H)^-1", from=1-3, to=1-4]
    			["F_∗", from=1-3, to=2-3]
    			["(F)", from=1-4, to=2-4]
    			["_FG", from=2-1, to=2-2]
    			["(Fϕ)^∗"', curve=height=45pt, from=2-1, to=2-4]
    			["_FH^FG", from=2-2, to=2-3]
    			["(_FH)^-1", from=2-3, to=2-4]

		Finally, the outermost square commutes by Lemma <ref>.
		A standard diagram chase gives the result. 
	
	
	Suppose that F→ is a monoidal functor between ∞-topoi and G_∙∈(). We have from Lemma <ref> that F preserves G-actions. On the other hand, as a monoidal functor, F preserves group morphisms ϕ H_∙→ G_∙. Therefore ϕ induces an H-action on G, and Fϕ induces an FH-action on FG. Therefore there are, a priori, two ways to take a group map ϕ H_∙→ G_∙ and produce an FH-action on FG. The following corollary, which follows immediately from Proposition <ref>, is essentially the statement that these two actions agree.
	
	
		Let ϕ G_∙→ H_∙ be a morphism of group objects in  and F→ a monoidal functor between ∞-topoi. If we write ^ϕ(G,H) for right H-action on G induced by ϕ and ^Fϕ(FG,FH) for the FH-action on FG induced by Fϕ then there is an equivalence:
		
    F∘^ϕ(G,H)≃^Fϕ(FG,FH)

	
	
	

 §.§ The Space Case

	
	Given an ∞-topos  and a group object G_∙∈(), we have two equivalent ways of thinking about G-actions: as right action objects in _G(); or as right modules in _G() for G thought of as a (grouplike) -algebra. In the case that =, the ∞-topos of ∞-groupoids, we have yet another way. Using Corollary <ref> we have that the category of G-actions _G() is equivalent to the slice category _/ G. Then the next result follows from the straightening/unstraightening equivalence of <cit.>.
	
	
		There is an equivalence of ∞-categories _G()_/ G( G,).
	
	
	
		Note that if (X,G)∈() we have that G is a pointed ∞-groupoid and the associated functor G→ takes the base point to X∈. Moreover, every morphism in G is invertible, so this functor factors through Aut_(X). Indeed, any action of any group object H on X corresponds to a morphism H→Aut_(X). In other words, the space of actions on X by group objects of  is equivalent to the space of group maps into Aut_(X). Of course it's true that (X,G)∈() corresponds to a group map G→Aut_(X) for any ∞-topos , for instance by the results of <cit.>, but the case of  is particularly easy to see.
	
	
	Recall that the application of the unstraightening functor Un to a functor F Z→ can be given by taking the pullback of the universal coCartesian fibration U→ along F (see, for instance, <cit.>). We can use this to prove that two natural notions of “quotient” by a group action are compatible with the equivalence of Proposition <ref>.
	
	
		Let (X,G)∈() have associated functor F_X G→. Then X/G is equivalent to the colimit of F_X. 
	
	
	
		From Remark <ref> we have that X/G fits into a fiber sequence X→ X/G→ G. Proposition <ref> implies that the unstraightening of the  functor F_X G→ associated to (X,G) corresponds to the given map X/G→ G. In other words, we have a composite pullback diagram
		
		
    X    X/G   𝒰
    ∗    G   [from=1-1, to=1-2]
    			[from=1-1, to=2-1]
    			["⌟"anchor=center, pos=0.125, draw=none, from=1-1, to=2-2]
    			[from=1-2, to=1-3]
    			[from=1-2, to=2-2]
    			["⌟"anchor=center, pos=0.125, draw=none, from=1-2, to=2-3]
    			[from=1-3, to=2-3]
    			[from=2-1, to=2-2]
    			[from=2-2, to=2-3, "F_X"]

		
		From <cit.>, the colimit in  of the bottom right horizontal functor is equivalent to X/G. 
	
	
	

 §.§ Monomorphisms and Epimorphisms of Group Objects

	
	In the category of discrete groups every morphism can be factored as a surjection followed by an injection. This lifts the usual surjection/injection factorization system on the 1-category of sets. In this section we extend this factorization system to () for any ∞-topos . Recall the following result from <cit.>.

	
	
		If n denotes the class of n-connective morphisms, e.g. effective epimorphisms in the case that n=0, in  and n denotes the (n-1)-truncated morphisms of , then (n,n) is a factorization system on  in the sense of <cit.>.
	
	
	Proposition <ref> also appears in <cit.> although some translation in terminology is necessary. 
	
	
		Let  be an ∞-topos equipped with the factorization system (n,n) of Proposition <ref>. Then there is a factorization system (n_∗,n_∗) on _∗ where n_∗ and n_∗ are the classes of morphisms in _∗ which are in n, respectively n, after forgetting to . 
	
	
	
		We check the conditions of <cit.>. The two classes of morphisms are closed under retracts because they are closed under retracts in . The usual formulae for mapping spaces in slice and coslice ∞-categories and standard manipulations with pullback squares   imply that morphisms of n_∗ are still left orthogonal to morphisms of n_∗. Now let f X→ Y be a morphism of _∗ which factors as X Z Y with respect to (n,n). By choosing the pointing of Z to be the composite ∗→ X Z, we obtain a factorization in _∗.
	
	
	
		The factorization system (n_∗,n_∗) of Proposition <ref> restricts to a factorization system (n_∗^≥ 1,n_∗^≥ 1) on _∗^≥ 1. 
	
	
	
		Suppose that f X→ Y is a morphism in _∗^≥ 1. It suffices to show that if f factors in _∗ as X Z Y then Z is connected. By again applying Lemma <ref> to ∗→ X→∗ and using that X→∗ is 1-connective we have that ∗→ X is 0-connective, i.e. an effective epimorphism in . We also of course have that XZ is 0-connective (for instance, by <cit.>). By composing effective epimorphisms, e.g. using <cit.>, we have that ∗→ Z must be an effective epimorphism from which it follows, again by Lemma <ref>, that Z→∗ is 1-connective, hence Z is connected.
	
	
	
		Suppose f X→ Y is a morphism of  with Y pointed and connected. Then the fiber (f) is n-connective if and only if f is n-connective.
	
	
	
		We have a pullback diagram
		
		
    (f)    X 
    ∗    Y
    			[from=1-1, to=1-2]
    			[from=1-1, to=2-1]
    			["⌟"anchor=center, pos=0.125, draw=none, from=1-1, to=2-2]
    			["f", from=1-2, to=2-2]
    			[from=2-1, to=2-2]

		in which, since Y is connected, the bottom horizontal arrow is an effective epimorphism. Since, by definition, (f)→∗ is n-connective exactly when (f) is n-connective, the result then follows from <cit.>.
	
	
	
		Let f X→ Y be a morphism in either , _∗ or _∗^≥ 1 with a factorization X→ Z→ Y in the (n,n), (n_∗,n_∗) or (n_∗^≥ 1,n_∗^≥ 1) factorization systems respectively. Then we refer to Z as the n-image of f and write it _n(f). 
	
	
	
		Note that _n(f) is uniquely determined up to a contractible space of choices by <cit.>.
	
	
	
		Let (_n-1^,_n-1^) denote the factorization system on () induced by transferring the factorization system (n_∗^≥ 1,n_∗^≥ 1) along the equivalence _∗^≥ 1(). 
	


	A morphism f H→ G of  () is in _n^ if and only if it is n-connective as a morphism on underlying objects of .



	By Corollary <ref>, the underlying morphism of f is Ω fΩ H→Ω G. Moreover, f∈_n^ if and only if the map H→ G is (n+1)-connective. Therefore it suffices to show that if X→ Y is a morphism of ^≥ 1_∗ then it is (n+1)-connective if and only if Ω X→Ω Y is n-connective in . 
	
	
	We inductively apply <cit.>. For the base case, take n=-1. Suppose that f X→ Y is a morphism of _∗^≥ 1. Simply by virtue of being a morphism, Ω fΩ X→Ω Y is always (-1)-connected. Therefore it is only necessary to show that f is 0-connective. This follows immediately from recalling that for any pointed connected object the pointing itself is an effective epimorphism and applying <cit.> to the following commutative diagram
	
	
    X 
    ∗       Y
    		[from=1-2, to=2-3]
    		[from=2-1, to=1-2]
    		[from=2-1, to=2-3]

	
	Now suppose that the statement holds for 0≤ k≤ n. Suppose that f X→ Y is a morphism of pointed connected objects. Then f is (n+1)-connective if and only if the diagonal map δ X→ X×_Y X is n-connective. But by the inductive hypothesis δ is n-connective if and only if ΩδΩ X→Ω(X×_Y X)≃Ω X×_Ω YΩ X is (n-1)-connective. This is the case if and only if Ω fΩ X→Ω Y is n-connective, completing the proof. 



	A morphism ϕ H→ G is in _n^ if and only if its underlying map in  is n-truncated.



	
	Similarly to the preceding proposition, it suffices to show that a morphism f X→ Y in _∗^≥ 1 is n-truncated if and only if Ω fΩ X→Ω Y is (n-1)-truncated. We can also inductively use <cit.> in a similar way to reduce to the case of n=-1. 
	
	First we show that a morphism f X→ Y in _∗^≥ 1 is (-1)-truncated if and only if it is an equivalence. We have a commutative diagram in which both pointings are effective epimorphisms: 
	
    ∗       Y 
        X
    		[two heads, from=1-1, to=1-3]
    		[two heads, from=1-1, to=2-2]
    		[ "f"', from=2-2, to=1-3]

	Because factorizations are unique by <cit.>, and because f is (-1)-truncated, it must be the case that the lower composite in the above diagram is the unique factorization of the pointing ∗→ Y in the factorization system (0,0) (or its lifts to _∗ or _∗^≥ 1). By <cit.> we have that f is also an effective epimorphism. Therefore f is an equivalence (by, for instance, <cit.>). Of course if f is an equivalence, i.e. (-2)-truncated, then it is also (-1)-truncated.
	
	Now we check that f X→ Y in _∗^≥ 1 is an equivalence if and only if Ω f is an equivalence. This follows immediately from <cit.>, which shows that Ω is conservative in any ∞-topos. Therefore f is (-1)-truncated if and only if Ω f is an equivalence, i.e. (-2)-truncated.





	Let f H→ G be a morphism of () with factorization H K G induced by (_n^,_n^). Then we will refer to K as the n-image of f. 


Note that by Propositions <ref> and <ref> there is no ambiguity between Definitions <ref> and <ref>.


	When n=0 then the left class of morphisms in any of the above factorization systems will be referred to as effective epimorphisms and the right class will be called monomorphisms. 





 §.§ An Orbit Stabilizer Theorem


The classical Orbit-Stabilizer Theorem for discrete groups states that if a group G acts on a set X then (i) the set Stab_G(x)={g∈ G:gx=x} is a subgroup of G and (ii) there is a canonical isomorphism between the quotient G/Stab_G(x) and the orbits Orb_G(x)={y∈ X:gx=y for some g∈ G}. We prove this theorem for group objects in ∞-topoi here with an important caveat. The first is that asking for Stab_G(X) to be a “subgroup” of G is a much weaker condition in our setting. We only exhibit a canonical morphism of groups Stab_G(x)→ G. Our definitions below agree with those of <cit.>.


	Suppose that (X,G) is a right action object in  for G∈() with corresponding fiber sequence X X/G G. Suppose x∗→ X is a point of X and write z∗→ X/G for its composition with q. Taking the fiber of q over z induces a fiber sequence 
    G XX/G G

	Define the orbits of x∈ X with respect to p X/G→ G, denoted Orb_p(x), to be im_0(i). Define the stabilizer of x∈ X with respect to p, denoted Stab_p(x), to be _+(∗ X/G)∈(). Note that the underlying object of Stab_p(x) is the loop space object Ω_z(X/G). 





	Given  a right action object (X,G) in  with defining morphism p X/G→ G  and a morphism x∗→ X there is a canonical equivalence G/Stab_p(x)≃ Orb_p(x). 




	
	
	First we are explicit about the construction of the morphism ϕ Stab_p(x)→ G. By definition it is _+ applied to p X/G→ G, but for the case of exposition it will be useful to think of this as a diagram Δ^×Δ^1→:
	
	
    ⋮   ⋮
    Ω_z(X/G)×Ω_z(X/G)   G× G
    Ω_z(X/G)    G 
    ∗   ∗
    X/G    G[shift right, from=1-1, to=2-1]
    		[shift left, from=1-1, to=2-1]
    		[shift right=3, from=1-1, to=2-1]
    		[shift left=3, from=1-1, to=2-1]
    		[shift right, from=1-2, to=2-2]
    		[shift left, from=1-2, to=2-2]
    		[shift left=3, from=1-2, to=2-2]
    		[shift right=3, from=1-2, to=2-2]
    		["ϕ_2", from=2-1, to=2-2]
    		[from=2-1, to=3-1]
    		[shift left=2, from=2-1, to=3-1]
    		[shift right=2, from=2-1, to=3-1]
    		[from=2-2, to=3-2]
    		[shift left=2, from=2-2, to=3-2]
    		[shift right=2, from=2-2, to=3-2]
    		["ϕ_1", from=3-1, to=3-2]
    		[shift right, from=3-1, to=4-1]
    		[shift left, from=3-1, to=4-1]
    		[shift left, from=3-2, to=4-2]
    		[shift right, from=3-2, to=4-2]
    		[from=4-1, to=4-2, "ϕ_0"]
    		["z"', from=4-1, to=5-1]
    		[from=4-2, to=5-2]
    		["p", from=5-1, to=5-2]

	
	By definition, ϕ Stab_p(x)→ G is obtained by taking the colimit of the non-augmented part of the above diagram. This factors the above diagram so that the following commutes:
	
	
    ⋮   ⋮
    Ω_z(X/G)    G 
    ∗   ∗
     Stab_p(x)    G
    X/G    G[from=1-1, to=2-1]
    		[shift left=2, from=1-1, to=2-1]
    		[shift right=2, from=1-1, to=2-1]
    		[from=1-2, to=2-2]
    		[shift left=2, from=1-2, to=2-2]
    		[shift right=2, from=1-2, to=2-2]
    		["ϕ_1", from=2-1, to=2-2]
    		[shift right, from=2-1, to=3-1]
    		[shift left, from=2-1, to=3-1]
    		[shift left, from=2-2, to=3-2]
    		[shift right, from=2-2, to=3-2]
    		["ϕ_0", from=3-1, to=3-2]
    		[from=3-1, to=4-1]
    		[from=3-2, to=4-2]
    		["ϕ", from=4-1, to=4-2]
    		[from=4-1, to=5-1]
    		[from=4-2, to=5-2,"≃"]
    		["p", from=5-1, to=5-2]

	
	Because  is an ∞-topos we have that ∗→ Stab_p(x) is an effective epimorphism, i.e. Stab_p(x) is connected. On the other hand, <cit.> implies that Stab_p(x)→ X/G is (-1)-truncated. Recall that we construct the quotient of G by Stab_p(x) via ϕ by taking the pullback of ∗→ G along ϕ Stab_p(x)→ G.
	
	
	Therefore we have a composite of pullback squares 
	
	
    G    G/Stab_p(x)    X    ∗
    ∗    Stab_p(x)   X/G    G[from=1-1, to=1-2]
    		[two heads, from=1-1, to=2-1]
    		["⌟"anchor=center, pos=0.125, draw=none, from=1-1, to=2-2]
    		[from=1-2, to=1-3]
    		[two heads, from=1-2, to=2-2]
    		["⌟"anchor=center, pos=0.125, draw=none, from=1-2, to=2-3]
    		[from=1-3, to=1-4]
    		[two heads, from=1-3, to=2-3]
    		["⌟"anchor=center, pos=0.125, draw=none, from=1-3, to=2-4]
    		[two heads, from=1-4, to=2-4]
    		[from=2-1, to=2-2]
    		[from=2-2, to=2-3]
    		["ϕ"', curve=height=12pt, from=2-2, to=2-4]
    		[from=2-3, to=2-4]

	in which each vertical morphism is an effective epimorphism because G is connected and effective epimorphisms are stable under pullback. Similarly, because ∗→ Stab_p(x) is an effective epimorphism so is G→ G/Stab_p(x). Finally, we have already seen that Stab_p(x)→ X/G is (-1)-truncated and therefore so is G/Stab_p(x)→ X by <cit.>. By uniqueness of factorizations (see <cit.>) we must have that G→ G/Stab_p(x)→ X is the 0-connective/(-1)-truncated factorization of G→ X. In other words, G/Stab_p(x) is the 0-image of G→ X in the (_0,_0) factorization system on . So there is an equivalence making the following diagram commute which is unique up to a contractible space of choices.
	
	
    Orb_p(x)
    
    		G        X 
       G/Stab_p(x)[from=1-2, to=2-3]
    		[from=2-1, to=1-2]
    		[from=2-1, to=3-2]
    		["≃"', from=3-2, to=1-2]
    		[from=3-2, to=2-3]






§ NORMAL MAPS OF GROUP OBJECTS





Going forward, we will write G for: a group object G_∙∈(); the underlying pointed object of that group object s_0^∗ G∈_∗; the underlying object G_1∈. Which category G is living in will be clear from context.

Suppose G is a group object in  and H≤ G is a subgroup. By composing with the translation action of G on itself, G inherits a right H-action and we may form the set of cosets, or orbits, G/H. The notion of normality is an answer to the question: when does G/H admit a group structure compatible with the quotient map (of sets) G→ G/H? That question can be answered in the affirmative precisely when H is a normal subgroup. There are many equivalent definitions of “normal,” some of which make sense in the setting of higher algebra and some of which do not. 

Our definition of normality below, essentially the same as that of Prasma <cit.>, defines normality of a “subgroup” H in G to be the property that the quotient of G by the induced right H-action admits a group structure. However, there are immediately two significant differences between our case and the classical case.

First, notice that “being a subgroup” is not a well-behaved concept in homotopy theory. For instance, we can clearly take S^1 to be a subgroup of ℂ in classical group theory, but up to homotopy the inclusion becomes the terminal map S^1→∗. Therefore, again following Prasma (who in turn is following Farjoun and Segev <cit.>), we consider normality for arbitrary morphisms of group objects H→ G. 

Secondly, we will soon see that if H→ G is a morphism of group objects in an ∞-topos (indeed, simply take =), then there may be more than one group structure on G/H so that the quotient map G→ G/H is a group morphism. This is in contrast to the discrete case, where the group structure on G/H, if it exists, will be unique. To deal with this added complexity, we make a change which is common in higher algebra and replace the property of being normal with structure or data. This leads us to the following definition of normality data for a morphism of group objects.



	Let  be an ∞-topos and let ϕ H→ G be a morphism in (). We say a morphism π G→ X in _∗^≥ 1 is a normality datum for ϕ if the fiber of π is equivalent to ϕ H→ G. If ϕ admits at least one normality datum, we will say that ϕ is a normal map of group objects. 



	Let F→ E→ B be a fiber sequence of connected pointed objects in an ∞-topos. Then Ω F→Ω E is normal with normality datum E→ B. Note that by assuming F is connected we guarantee that E→ B is 1-connective and so Ω E→Ω B is an effective epimorphism, which also allows us to apply Proposition <ref>. Note that every normality datum is of this form.


The following proposition makes it clear that, indeed, a normality datum induces a group structure on the quotient G/H.


	Let ϕ H→ G be a morphism of () with a normality datum π G→ X. Then the underlying object in  of  (X) is equivalent to the quotient G/H of Definition <ref>. 



	Recall from Example <ref> that the right action of H on G induced by ϕ is obtained by taking the translation action of G on itself, corresponding to the pointing ∗→ G, and pulling it back along ϕ H→ G. In other words, G/H, which is the geometric realization of the right H-action on G induced by ϕ, is equivalent to the fiber of ϕ. Because there is a fiber sequence H G X we have that G/H≃Ω X. On the other hand, (X) is obtained, by definition, by taking the Čech nerve of the pointing ∗→ X. The arguments in the proof of Corollary <ref> make it clear that the underlying object of (X), in the sense of Definition <ref>, is equivalent to Ω X. 



	There is potentially some ambiguity in Definition <ref>, in that we do not specify in which category we are forming the fiber of π. In general, the fiber in _∗^≥ 1 of a morphism of pointed connected objects need not be equivalent to the fiber taken in _∗. The former is obtained from the latter by deleting everything which is not in the same “connected component” of the base point (see Corollary <ref>). However, by demanding that the fiber of π be equivalent to H, which is necessarily connected, we have that the two fibers are in agreement. 




We reiterate that a morphism of group objects can be normal in more than one way. This is illustrated by the following two examples, the first of which is due to Kiran Luecke. 	See also Example <ref> for another case in which the same group map, in this case the trivial map ℤ→Ω S^2, can have more than one normality datum.


	Let f K(,2)→ K(,4) generate the degree 4 integral cohomology of K(,2). Let F be the fiber of f and note that because f is not null it is not the case that F splits as a product. Taking loops gives a map Ω f K(,1)→ K(,3) which must be null because K(,1) has no integral third cohomology. Therefore we must have that Ω F→ K(,1)→ K(,3) is a fiber sequence, whence Ω F≃ K(,1)× K(,3). Therefore we have a fiber sequence K(,1)× K(,3)→∗→ F. On the other hand, we have a fiber sequence K(,1)× K(,3)→∗→ K(,2)× K(,4). But F is not equivalent to K(,2)× K(,4) so we have two inequivalent normality data for the map K(,1)× K(,3)→∗.



	For another example, let G be an object of () which admits two distinct _2-algebra structures lifting its _1-structure. Each of these _2-structures will produce distinct deloopings, say BG and BG'. Then ∗→ BG and ∗→ BG' are distinct normality data for the same map, namely G→∗.  



	Following classical group theory, one might wish to say that a map of group objects ϕ H→ G is normal whenever the dotted arrow below is an equivalence (where the cofiber and the fiber are both taken in ()):
	
	
    G 
       ((ϕ))    H 
       ∗   (ϕ)
    		[dashed, from=1-1, to=2-2]
    		[curve=height=-12pt, from=1-1, to=2-3]
    		[curve=height=12pt, from=1-1, to=3-2]
    		[from=2-2, to=2-3]
    		[from=2-2, to=3-2]
    		["⌟"anchor=center, pos=0.125, draw=none, from=2-2, to=3-3]
    		["(ϕ)", from=2-3, to=3-3]
    		[from=3-2, to=3-3]

	We will see in Section <ref> that (ϕ) does behave something like a normal closure. However, in the setting of an ∞-topos, where fibers and cofibers are inherently homotopical, this definition is too strong. 
	
	To see this, consider the multiplication by 2 map 2→. This deloops to the index 2 map 2 S^1→ S^1 whose cofiber (in _∗^≥ 1) is equivalent to ℝP^2. If the above definition recovered classical normality then we would require S^1→(S^1→ℝP^2) to be an equivalence. However, a cursory examination of the long exact sequence in homotopy groups reveals that this cannot be the case (for instance, π_2((S^1→ℝP^2))≅). So the preceding definition of normality does not contain classical normality as a special case (unlike Definition <ref>). It is not hard to check that the above definition of normality also makes Noether's First Isomorphism Theorem false. 





 §.§ A Category of Normality Data


Clasically, e.g. in Galois theory, one often considers the lattice of normal subgroups of a discrete group G. This lattice is closely related to the set of quotient groups of G and one passes from the latter to the former by taking kernels of quotient maps. We do something similar here, but replace the lattice of normal subgroups with a category comprising all normality data associated to a group object G. First we note an important property of all normality data.


	Let ϕ H→ G be a morphism of () and let π G→ X be a normality datum for ϕ. Then G→ X is 1-connective. 
 


	If G→ X  is a normality datum for ϕ then there is a fiber sequence H G→ X. This corresponds to a pullback square:
	
	
    H    G
    ∗    X
    		["ϕ", from=1-1, to=1-2]
    		[from=1-1, to=2-1]
    		["⌟"anchor=center, pos=0.125, draw=none, from=1-1, to=2-2]
    		["π", from=1-2, to=2-2]
    		[from=2-1, to=2-2]

	Because X is connected, the pointing ∗→ X is an effective epimorphism. Because H is connected, i.e. 1-connective, the terminal map H is 1-connective. Then from <cit.> we have that π must also be 1-connective. 



	In the case that =, Lemma <ref> implies that the normality datum G→ X is a surjection on π_1. It follows from this, e.g. by the long exact sequence in homotopy, that the associated quotient map is a surjection on π_0. This should not be surprising as one would expect “quotient maps” to be effective epimorphisms in general. This is generalized, and made more precise, by Proposition <ref>.



	Let  be an ∞-topos and G∈().
	
		
  * Define the ∞-category (G) to be the full subcategory of the slice category (_∗^≥ 1)^\ G spanned by the 1-connective morphisms. 
		
  * Define the underlying group functor U_G(G)→()_/G to be the composite of (G)→ (_∗^≥ 1)_/ G with the equivalence _/ G (_∗^≥ 1)_/ G→()_/G. 
		
  * If ϕ H→ G is a morphism of (), define the ∞-category of normality data for ϕ, denoted _ϕ(G), to be the following pullback:
		
		
    _ϕ(G)   (G)
    {ϕ}   ()_/G[from=1-1, to=1-2]
    			[from=1-1, to=2-1]
    			["⌟"anchor=center, pos=0.125, draw=none, from=1-1, to=2-2]
    			[from=1-2, to=2-2, "U_G"]
    			[from=2-1, to=2-2]

	 


Note that any morphism G→ X in _∗^≥ 1 can be used to produce a normality datum. Specifically, one takes a the fiber of G→ X in _∗ to obtain a fiber sequence F→ G→ X. Beacuse the fiber was taken in _∗ (rather than _∗^≥ 1), it need not be the case that F is connected and therefore, by definition, G→ X cannot be a normality datum for any group map. However, we may apply _+_∗→() to obtain a fiber sequence in (). For the sake of exposition, let us be imprecise and write Ω instead of  and denote this fiber sequence by Ω F→ G→Ω X. Applying _∗ we obtain a new fiber sequence Ω F→ G→Ω X in _∗^≥ 1. Because X is connected, we have that Ω X≃ X (see Lemma <ref>). Hence we now have a normality datum for the group map Ω F→ G given by an equivalent map G→ X. By restricting to 1-connective morphisms, we remove the “overcounting” caused by such ambiguity. 


	The fully faithful inclusion (G)↪ (_∗^≥ 1)^\ G followed by ^\ G (_∗^≥ 1)^\ G→()^\ G, which is an equivalence, has essential image the full subcategory spanned by group morphisms G→ Q which are effective epimorphisms on underlying objects of .



	It suffices to show that a group morphism ϕ G→ Q is an effective epimorphism if and only if G→ Q is 1-connective. Given that, as a morphism in , ϕ≃Ωϕ, this follows immediately from Proposition <ref>.


This characterization of normality allows one to produce examples more readily.


	Let ϕ R→ S be a morphim of _1-ring spectra, as in <cit.>, with fiber (ϕ)→ R (which is a morphism of _1-rings as well).  Suppose that π_0(ϕ)π_0(R)→π_0(S) is surjective and π_0(R) is left Artinian. Applying the units functor GL_1__1()→(), which is a right adjoint, gives a fiber sequence in () 
    GL_1((ϕ))→GL_1(R)→GL_1(S)
 in which the second map is an effective epimorphism by <cit.>. It follows that the first map has a normality datum given by by BGL_1(R)→BGL_1(S). It would be interesting to know some non-trivial normal maps over groups of units like GL_1(). 



	Let ϕ H→ G be a morphism of (). Then the category of normality data for ϕ is equivalent to the pullback of the cospan 
	
    {ϕ}   ()_/G   ()^\ G_epi[from=1-1, to=1-2]
    		["", from=1-3, to=1-2]

	where ()^\ G_epi denotes the full subcategory of ()^\ G spanned by the epimorphisms. 
	



	Every group object G∈() is a quotient of a free group object.



	Recall the free group object functor Σ_∗→() of Lemma <ref> which is left adjoint to the forgetful functor s_0^∗()→_∗. This adjunction necessarily has a counit ϵ_GΣ s_0^∗ G→ G. The triangle identities of the adjunction give a commutative triangle
	
	
    s_0^∗ G      s_0^∗ G
       s_0^∗Σ s_0^∗ G["id_s_0^∗ G", from=1-1, to=1-3]
    		["η_s_0^∗ G"', from=1-1, to=2-2]
    		["s_0^∗(ϵ_G)"', from=2-2, to=1-3]

	By <cit.> and <cit.> it must be the case that s_0^∗(ϵ_G) is also an effective epimorphism. Hence ϵ_GΣ s_0^∗ G→ G is a normality datum, i.e. G≃Σ s_0^∗ G/(ϵ_G).


We will need Lemma <ref> in the next section but first we prove a preliminary technical result.


	Let K be a sifted ∞-category and suppose that X_K K→ is a constant functor for some X∈. Then (X_K)≃ X.



	From (the proof of) <cit.>, we have that the terminal morphism K→Δ^0 is right cofinal, in the sense of <cit.>. It follows that pulling back along K→Δ^0 preserves colimits. Every constant functor out of K is obtained by pulling back in this way, so the colimit of X_K is equivalent to the colimit of XΔ^0→, which of course is X itself.



	The category (G) is closed under sifted colimits. 


	
	Let K be a sifted ∞-category and let F K→(G) a functor. For the purposes of this proof, write U^≥ 1_∗ (_∗^≥ 1)^\ G≃(G)→_∗^≥ 1 and U(G)→ for the forgetful functors. Write K^◃ for the ∞-category K with an initial object, {-∞} adjoined.
	
	By writing (^≥ 1_∗)^\ G as the pullback of the cospan { G}↪_∗^≥ 1(_∗^≥ 1)^Δ^1, where s is restriction to {0}∈Δ^1, and applying (K,-), we see that F is equivalent to the data of a functor F' K^◃→_∗^≥ 1 with F'(-∞)≃ G.  It is “standard” that U_∗^≥ 1((F))≃(F') and that the structure morphism of this colimit, as an object of the undercategory, is the composite F'(-∞)≃ G⇒ U_∗^≥ 1F⇒(F') (see, for instance, <cit.>). By Lemma <ref> we may equivalently take this to be the morphism G≃( G_K)→(F'). To ensure that G→(F') determines an object of (G), we need to show that it is 1-connective. We will write ζ G→(F') for this morphism. By Lemma <ref> it suffices to show that (ζ) is connected. By Lemma <ref> we have that the inclusion _∗^≥ 1↪_∗ preserves colimits, so we may take the colimit in _∗.
	
	By another application of <cit.>, this time to F' K^◃→_∗, we obtain a commutative triangle of natural transformations of functors K→:
	
	
    ∗_K       G_K
        UF
    		[Rightarrow, from=1-1, to=1-3]
    		[Rightarrow, from=1-1, to=2-2]
    		[Rightarrow, from=1-3, to=2-2]

	or, equivalently, a functor Δ^2→(K,). Restricting to the 2-horn gives a cospan Λ_2^2→(K,) or equivalently a functor K→(Λ_2^2,). From <cit.> we have that the limit of ∗≃(∗_K)→(UF)←( G)≃ G is equivalent to the colimit of limits ∗→ UF(k)← G for each k∈ K. But each G→ UF(k), by assumption, is 1-connective and therefore each fiber over UF(k) is connected. It remains to show that sifted colimits of connected objects are connected. First recall from <cit.> that the projection _∗≃^\∗→ is a left fibration. We also have, again from the proof of <cit.>, that K is weakly contractible. Hence the colimit of the fibers G→ UF(k) is connected as a result of <cit.>. This completes the proof.


		
		

 §.§ Normality Data is Monadic

		
		Recall that in classical group theory one can take the “normal closure” of a subgroup H≤ G. Explicitly, this is the smallest normal subgroup of G containing H. If we write ncl_G(H) for the normal closure of H in G, then we have the following standard fact:
		
		
			Let H≤ G be a subgroup. Then the pushout of the cospan 0 ← H→ G in the category of discrete groups is isomorphic to G/ncl_G(H). 
		
		
		In the case of discrete groups, the above proposition allows one to characterize normal subgroups H⊴ G as those with the property that the canonical morphism H→(G→ G/ncl_G(H)) is an isomorphism. We have already seen in Remark <ref> that this definition is not well-behaved in higher group theory. Nonetheless, there is still a useful notion of normal closure for morphisms of group objects which produces the “free normality datum” generated by a morphism H→ G in ().
		
		
			If f X→ Y is a morphism in _∗^≥ 1 then (f) Y→(f) is 1-connective in . In particular, the cofiber functor (_∗^≥ 1)_/ G→ (_∗^≥ 1)^\ G factors through (G).
		
		
		
			Recall that, by definition, the terminal morphism X→∗ is 1-connective since X is connected. By <cit.> we know that 1-connective morphisms are preserved by pushout, so the result follows from considering the defining pushout diagram of the cofiber:
			
			
    X     Y 
    ∗   (f)["f", from=1-1, to=1-2]
    				[from=1-1, to=2-1]
    				[from=1-2, to=2-2]
    				[from=2-1, to=2-2]
    				["⌟"anchor=center, pos=0.125, rotate=180, draw=none, from=2-2, to=1-1]

		
		
		
			Let G∈(). Then we define the normal closure in G functor to be the following composite:
			
    _G()_/G (_∗^≥ 1)_/ G(G)

			By applying the underlying group functor U_G(G)→()_/G we obtain a morphism of group objects ((ϕ))≃((ϕ))→ G.
		
		
		
			Taking G=∗, we see that the normal closure of the terminal group map H→∗ is the terminal morphism ΩΣ H→∗ or, on underlying objects, Ω^2Σ H→∗. This is consistent with the fact that normal morphisms to ∗ in () are the same as double loop spaces and that Ω^2Σ is the “free double loop space on a loop space” functor (not the be confused with the “free double loop space on a pointed space” functor which is given by Ω^2Σ^2). 
		
		
		
			There is a monadic adjunction between ()_/G and (G) whose left adjoint is _G and whose right is the underlying group functor U_G.
		
		
		
			First we show the existence of the adjunction. We begin with the adjunction 
			
    (_∗)_/BG      (_∗)^\ BG[""name=0, anchor=center, inner sep=0, "", curve=height=-6pt, from=1-1, to=1-3]
    				[""name=1, anchor=center, inner sep=0, "", curve=height=-6pt, from=1-3, to=1-1]
    				["⊣"anchor=center, rotate=-90, draw=none, from=0, to=1]
 described in Lemma <ref>.  Using the definition of adjunctions given in <cit.> it is not hard to check that, after restricting on the left to the full subcategory (_∗^≥ 1)_/BG and on the right to (G), the unit and counit transformations descend to unit and counit transformations for these restricted functors and still satisfy the triangle identities (since we have restricted to full subcategories). Therefore we have another adjunction 
			
    (_∗^≥ 1)_/BG      (G)[""name=0, anchor=center, inner sep=0, "", curve=height=-6pt, from=1-1, to=1-3]
    				[""name=1, anchor=center, inner sep=0, "", curve=height=-6pt, from=1-3, to=1-1]
    				["⊣"anchor=center, rotate=-90, draw=none, from=0, to=1]

			
			We now check that this adjunction is monadic via the Barr-Beck Theorem of <cit.>. Checking conservativity requires showing that (α) in the following diagram
			
			
    (ϕ)      (ψ)
    
    				BG        BG 
    
    				X        Y
    				["ϕ"', from=2-1, to=3-1]
    				["α", from=3-1, to=3-3]
    				[from=1-1, to=2-1]
    				[from=1-3, to=2-3]
    				["ψ", from=2-3, to=3-3]
    				[shift left=1, no head, from=2-1, to=2-3]
    				[no head, from=2-1, to=2-3]
    				["(α)", from=1-1, to=1-3]

			is an equivalence if and only if α is. Of course, taking fibers is functorial and therefore preserves equivalences, so the “if” direction is immediate. For the “only-if” direction, note that ψ and ϕ being 1-connective (by assumption) implies that α is also 1-connective. Therefore it suffices to show that α is also 0-truncated whenever (α) is an equivalence. Note that, by definition, α is 0-truncated if and only if (Z,α)(Z,X)→(Z,Y) is 0-truncated for all Z∈. Now apply (Z,-) to the morphism of pullback diagrams (using the fact that (Z,-) preserves limits) to obtain a morphism of pullback diagrams in :
			
			
    (ϕ_∗)   (ψ_∗)
    (Z, G)   (Z, G)
    (Z,X)   (Z,Y)[from=1-1, to=1-2]
    				[from=1-1, to=2-1]
    				[from=1-2, to=2-2]
    				[shift left=1, no head, from=2-1, to=2-2]
    				[ no head, from=2-1, to=2-2]
    				["ϕ_∗"', from=2-1, to=3-1]
    				["ψ_∗", from=2-2, to=3-2]
    				["α_∗"', from=3-1, to=3-2]

			
			Applying the 5-lemma to the morphism of long exact sequences in homotopy for the two fiber sequences proves the result in . In other words, (Z,X)→(Z,Y) is 0-truncated when (ϕ_∗)→(ψ_∗) is an equivalence. This holds for all Z∈ so α is 0-truncated by definition, and therefore an equivalence.
			
			Suppose we are given a simplicial object of (G). By Lemma <ref> we know that this diagram admits a colimit so it suffices to show that taking fibers preserves this colimit, but this was already shown in the proof of Lemma <ref>. The result follows from composing with the equivalence ()_/G≃ (_∗^≥ 1)_/BG.
		
		
		Another way to state Theorem <ref> is to say that normal maps to G are the algebras for the normal closure monad. By taking monadic resolutions this gives an alternative proof of Corollary <ref>. Noting that normality of the terminal morphism G→∗ in () corresponds to a grouplike _2-algebra structure on G, also gives a description of grouplike _2-algebras in  as algebras for the ΩΣ monad, which can be thought of as an expression of Dunn additivity.
		
		
			Note that the normal closure of a normal map ϕ H→ G is not generally equivalent to ϕ. This corresponds to the fact that taking normal closure corresponds to freely adding normality data which necessarily makes ϕ “bigger.” Alternatively, note that because normality data is not generally unique any normal closure functor cannot “know” which normality datum to append to a given map, whether or not that map is normal. As such, it is forced to add new data. 
		
		
		
		
		

 §.§ Normality is Preserved by Monoidal Functors

		
		In <cit.>, Prasma shows that a homotopy monoidal endofunctor of the category of topological spaces preserves normal maps of loop spaces. In this section we will give a slight generalization of his result by showing that any monoidal functor of ∞-topoi preserves normal maps of group objects. Note that monoidal maps do not preserve quotients. To clarify, let ϕ H→ G be normal in () and let F→ be a monoidal functor of ∞-topoi. We will show that F(ϕ) F(H)→ F(G) remains a normal morphism of group objects. The fact that F is monoidal and G/H has a group structure imply that F(G/H) has a group structure as well. However, what is not generally true is that if F is monoidal then the group F(G/H) is equivalent to the group F(G)/F(H) (where the latter has a group structure coming from F preserving normality data). This is true almost by definition of course if F preserves pullbacks or geometric realizations, but we wish to recover Prasma's more general theorem. 
		
		
			Suppose that a morphism ϕ H→ G in () admits a normality datum π G→ (G/H). Then _∙(G,H)Δ^→, the bar construction for the H-action on G induced by ϕ, factors through (). 
		
		
		
			By assumption there is a fiber sequence in ^≥ 1_∗: 
    H G (G/H)
 Now take the Čech nerve of π, followed by  (applied levelwise) to get a simplicial object of (). Noticing that _∗^≥ 1→() and U_()→ are both right adjoints which preserve right Kan extensions we have that U_∘ applied to the Čech nerve of π is the Čech nerve of G≃Ω G→Ω (G/H)≃ G/H. From this we have that the Čech nerve of G→ G/H lifts to a simplicial group object.
			
			Now from the discussion following <cit.> we recall that groupoid resolutions of effective epimorphisms are unique. We know by Lemma <ref> that _∙(G,H) is a groupoid resolution of G→ G/H and therefore is equivalent, as a simplicial object of , to the Čech nerve of G→ G/H. It follows that it factors through ().
		
		
		
			If ϕ H→ G is a normal map in () and F→ is a monoidal functor of ∞-topoi then F(ϕ) F(H)→ F(G) is normal in (). 
		
		
		
			From Corollary <ref>, we have that F applied to the action object of H on G (induced by ϕ) is the action object of F(H) on F(G) (induced by F(ϕ)). By Lemma <ref>, we have that _∙(H,G)∈()^Δ^ and therefore _∙ (FG,FH)∈()^Δ^. Note also that Lemma <ref> implies that _∙(FG,FH) is a groupoid resolution, by definition, of F(G)/F(H). 
			
			Now take the colimit of _∙(FG,FH) in (). The forgetful functor U_()→ preserves sifted colimits (as shown, e.g., in the proof of <cit.>) so there is an effective epimorphism of group objects F(G)→ F(G)/F(H). It follows from Proposition <ref> that F(ϕ) F(H)→ F(G) is normal, with normality datum F(G)→ (F(G)/F(H)).
		
		
		Since a map of group objects H→ G being normal is the same as the existence of a fiber sequence H→ G→ X in , we also have that monoidal functors of ∞-topoi preserve the property of being a fiber, at least for connected objects. This is similar to <cit.> which recovers a result of Dwyer and Farjoun <cit.> regarding the preservation of principal fibrations by localization functors.
		
		
			If  F→ is a monoidal functor of ∞-topoi and X→ Y→ Z is a fiber sequence of connected objects in  then FX→ FY is the fiber of a morphism FY→ W in . 
		
		
		
		
		

§ NOETHER'S ISOMORPHISM THEOREMS IN ∞-TOPOI

		
		In this section we show that Noether's First Isomorphism Theorem holds for group objects in ∞-topoi and that her Second and Third Isomorphism Theorems do not hold (indeed, the third can barely be stated). 
		
		

 §.§ The First Isomorphism Theorem

		
		Noether's First Isomorphism Theorem for discrete groups can be stated as follows:
		
			Let ϕ H→ G be a morphism of discrete groups. Then:
			
				
  * The image im(ϕ)⊆ G is a subgroup of G.
				
  * The kernel (ϕ)⊆ H is a normal subgroup of H.
				
  * There is an isomorphism of groups H/(ϕ)≅ im(ϕ). 
			 
		
		
		We now state the analogous theorem for group objects in ∞-topoi. Note that for group objects in ∞-topoi the statement is shorter because providing a normality datum simultaneously implies the second and third points of the classical theorem.
		
		
			Let ϕ H→ G be a morphism in (). Then H→_0(ϕ) is a normality datum for (ϕ)→ H. 
		
		
		
			By construction the morphism H→_0(ϕ) is an effective epimorphism in (), so by Proposition <ref> it only remains to show that (H→_0(ϕ))≃(ϕ). 
			
			By Proposition <ref> we know that _0(ϕ)→ G is (-1)-truncated.  It follows from <cit.> that the diagonal _0(ϕ)→_0(ϕ)×_G_0(ϕ) is (-2)-truncated, i.e. an equivalence. Now consider the commutative diagram
			
			
    _0(ϕ)
       _0(ϕ)×_G_0(ϕ)   _0(ϕ)
       _0(ϕ)    G
    				["≃"description, from=1-1, to=2-2]
    				["≃"description, curve=height=-12pt, from=1-1, to=2-3]
    				["≃"description, curve=height=18pt, from=1-1, to=3-2]
    				[from=2-2, to=2-3]
    				[from=2-2, to=3-2]
    				["⌟"anchor=center, pos=0.125, draw=none, from=2-2, to=3-3]
    				[from=2-3, to=3-3]
    				[from=3-2, to=3-3]
 in which every morphism is (-1)-truncated and the indicated morphisms are equivalences. It follows that both projections _0(ϕ)×_G_0(ϕ)→_0(ϕ) are equivalences. Because the horizontal fibers of the above pullback square must be equivalent we have that (_0(ϕ)→ G)≃∗. Therefore we have the composite pullback square
			
			
    (ϕ)   ∗   ∗
    
    				H    _0(ϕ)    G
    				[from=1-1, to=1-2]
    				[from=1-1, to=2-1]
    				["⌟"anchor=center, pos=0.125, draw=none, from=1-1, to=2-2]
    				[from=1-2, to=1-3]
    				[from=1-2, to=2-2]
    				["⌟"anchor=center, pos=0.125, draw=none, from=1-2, to=2-3]
    				[from=1-3, to=2-3]
    				[from=2-1, to=2-2]
    				[from=2-2, to=2-3]

			implying that (ϕ)≃(H→_0(ϕ)).
		
		
		
			It is worth noting that in the case =, the fact that _0(ϕ)→ G is (-1)-truncated implies that it is an injection on π_0. Therefore the above theorem recovers the classical First Isomorphism Theorem after taking π_0. 
		
		
		

 §.§ The Second Isomorphism Theorem

		
		Noether's Second Isomorphism Theorem for discrete groups can be stated as follows:
		
		
			Let H, K ≤ G be subgroups with K ⊴ G normal in G. Then,
			
				
  * H ∩ K ⊴ H is normal;
				
  * K ⊴ HK is normal;
				
  * There exists an isomorphism of groups
				
    H/H ∩ K ≅ HK/K.

			
		
		
		To prove this statement for group objects in an ∞-topos, we first need to make sense of the notions of intersection and product for morphisms K→ G and H→ G in (). We generalize these constructions in the “obvious” way. Taking π_0 recovers the classical definitions. 
		
		
			Let ϕ H→ G and ψ K→ G be morphisms in (). Then we define:
			
				
  * The intersection of ϕ and ψ, denoted ϕ∩ψ H∩ K→ G is the following pullback:
				
				
    H∩ K    K 
    
    					H     G
    					[from=1-1, to=1-2]
    					[from=1-1, to=2-1]
    					["⌟"anchor=center, pos=0.125, draw=none, from=1-1, to=2-2]
    					["ψ", from=1-2, to=2-2]
    					["ϕ"', from=2-1, to=2-2]

				
  * The product of ϕ and ψ, which we will denote by ϕψ HK→ G, is the following pushout:
				
				
    H∩ K    K 
    
    					H     HK 
           G
    					[from=1-1, to=1-2]
    					[from=1-1, to=2-1]
    					[from=1-2, to=2-2]
    					[curve=height=-6pt, from=1-2, to=3-3]
    					[from=2-1, to=2-2]
    					[curve=height=6pt, from=2-1, to=3-3]
    					["⌟"anchor=center, pos=0.125, rotate=180, draw=none, from=2-2, to=1-1]
    					["ϕψ"description, from=2-2, to=3-3]

				
			The pullback and pushout above are both taken in (). 
			
		
		
		The following is an counterexample to the statement that H/H∩ K≃ HK/K. 
		
		
			Take K = H = ∗ and G =  S^2 in =. It is clear that the trivial map K → G has a normality datum with G/K = G. Since  is a right adjoint we have that H ∩ K ≃Ω S^2. If we then deloop, we obtain
			
    (H ∩ K) ≃Ω S^2 ≃Ω S^2.

			As for the product,  preserves colimits and so (HK) ≃ΣΩ S^2.
			
			
    Ω S^2   ∗
    ∗   ΣΩ S^2[from=1-1, to=2-1]
    				[from=2-1, to=2-2]
    				[from=1-2, to=2-2]
    				[from=1-1, to=1-2]
    				["⌟"anchor=center, pos=0.125, rotate=180, draw=none, from=2-2, to=1-1]

			We then have the following equivalences of fiber sequences in .
			
			
    H/H ∩ K    (H ∩ K)    H       HK/K    K     (HK) 
    Ω S^2   Ω S^2   ∗      ΩΣΩ S^2   ∗   ΣΩ S^2["≃"', from=1-3, to=2-3]
    				["≃"', from=1-2, to=2-2]
    				["≃"', from=1-1, to=2-1]
    				[from=1-1, to=1-2]
    				[from=1-2, to=1-3]
    				[from=2-1, to=2-2]
    				[from=2-2, to=2-3]
    				["≃"', from=1-5, to=2-5]
    				["≃"', from=1-6, to=2-6]
    				["≃"', from=1-7, to=2-7]
    				[from=1-5, to=1-6]
    				[from=1-6, to=1-7]
    				[from=2-5, to=2-6]
    				[from=2-6, to=2-7]

			Hence, H/H ∩ K ≄HK/K.
		
		
		

 §.§ The Third Isomorphism Theorem

		
		One might state the classical Third Isomorphism Theorem as follows:
		
		
			Let G be a group with normal subgroups H and K such that K⊆ H. Then the quotient group (G/K)/(H/K) is isomorphic to G/H. 
		
		
		Note, however, that this statement has some implicit assumptions which, while always true for discrete groups, need not be true for group objects in an arbitrary ∞-topos. First, if we want to write down the quotient (G/K)/(H/K), we need that K be normal in H. Otherwise, we do not have that H/K is necessarily a group at all. Even assuming that we have K normal in H, we still need that H/K be a normal subgroup of G/K to make the final statement. 
		
		The first condition above is sometimes phrased as “normality satisfies the intermediate subgroup condition.” In other words, if K⊴ G is a normal subgroup and H≤ G is a subgroup containing K, then K is normal in H. As soon as we replace normal subgroups with normality data this property fails. The following counterexample was suggested to us by Kiran Luecke.
		
		
			
			Consider the Hopf fibration sequence
			
    S^1→ S^3→ S^2→ℂP^∞→ BS^3
 where S^3 is equipped with the group structure coming from the equivalence S^3≃ SU(2). It is classical that, with respect to this, S^1→ S^3 is a map of group objects in  (for instance as the inclusion of a certain class of diagonal matrices). Moreover, we know that S^2 does not admit the structure of a group object in  (if it did, it would admit an H-space structure, which it cannot by <cit.>). Now note that we have a commutative triangle in ():
			
			
    S^1      ∗
       S^3[from=1-1, to=2-2]
    				[from=1-1, to=1-3]
    				[from=2-2, to=1-3]

			where ∗ is contractible. Because S^1 is an infinite loop space, the trivial map S^1→∗ admits a normality datum ∗→ K(,3). However, a normality datum for S^1→ S^3 would endow S^2 with the structure of a group object, which is impossible.
		
		In fact there are many counterexamples even for discrete ∞-groups. In other words, the issue is with replacing normal subgroups with normality data, not necessarily replacing groups with loop spaces. The following counterexample is due to David Wärn.
		
			Let S_3 denote the symmetric group on {1,2,3} and let /2↪ S_3 be the inclusion of the subgroup generated by the transposition (1,2). Therefore we have a commutative triangle of groups:
			
			
    ℤ/2      ∗
       S_3["(1,2)"', from=1-1, to=2-2]
    				[from=1-1, to=1-3]
    				[from=2-2, to=1-3]

			where ∗ is the trivial group. Because ℤ/2 is Abelian the trivial map /2→∗ admits a normality datum ∗→ B^2/2. Asking for the inclusion /2↪ S_3 to admit a normality datum is equivalent to asking for it to be the fiber of a map of group objects S_3→ G in (). One can check that G must be discrete, in which case /2 would need to be the kernel of a discrete group map S_3→ G. This would imply that /2 is normal in S_3. However, S_3 only has three normal subgroups: the trivial group, the entire group, and A_3.
		
		
		
		To better understand this problem, let's first enumerate the data required to even state the Third Isomorphism Theorem, as written above. Given a commutative triangle in ()
		
    K        G 
        H
    			["ϕ", from=1-1, to=1-3]
    			["ρ"', from=1-1, to=2-2]
    			["ψ"', from=2-2, to=1-3]
  which we think of as the sequence of subgroup inclusions K≤ H≤ G, let us assume only what the classical theorem must assume; namely, we assume that we have normality data π_ϕ BG→ B(G/K) and π_ψ BG→ B(G/H) for ϕ and ψ respectively. Classically, this suffices for stating the Third Isomorphism Theorem (due to what is sometimes called the Fourth Isomorphism Theorem or Lattice Theorem cf. Propositions 8.9 and 8.10 of <cit.>), but we will need to assume more:
		
			
  * We need a normality datum π_ρ BH→ B(H/K) for ρ so that H/K can be lifted to an object of ().
			
  * Assuming the existence of the above, we also need a morphism θ H/K→ G/K and a normality datum B(G/K)→ B((G/K)/(H/K)) so that (G/K)/(H/K) has a group object structure.
			
  * Again, assuming both of the above data, we moreover need a ∞-group morphism ω G/H→ (G/K)/(H/K) to serve as a candidate for the equivalence. 
		
		
		In the case that we suppose the existence of all of the above data, the Third Isomorphism Theorem follows from checking that a simple diagram commutes:
		
		
			Suppose we have a commutative triangle in () 
			
    K        G 
        H
    				["ϕ", from=1-1, to=1-3]
    				["ρ"', from=1-1, to=2-2]
    				["ψ"', from=2-2, to=1-3]
 along with the following data:
			
				
  * Normality data for ϕ, ψ and ρ:
				
					
  * π_ϕ G→ (G/K),
					
  * π_ψ G→ (G/H),
					
  * π_ρ H→ (H/K).
				
				
  * A morphism θ H/K→ G/K in ().
				
  * A normality datum π_θ (G/K)→ ((G/K)/(H/K)) for θ.
				
  * A morphism ω G/H→ (G/K)/(H/K) in ().
			
			Then the morphism ω G/H (G/K)/(H/K) is an equivalence whenever the following diagram commutes:
			
			
    H     (H/K)
     G     (G/K)
     (G/H)    ((G/K)/(H/K))["π_ρ", from=1-1, to=1-2]
    				["ψ"', from=1-1, to=2-1]
    				["θ", from=1-2, to=2-2]
    				["π_ϕ"', from=2-1, to=2-2]
    				["π_ψ"', from=2-1, to=3-1]
    				["π_θ", from=2-2, to=3-2]
    				["ω"', from=3-1, to=3-2]

		
		
		
			Suppose the given diagram commutes. Then we may take fibers along the top two horizontal arrows to obtain a commutative diagram in which the top two rows are fiber sequences:
			
			
    K     H     (H/K)
     K     G     (G/K)
        (G/H)    ((G/K)/(H/K))[from=1-1, "ρ", to=1-2]
    				["id_ K"', from=1-1, to=2-1]
    				["π_ρ", from=1-2, to=1-3]
    				["ψ"', from=1-2, to=2-2]
    				["θ", from=1-3, to=2-3]
    				[from=2-1, "ϕ", to=2-2,swap]
    				["π_ϕ"', from=2-2, to=2-3]
    				["π_ψ"', from=2-2, to=3-2]
    				["π_θ", from=2-3, to=3-3]
    				["ω"', from=3-2, to=3-3]

			Further taking fibers along the columns gives a commutative diagram in which the top three rows are fiber sequences:
			
			
    ∗   G/H   (G/K)/(H/K)
     K     H     (H/K)
     K     G     (G/K)
        (G/H)    ((G/K)/(H/K))[from=1-1, to=1-2]
    				[from=1-1, to=2-1]
    				["≃", from=1-2, to=1-3]
    				[from=1-2, to=2-2]
    				[from=1-3, to=2-3]
    				[from=2-1, "ρ", to=2-2]
    				["id_ K"', from=2-1, to=3-1]
    				["π_ρ", from=2-2, to=2-3]
    				["ψ"', from=2-2, to=3-2]
    				["θ", from=2-3, to=3-3]
    				[from=3-1, "ϕ", to=3-2, swap]
    				["π_ϕ"', from=3-2, to=3-3]
    				["π_ψ"', from=3-2, to=4-2]
    				["π_θ", from=3-3, to=4-3]
    				["ω"', from=4-2, to=4-3]

It follows that the upper right morphism is an equivalence. But this morphism is Ωω=s_0^∗ω and s_0^∗ is conservative (as shown in the proof of <cit.>). Therefore ω≃ω is an equivalence of group objects.
		
		
		
			Note that weaker assumptions would give an equivalence in  of the form G/H→ (G/K)/(H/K). Specifically, suppose we had not assumed the existence of the morphism ω in the above theorem and had only assumed commutativity of the following diagram:
			
			
    H     (H/K)
     G     (G/K)["π_ρ", from=1-1, to=1-2]
    				["ψ"', from=1-1, to=2-1]
    				["θ", from=1-2, to=2-2]
    				["π_ϕ"', from=2-1, to=2-2]

			Then essentially the same argument would show that a morphism G/H→ (G/K)/(H/K) existed and was an equivalence of objects in . This map does not, however, necessarily necessary lift to a morphism of ().
		
		
		
		Clearly the statement and “proof” of the Third Isomorphism Theorem for () requires the input of much more data than the classical version. We do not believe that any of that data comes “for free” from the initial assumptions of the classical case, but we do not have counterexamples for every possible configuration. However, the following example shows that even if we have a normal morphism which is a composite of normal morphisms K→ H→ G (i.e. every morphism in the commutative triangle is normal) it need not be the case that the Third Isomorphism Theorem holds. The following counterexample is due to Kiran Luecke.
		
			Let S^2_t denote the 2-sphere equipped with the trivial S^1-action and let S^2_r denote the 2-sphere with the S^1-action that rotates it about an axis through the poles. We have (homotopy) quotients S^2_t/S^1≃ S^2×ℂP^∞ and S^2_r/S^1≃ℂP^∞∨ℂP^∞. These give fiber sequences S^1→ S^2_t→ S^2×ℂP^∞ and S^1→ S^2_r→ℂP^∞∨ℂP^∞. Therefore, we have a commutative diagram in which each straight composite is a fiber sequence: 
			
    ∗
    S^1      S_r^2   ℂP^∞∨ℂP^∞
       S^2_t
          S^2×ℂP^∞[from=2-1, to=2-3]
    				[from=2-1, to=3-2]
    				[from=2-3, to=1-4]
    				[from=2-3, to=2-4]
    				["id"', from=3-2, to=2-3]
    				[from=3-2, to=4-3]

			Now, take loops on this diagram to get a commutative triangle in ():
			
			
    ∗
    ℤ      Ω S_r^2   Ω(ℂP^∞∨ℂP^∞)
       Ω S^2_t
          Ω S^2× S^1[from=2-1, to=2-3]
    				[from=2-1, to=3-2]
    				[from=2-3, to=1-4]
    				[from=2-3, to=2-4]
    				["id"', from=3-2, to=2-3]
    				[from=3-2, to=4-3]

			Because we have taken loops on fiber sequences, each of the group maps above is normal. For clarity we explain what each piece of the above pair of diagrams is in terms of the terminology of Theorem <ref>. We have:
			
				
  * K=, H=Ω S^2_t, G=Ω S^2_r.
				
  * G/K=Ω(ℂP^∞∨ℂP^∞), G/H=∗, H/K=Ω S^2× S^1. 
			
			For these groups to satisfy the Third Isomorphism Theorem we would first need a group map Ω S^2× S^1→Ω(ℂP^∞∨ℂP^∞). This would deloop to a map of spaces S^2×ℂP^∞→ℂP^∞∨ℂP^∞, and we would need a normality datum πℂP^∞∨ℂP^∞→ X whose fiber is S^2×ℂP^∞. Let us suppose such data existed. Then we would have Ω X as our model of (G/K)/(H/K). But G/H=Ω S^2/Ω S^2 is contractible, so we would need Ω X≃∗, and thus X≃∗. But if X is contractible then the fiber sequence above yields S^2×ℂP^∞≃ℂP^∞∨ℂP^∞, which is not true (for instance, they have non-isomorphic cohomology rings).
		
		
		The third isomorphism does however hold for any commutative triangle of _n-group objects (i.e. _n-algebras of  whose underlying _1-algebra structure is grouplike) when n>1. We first need a lemma. 
		
		
			Let K→ H→ G be a composite of morphisms of (). Then there is a fiber sequence H/K→ G/K→ G/H. 
		
		
		
			By definition there is a pullback square: 
			
    G/H   ∗
     H     G
    				[from=1-1, to=1-2]
    				[from=1-1, to=2-1]
    				["⌟"anchor=center, pos=0.125, draw=none, from=1-1, to=2-2]
    				[from=1-2, to=2-2]
    				[from=2-1, to=2-2]

			By pasting, we can extend this to the following composite pullback diagram:
			
			
    G/K   G/H   ∗
     K     H     G
    				[from=1-1, to=1-2]
    				[from=1-1, to=2-1]
    				["⌟"anchor=center, pos=0.125, draw=none, from=1-1, to=2-2]
    				[from=1-2, to=1-3]
    				[from=1-2, to=2-2]
    				["⌟"anchor=center, pos=0.125, draw=none, from=1-2, to=2-3]
    				[from=1-3, to=2-3]
    				[from=2-1, to=2-2]
    				[from=2-2, to=2-3]

			Finally, taking the fiber of the upper left horizontal morphism and using that there is a fiber sequence H/K→ BK→ BH, we obtain the desired fiber sequence in the uppermost square of the following diagram:
			
			
    H/K   ∗
    G/K   G/H   ∗
     K     H     G
    				[from=1-1, to=1-2]
    				[from=1-1, to=2-1]
    				["⌟"anchor=center, pos=0.125, draw=none, from=1-1, to=2-2]
    				[from=1-2, to=2-2]
    				[from=2-1, to=2-2]
    				[from=2-1, to=3-1]
    				["⌟"anchor=center, pos=0.125, draw=none, from=2-1, to=3-2]
    				[from=2-2, to=2-3]
    				[from=2-2, to=3-2]
    				["⌟"anchor=center, pos=0.125, draw=none, from=2-2, to=3-3]
    				[from=2-3, to=3-3]
    				[from=3-1, to=3-2]
    				[from=3-2, to=3-3]

		
		
		
			Suppose we have a commutative diagram of  _n-group objects for n>1 (in which all maps are morphisms of  _n-groups):
			
			
    K        G 
        H
    				["ϕ", from=1-1, to=1-3]
    				["ρ"', from=1-1, to=2-2]
    				["ψ"', from=2-2, to=1-3]
 
			Then there are canonical choices for all data in the statement of Theorem <ref> such that the Third Isomorphism Theorem holds.
		
		
		
			Because K, H and G are all at least _2-algebras, we may form the pullback diagrams of the proof of Lemma <ref> entirely within __1(). It follows that there is a fiber sequence of group objects H/K G/K G/H. We take the first morphism of this composite as the θ required by Theorem <ref>. This has normality datum π_θ=λ (G/K)→ (G/H), the delooping of the second morphism in the last fiber sequence. It follows that we can take ω to be the identity morphism G/H→ G/H. It only remains to check that the required diagram of Theorem <ref> commutes. We prove that the diagram commutes after applying the equivalence Ω, which gives the result. This diagram is as follows:
			
			
    H    H/K
    
    				G    G/K
    G/H   G/H["Ωπ_ρ", from=1-1, to=1-2]
    				["ψ"', from=1-1, to=2-1]
    				["θ", from=1-2, to=2-2]
    				["Ωπ_ϕ", from=2-1, to=2-2]
    				["Ωπ_ψ"', from=2-1, to=3-1]
    				["λ", from=2-2, to=3-2]
    				["id"', from=3-1, to=3-2]

			To see that the top square commutes, note that we have a commutative square obtained by applying  to the initial commutative triangle:
			
			
    BK     BH 
    
    				BK     BG
    				["Bρ", from=1-1, to=1-2]
    				["id"', from=1-1, to=2-1]
    				["Bψ", from=1-2, to=2-2]
    				["Bϕ"', from=2-1, to=2-2]

			Taking fibers twice in the horizontal direction gives the commutativity of the top square of the diagram in question. The commutative lower square is obtained by taking fibers twice in the horizontal direction of the following commutative diagram:
			
			
    BK     BG 
    
    				BH     BG
    				["Bϕ", from=1-1, to=1-2]
    				["Bρ"', from=1-1, to=2-1]
    				["id", from=1-2, to=2-2]
    				["Bψ"', from=2-1, to=2-2]

			
			
		
		
		
			The above theorem uses that any morphism of _n-groups, for n>1, admits a canonical normality datum. We warn the reader however that even in the Abelian case, e.g. n=∞, this normality datum is not necessarily unique. One example of an infinite loop map with more than one normality datum is given by the terminal morphism K(,1)× K(,3)→∗ of Example <ref>.
		
		
		For H a normal subgroup of a discrete group G, the Third Isomorphism Theorem assumes that there is an order reversing correspondence between subgroups of G containing H and subgroups of G/H. We have seen above that this need not be the case in our more general context. Nonetheless, if ϕ H→ G has a normality datum, we can classify normal maps to G/H in terms of morphisms, in a slice category, out of the normality datum itself. 
		
		
			Let ϕ H→ G be a normal map of group objects in  with normality datum π G→ (G/H). Then there is an equivalence of ∞-categories 
    (G)^\π≃(G/H).

		
		
		
			Recall the equivalence of Proposition <ref> which identifies (G) with the full subcategory of ()^\ G spanned by the effective epimorphisms G→ Q. Write ()^\ G_epi for this ∞-category. Therefore we have an equivalence (G)^\π≃ (()^\ G_epi)^\π. This last category has as objects commutative triangles of the following form:
			
			
    G    G/H
    
    				Q
    				["π", from=1-1, to=1-2]
    				[two heads, from=1-1, to=2-1]
    				[from=1-2, to=2-1]

			where G↠ Q is an effective epimorphism. But this implies that G/H→ Q must also be an effective epimorphism (by <cit.>). Therefore this category is equivalent to the full subcategory of ()^\π spanned by commutative triangles whose two edges to the cone point are effective epimorphisms. But by <cit.> (more concretely, we are using the fact that the inclusion {1}↪Δ^1 is right anodyne along with <cit.>), there is an equivalence ()^\π≃()^\ G/H. Therefore the category (()^\ G_epi)^\π is equivalent to ()^\ G/H_epi which is, by another application of Proposition <ref>, equivalent to (G/H). 
		

			
			

§ QUESTIONS

			
			We record some questions that arise naturally from the above considerations.
			
			

 §.§ An Alternative Characterization of Normality

			
			Normality can classically be determined by comparing right and left cosets. In this paper we focused exclusively on right action objects, but there is a dual notion of left action object which behaves identically. It then follows readily from the work of Section <ref> that right actions by G are in bijection with left actions of G (this is not the case for monoid actions). This is in analogy with the classical case in which G-actions on a set are in bijection with G^-actions. We can therefore form the left quotient and right quotients of an action by a group object.
			
			In the case of a morphism of group objects ϕ H→ G we can also read off these left and right actions from <cit.>, which implies that the data of ϕ is equivalent to that of an algebra structure on G in the category of H-bimodules. Moreover, the quotient by the left action induced by ϕ and the quotient by the right action induced by ϕ must be equivalent as objects of , since both are the fiber of ϕ H→ G.
			
			
				Let ϕ H→ G be a morphism in () so that 
    G⊗_H∗≃ G/H≃(ϕ)≃ H\ G≃∗⊗_HG
 Noticing that G⊗_H ∗ has a residual left H-action and ∗⊗_H G has a residual right H-action, we can take further quotients to obtain double coset objects. Is there a bijection between normality data for ϕ and equivalences in  of the form ∗⊗_H(G⊗_H∗)(∗⊗_H G)⊗_H ∗? Recall that, in the classical case, H is a normal subgroup of G if and only if the left (or right) cosets of H in G are isomorphic to the double cosets. 
			
			
			

 §.§ Subgroups via Monomorphisms

			
			Clearly, and unsurprisingly, group objects in ∞-topoi are not as rigid as discrete groups. Replacing normal subgroups with normality data significantly expands the theory, even in the case of discrete groups (cf. <cit.>). There is, however, a restriction one could put on normality data that would make it behave more like the classical example.  
			
			
				Suppose we say that a morphism of group objects ϕ H→ G in an ∞-topos  is a subgroup if it is (-1)-truncated. For example if = then this is equivalent to saying that π_0(ϕ) is injective. We might then restrict our discussion to the case of normality structures on subgroups H≤ G, rather than arbitrary morphisms. Is it the case that the Second and Third Isomorphism Theorems hold in this setting? 
			
			
			A piece of partial evidence that restricting subgroups might be effective is that it would rule out the counterexample to the Second Isomorphism Theorem in Example <ref>. Example <ref> still gives a counterexample to normality satisfying the intermediate subgroup property (since S^1 and S^3 are both connected) but Example <ref> is no longer valid since the trivial map ℤ→Ω S^2 is not injective on π_0. 
			
			We reiterate however that requiring morphisms of group objects to be (-1)-truncated can result in some strange behavior. It essentially says that H⊆ G should be the inclusion of a collection of connected components of G (which must include the unit component). As a result, connected and non-contractible group objects, like S^1, do not have the trivial group as a subgroup. 
			
			

 §.§ Galois Theory

			
			In <cit.>, Rognes introduced Galois extensions of ring spectra for what he called stably dualizable groups. Examples of stably dualizable groups include compact Lie groups and Eilenberg-MacLane spaces (after localizing at the Morava K-theories). In practice, Galois extensions of ring spectra in which the Galois group is not discrete, and typically finite, seem uncommon. Rognes does however give examples of Galois extensions for more general groups by taking 𝔽_p-valued cochain spectra on principal G-bundles <cit.>. Rognes also describes a Galois correspondence in <cit.> and <cit.>, but under somewhat strict conditions. For instance, his result going from intermediate extensions to subgroups requires G to be finite and discrete. Given that the Galois correspondence is extremely complicated (and rarely a bijection) even in the case of extensions of discrete rings, we cannot expect to obtain a perfect correspondence in this significantly more general setting, but one might still ask how far the analogy can be taken.
			
			
				Let G∈() be a group object in an ∞-topos. For a morphism of algebra objects A→ B in Stab(), the stabilization of , with G acting on B by equivalences of A-bimodules, we can repeat Rognes' definition of A→ B being a G-Galois extension. Given a normal map H→ G, perhaps one which is additionally a monomorphism, can we produce a composite A→ C→ B where C→ B is an H-Galois extension and A→ C is a G/H-Galois extension? On the other hand, given a factorization A→ C→ B of a G-Galois extension A→ B, when is it possible to produce a normal map CC(A,A)^×→ G?
			 
			
			In the slightly more general case of so-called Hopf-Galois extensions, in which one replaces groups with Hopf-algebras and G-actions with H-coactions, partial results can be obtained by combining <cit.>. 
			
			
			

 §.§ Representation Theory

			
			The standard approach to groups, especially as symmetries of geometric objects, is to study their representations. Representation theory is already very hard (especially over ) of course, but one should never miss an opportunity to generalize an existing subject and make it exponentially more difficult. One potential foothold in understanding the representation theory of higher groups is to restrict to group objects in (), i.e. loop spaces, and consider their representations in the K(n)-local or T(n)-local categories of spectra. In these two cases, the ambidexterity results of <cit.> and  <cit.> suggest that representation theory over K(n) and T(n) is, in some ways, similar to classical representation theory in characteristic zero. It may be the case that similar methods can be used to decompose representations of (π-finite) group objects in . 
			
			

§ FIBER/COFIBER ADJUNCTIONS

			
			We briefly recall the adjunction between slice and coslice categories whose left adjoint is given by taking cofibers and whose right adjoint is given by taking fibers. The proof of the following lemma is most obvious when written in the language of ∞-cosmoi, as developed in <cit.>. The ∞-category of (small) ∞-categories, , is indeed an ∞-cosmos by <cit.>. 
			
			
				Let  be a pointed ∞-category, with zero object ∗, that has all pullbacks and pushouts, and X an object of . Then there is an adjunction between _/X and ^\ X whose left adjoint _/X→^\ X is given on objects by taking the right vertical map in the below pushout square
				
    Y     X 
    ∗    P
    					[from=1-1, to=1-2]
    					[from=1-2, to=2-2]
    					[from=1-1, to=2-1]
    					[from=2-1, to=2-2]
    					["⌟"anchor=center, pos=0.125, rotate=180, draw=none, from=2-2, to=1-1]
 which we will refer to as taking the cofiber of  Y→ X.
				The right adjoint of the adjunction ^\ X→_/X is given on objects by taking the upper horizontal leg of the below pullback square
				
    F     X 
    ∗    Y
    					[from=2-1, to=2-2]
    					[from=1-1, to=2-1]
    					[from=1-1, to=1-2]
    					[from=1-2, to=2-2]
    					["⌟"anchor=center, pos=0.125, draw=none, from=1-1, to=2-2]

				which we will refer to as taking the fiber of X→ Y.
			
			
			
				Let  denote the cospan category {1← 0→ 2} and  let  denote the commutative square category depicted below: 
				
    0     1 
    
    					2     3
    					[from=1-1, to=1-2]
    					[from=1-1, to=2-1]
    					[from=1-2, to=2-2]
    					[from=2-1, to=2-2]
 
				These categories fit into a commutative diagram of inclusion functors:
				
    {1}∐{2}[from=1-1, to=1-3]
    					[from=2-2, to=1-1]
    					[from=2-2, to=1-3]

				
				By functoriality of the cotensoring of ∞-categories over simplicial sets there is a commutative triangle of ∞-categories
				
				
    ^      ^
       ^{1}∐{2}[from=1-3, to=1-1]
    					[from=1-1, to=2-2]
    					[from=1-3, to=2-2]

				in which each functor is given by precomposition. Because  is pointed and admits pushouts each of these functors has a left adjoint given by left Kan extension. Because Kan extensions paste, the triangle of right adjoints and triangle of left adjoints below each respectively commute.
				
				
    ^      ^
       ×[""name=0, anchor=center, inner sep=0, shift left=4, from=1-1, to=1-3]
    					[""name=1, anchor=center, inner sep=0, shift right=4, from=1-1, to=2-2]
    					[""name=2, anchor=center, inner sep=0, shift left=4, from=1-3, to=2-2]
    					[""name=3, anchor=center, inner sep=0, from=2-2, to=1-1]
    					[""name=4, anchor=center, inner sep=0, from=2-2, to=1-3]
    					[""name=5, anchor=center, inner sep=0, from=1-3, to=1-1]
    					["⊣"anchor=center, rotate=-90, draw=none, from=0, to=5]
    					["⊣"anchor=center, rotate=-54, draw=none, from=4, to=2]
    					["⊣"anchor=center, rotate=-128, draw=none, from=3, to=1]

				Note that horizontal left adjoint above is given by forming the pushout. Ignoring the diagonal left adjoints, we obtain a commutative diagram
				
				
    ^      ^
       ×[""name=0, anchor=center, inner sep=0, shift left=3, from=1-1, to=1-3]
    					[from=1-1, to=2-2]
    					[from=1-3, to=2-2]
    					[""name=1, anchor=center, inner sep=0, shift left=1, from=1-3, to=1-1]
    					["⊣"anchor=center, rotate=-90, draw=none, from=0, to=1]

				Since ∞-categories form an ∞-cosmos by <cit.>, part (ii) of Definition 1.2.1 of ibid. implies that cotensoring with monomorphisms gives isofibrations. As a reuslt, the diagonal maps above define objects of the slice ∞-cosmos ()_/×. Moreover it follows in a straightforward manner from definitions (cf. <cit.>) that the above adjunction between ^ and ^ lifts to an adjunction in the slice ∞-cosmos. 
				
				Now consider the functor (∗,X)𝕀→×. Pulling back along any functor in an ∞-cosmos is a so-called cosmological functor by <cit.> and therefore preserves adjunctions. As a result we have an adjunction in ()_/𝕀≃
				
				
    ^_∗,X      ^_∗,X[""name=0, anchor=center, inner sep=0, shift left=3, from=1-1, to=1-3]
    					[""name=1, anchor=center, inner sep=0, shift left=1, from=1-3, to=1-1]
    					["⊣"anchor=center, rotate=-90, draw=none, from=0, to=1]

				in which ^_∗,X and ^_∗,X denote the diagram categories in which 1 and 2 are always taken to ∗ and X respectively. 
				
				We check that when  is pointed there is an equivalence ^_∗,X≃_/X. Consider the inclusion of {0→ 1} into . There are no morphisms in  out of 2 (besides the identity) so the pointwise formula for right Kan extensions extends a functor out of {0→1} to one which takes 2 to the limit over the empty set, which is the zero object ∗. By <cit.> this induces an equivalence between functors out of {0→ 1} and functors out of  that take 2 to ∗.  
				
				Now, using that every equivalence is both a left and right adjoint, we have a composite adjunction
				
				
    _/X   ^_∗,X      ^_∗,X[""name=0, anchor=center, inner sep=0, shift left=3, from=1-2, to=1-4]
    					[""name=1, anchor=center, inner sep=0, shift left=2, from=1-4, to=1-2]
    					[""name=2, anchor=center, inner sep=0, shift left=3, from=1-1, to=1-2]
    					[""name=3, anchor=center, inner sep=0, shift left=2, from=1-2, to=1-1]
    					["⊣"anchor=center, rotate=-90, draw=none, from=0, to=1]
    					["≃"description, draw=none, from=2, to=3]

				in which the left adjoint is given by taking the cofiber of a morphism Y→ X.
				
				Writing .7em for the category 1→ 3← 2 and repeating the above arguments mutatis mutandem, we also obtain and an adjunction
				
				
    ^_∗,X      ^_∗,X   ^\ X[""name=0, anchor=center, inner sep=0, shift left=2, from=1-1, to=1-3]
    					[""name=1, anchor=center, inner sep=0, shift left=3, from=1-3, to=1-1]
    					[""name=2, anchor=center, inner sep=0, shift left=2, from=1-3, to=1-4]
    					[""name=3, anchor=center, inner sep=0, shift left=3, from=1-4, to=1-3]
    					["⊣"anchor=center, rotate=-90, draw=none, from=0, to=1]
    					["≃"description, draw=none, from=2, to=3]

				in which the right adjoint is given by taking the pullback and the left adjoint is precomposition. Composing gives the desired adjunction 
				
				
    _/X      ^\ X[""name=0, anchor=center, inner sep=0, "cof", shift left=2, from=1-1, to=1-3]
    					[""name=1, anchor=center, inner sep=0, "fib", shift left=2, from=1-3, to=1-1]
    					["⊣"anchor=center, rotate=-90, draw=none, from=0, to=1]

				
			
			
			
			

 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>.
                                                            plain


 Paper Title: Quantale-valued maps and partial maps.

































shenlili@scu.edu.cn


tang.xiaoye@qq.com

[cor]Corresponding author.
School of Mathematics, Sichuan University, Chengdu 610064, China

§ ABSTRACT 
Let 𝖰 be a commutative and unital quantale. By a 𝖰-map we mean a left adjoint in the quantaloid of sets and 𝖰-relations, and by a partial 𝖰-map 
we refer to a Kleisli morphism with respect to the maybe monad on the category 𝖰-𝐌𝐚𝐩 of sets and 𝖰-maps. It is shown that every 𝖰-map is symmetric if and only if 𝖰 is weakly lean, and that every 𝖰-map is exactly a map in 𝐒𝐞𝐭 if and only 𝖰 is lean. Moreover, assuming the axiom of choice, it is shown that the category of sets and partial 𝖰-maps is monadic over 𝖰-𝐌𝐚𝐩.




Category Theory Quantale Quantaloid 𝖰-map Partial 𝖰-map




[2020] 18F75 18B10 18C20













§ INTRODUCTION


Given (crisp) sets X and Y, a partial map f from X to Y is a map from a (possibly empty) subset X'⊆ X to Y. For the category

    ^∂
 
of (crisp) sets and partial maps, the following results are well known <cit.>:


  * ^∂ is equivalent to the coslice category {⋆}/, where {⋆} is the singleton set.

  * The forgetful functor {⋆}/→ admits a left adjoint, which carries a set X to the inclusion map

    {⋆} →/^(->/ X⨿{⋆},

where X⨿{⋆} refers to the disjoint union of X and the singleton set.

  * The induced monad on  is called the maybe monad, whose Eilenberg-Moore and Kleisli categories are {⋆}/ and ^∂, respectively (cf. <cit.>). 

Therefore, the Eilenberg-Moore and Kleisli categories of the maybe monad on  are equivalent. In particular, {⋆}/ and ^∂ are both monadic over .

It is then natural to ask whether it is possible to formulate the notion of partial map, and establish the above results in the fuzzy setting. To achieve this goal, we note that there is a very general framework in category theory for partial maps; see, e.g., <cit.>. Explicitly, let  be a category with finite coproducts. For every -object A, the forgetful functor from the coslice category A/ to , i.e.,

    A/→,   (A C)↦ C,

admits a left adjoint, given by

    C↦(A A⨿ C).

It is not difficult to see that the Eilenberg-Moore category of the induced monad is isomorphic to A/, and thus A/ is monadic over . The maybe monad on  is simply obtained by setting = and A={⋆}.

Considering a non-trivial, commutative and unital quantale  <cit.>  as the table of truth values and following the terminology of <cit.>, by a -map (i.e., a map valued in the quantale ) we mean a left adjoint in the quantaloid <cit.>

    

of sets and -relations; that is, a -relation ζ X Y admitting a right adjoint in , with the value

    ζ(x,y)

interpreted as the extent of y being the image of x under the map ζ (see Remark <ref> for detailed elaboration). In Section <ref> we prove the following results (Theorems <ref> and <ref>), which reveal that for a general quantale , -maps are essentially different from crisp maps between sets:


  * Every -map is symmetric (in the sense that the right adjoint is given by its opposite) if, and only if,  is weakly lean. This is a discrete counterpart of <cit.>.

  * Every -map is the graph of a map in  (so that the category  of sets and -maps is isomorphic to ) if, and only if,  is lean. This is a generalization of <cit.> to the non-integral setting.


With the above preparations we are now able to postulate the notion of partial -map through the maybe monad on the category , which is obtained by setting = and A={⋆} in (<ref>) and (<ref>). Explicitly, by a partial -map ζ from X to Y we refer to a -map

    ζ X Y⨿{⋆},

which is precisely a morphism in the Kleisli category of the maybe monad on , with the value 

    ζ(x,⋆)

interpreted as the degree that x has no image under the map ζ (see Remark <ref> for detailed elaboration). The main result of Section <ref>, Theorem <ref>, is to show that assuming the axiom of choice, the Eilenberg-Moore category (i.e., the coslice category {⋆}/) and the Kleisli category (i.e., the category  of sets and partial -maps) of the maybe monad on  are equivalent, so that  and  are both monadic over . It is noteworthy to point out that this result may not hold in the general categorical setting (i.e., the monad induced by (<ref>) and (<ref>) for a general category ), and an equivalent statement of the axiom of choice given in <cit.> (i.e., every non-empty set admits a group structure) plays a vital role in its proof.















































































































§ QUANTALES


Throughout, let 

    =(,,k)

denote a non-trivial, commutative and unital quantale <cit.>, which consists of a complete lattice  (with the top and bottom elements denoted by ⊤ and , respectively), an element k∈ and a binary operation  on , such that 


  * <k,

  * (,,k) is a commutative monoid with unit k, and

  * p(_i∈ Iq_i)=_i∈ Ip q_i
for all p, q_i∈ (i∈ I).


The right adjoint induced by the multiplication , denoted by ,

    (p -)⊣(p -)→,

satisfies

    p q≤ r p≤ q r

for all p,q,r∈. 

We say that  is integral if k=⊤, and  is divisible <cit.> if 

    d≤ q∃ p∈ d=p q

for all d,q∈. In particular, divisible quantales are necessarily integral (see, e.g., <cit.>).

 
We list here some examples of commutative and unital quantales that will be concerned later:



  *  On the three-chain C_3={,k,⊤} we have the quantale (C_3,,k), with 

    ⊤⊤=⊤

and the other multiplications being trivial.

  *  Every frame is a divisible quantale. In particular, the partially ordered sets 

    F_1={,p,q,⊤}  and   F_2={,p,q,r,⊤}

illustrated by the Hasse diagrams

    /-`-`/<200,200>[⊤`p`q;“]
    (0,-200)/`-`-/<200,200>[p`q`;“]
    (800,0)[and]
    (1200,0)/-`-`/<200,200>[r`p`q;“]
    (1400,200)/-/<0,200>[r`⊤;]
    (1200,-200)/`-`-/<200,200>[p`q`;“]
    (1700,0)[,]

respectively, are both frames.

  * 
On the diamond lattice M_3 given by the Hasse diagram

    /-`-`-“/<250,250>[⊤`a`b`k;““]
    (0,-250)/“-`-`-/<250,250>[a`b`k`;““]









we have the quantale (M_3,,k), with

    a a=b,   b b=a,   a b=k  and   a⊤=b⊤=⊤.


  *  Every continuous t-norm <cit.> * on the unital interval [0,1] gives rise to a divisible quantale ([0,1],*,1). 

  *  Let [-∞,∞] be the extended real line equipped with the order “≥”. Then ([-∞,∞],+,0) is a quantale, which includes the Lawvere quantale <cit.> ([0,∞],+,0) as a divisible subquantale.


  *  Every commutative monoid (M,,k) induces a free quantale (,,{k}), where  is the powerset of M, and

    A B={a b| a∈ A, b∈ B}

for all A,B⊆ M.



A quantaloid <cit.>  is a category with a class of objects _0 in which (p,q) is a complete lattice for all p,q∈_0, such that the composition ∘ of morphisms preserves joins on both sides, i.e.,

    v∘(_i∈ Iu_i)=_i∈ I(v∘ u_i)  and  (_i∈ I v_i)∘ u=_i∈ I(v_i∘ u)

for all u,u_i∈(p,q), v,v_i∈(q,r) (i∈ I). The corresponding right adjoints induced by the compositions 

    (-∘ u)(- u)(p,r)→(q,r)  and  (v∘ -)(v -)(p,r)→(p,q)

satisfy 

    v∘ u≤ w v≤ w u u≤ v w
 
for all -arrows u p→ q, v q→ r, w p→ r.

An adjunction in a quantaloid  is a pair of -arrows u p→ q and v q→ p, denoted by u v, such that

    1_p≤ v∘ u  and   u∘ v≤ 1_q.


 
If u v q→ p in a quantaloid , then

    v=u 1_q  and   u=1_q v.




Note that 1_p≤ v∘ u implies that u 1_q≤ v∘ u∘(u 1_q)≤ v, and the reverse inequality is an immediate consequence of u∘ v≤ 1_q. Therefore v=u 1_q, and u=1_q v can be proved analogously.


Therefore, the right adjoint of a -arrow, when it exists, is necessarily unique. In what follows we define 

    u^*:=u 1_q q→ p

for each -arrow u p→ q. u is called a map in  <cit.> if u u^*, and we denote by

    ()

the subcategory of  whose objects are the same as , and whose morphisms are maps in .

Since it always holds that u∘ u^*=u∘(u 1_q)≤ 1_q, we immediately obtain the following lemma:

 
A -arrow u p→ q is a map in  if, and only if, 1_p≤ u^*∘ u.


Given sets X, Y, a -relation ϕ X Y is a function ϕ X× Y→. We denote by

    ϕ^:Y X,  ϕ^(y,x)=ϕ(x,y)

the opposite of a -relation ϕ X Y. Sets and -relations constitute a quantaloid

    ,

in which the local order is inherited from , and 

    ψ∘ϕ X Z,  (ψ∘ϕ)(x,z)=_y∈ Yψ(y,z)ψ(x,y),
       ξϕ Y Z,   (ξϕ)(y,z)=_x∈ X(ϕ(x,y)ξ(x,z)),
       ψξ X Y,   (ψξ)(x,y)=_z∈ Zψ(y,z)ξ(x,z)

for all -relations ϕ X Y, ψ Y Z, ξ X Z; the identity -relation on X is given by

    𝕀 X X,  𝕀_X(x,y)=
    k    if x=y,
       else.

Given -relations ϕ X Y and ψ Z W, their disjoint union 

    ψ⨿ϕ X⨿ Z Y⨿ W

is given by

    (ψ⨿ϕ)(x,y)=ϕ(x,y),  (ψ⨿ϕ)(z,w)=ψ(z,w)  and  (ψ⨿ϕ)(x,w)=(ψ⨿ϕ)(z,y)=

for all x∈ X, z∈ Z, y∈ Y, w∈ W. It is straightforward to verify that

    (ψ∘ϕ)⨿ξ=(ψ⨿ξ)∘(ϕ⨿ξ)

for all -relations ϕ X Y, ψ Y Z, ξ Z W.
























§ QUANTALE-VALUED MAPS
 

 
Let X, Y be (crisp) sets. A -map ζ from X to Y is a map ζ X Y in the quantaloid .


Sets and -maps constitute a category 

    :=().


 
To justify Definition <ref>, let ζ X Y be a -map. First of all, the value

    ζ(x,y)

is interpreted as the extent of y being the image of x under the map ζ. By Equations (<ref>) and (<ref>), the value

    ζ^*(y,x)=(ζ𝕀_Y)(y,x)=_z∈ Yζ(x,z)𝕀_Y(y,z)

also represents the extent of y being the image of x under the map ζ, since the last expression of (<ref>) may be understood as:


  * For each z∈ Y, if z is the image of x under ζ, then z is equal to y.
  
Therefore, the adjunction ζζ^* can be translated as follows:


  * For every x∈ X, there exists y∈ Y such that y is the image of x under ζ; because 𝕀_X≤ζ^*∘ζ means that

    k≤_y∈ Yζ^*(y,x)ζ(x,y)

for all x∈ X.

  * If y,z∈ Y are both the images of x under ζ, then y is equal to z; because ζ∘ζ^*≤𝕀_Y means that

    _x∈ Xζ(x,z)ζ^*(y,x)≤𝕀_Y(y,z)

for all y,z∈ Y.




The notion of -map should be carefully distinguished from related notions in the literature:


  * Fuzzy mappings in <cit.> and fuzzifying functions in <cit.> are precisely our -relations.

  * When  is an integral quantale, every -map is a fuzzy function with respect to 𝕀_X and 𝕀_Y in the sense of <cit.> or <cit.>, but not vice versa.

  * When  is a frame, -maps are exactly -valued fuzzy functions in the sense of <cit.>.



 
Let ζ X Y be a -map. Then

    (ζ^*∘ζ)(x,x)=k  and  ζ(x,z)ζ^*(y,x)=

for all x∈ X, y,z∈ Y with y≠ z.



The first equality follows from

    k   =𝕀_X(x,x)≤(ζ^*∘ζ)(x,x)=_y∈ Yζ^*(y,x)ζ(x,y)
       =_y∈ Yζ(x,y)ζ^*(y,x)≤_y∈ Y(ζ∘ζ^*)(y,y)≤_y∈ Y𝕀_Y(y,y)=k,

and the second equality holds because

    ζ(x,z)ζ^*(y,x)≤(ζ∘ζ^*)(y,z)≤𝕀_Y(y,z)=.



Given a -map ζ X Y, it is obvious that the restriction of ζ on any (crisp) subset W⊆ X induces a -map ζ|W W Y. Moreover, ζ can also be extended to any (crisp) superset Z⊇ Y:

 
Let ζ X Y be a -map, and let Z⊇ Y. Then

    ζ̃ X Z,  ζ̃(x,z)=ζ(x,z)    if z∈ Y,
       if z∉Y

is the unique -map from X to Z satisfying 

    ζ̃(x,y)=ζ(x,y)  and  (ζ̃)^*(y,x)=ζ^*(y,x)

for all x∈ X, y∈ Y.



It is easy to see that ζ̃ and

    η Z X,  η(z,x)=ζ^*(z,x)    if z∈ Y,
       if z∉Y

satisfy (<ref>) and (<ref>), and thus ζ̃η, making ζ̃ a -map. For the uniqueness of ζ̃, let ζ' X Z be another -map satisfying (<ref>). Then for any x∈ X and z∈ Z∖ Y,

    ζ'(x,z)   ≤ζ'(x,z)(_y∈ Yζ^*(y,x)ζ(x,y))   (Inequality (<ref>))
       =_y∈ Yζ'(x,z)ζ^*(y,x)ζ(x,y)
       =_y∈ Yζ'(x,z)ζ'^*(y,x)ζ(x,y)   (Equations (<ref>))
       ≤_y∈ Y𝕀_Z(y,z)ζ(x,y)   (Inequality (<ref>))
       =_y∈ Yζ(x,y)
       =,

which shows that ζ'=ζ̃.


































Given (crisp) sets X, Y, we say that a -map ζ X Y is symmetric <cit.> if ζ^*=ζ^, i.e., if

    ζ^*(y,x)=ζ(x,y)

for all x∈ X, y∈ Y. In particular, every map f X→ Y between (crisp) sets induces a symmetric -map

    f_∘ X Y,   f_∘(x,y)=
    k    if y=f(x),
       else,

called the graph of f. Thus we have a faithful functor

    (-)_∘→.


 
Let ζ X Y be a -map. Then ζ is symmetric if, and only if, the -relation

    ζ_ X Y,  ζ_(x,y)=ζ(x,y)∧ζ^*(y,x)

is a symmetric -map.



If ζ is symmetric, then ζ^*(y,x)=ζ^(y,x)=ζ(x,y), and consequently ζ_(x,y)=ζ(x,y) for all x∈ X, y∈ Y. Thus ζ_=ζ is a symmetric -map. Conversely, if ζ_ is a symmetric -map, then 

    ζ(x,y)   ≤(ζ∘ζ_^*∘ζ_)(x,y)   (ζ_ζ_^*)
       =_w∈ X_z∈ Yζ(w,y)ζ_(w,z)ζ_(x,z)   (ζ_ is symmetric)
       ≤_w∈ X_z∈ Yζ(w,y)ζ^*(z,w)ζ_(x,z)
       ≤_z∈ Y𝕀_Y(z,y)ζ_(x,z)   (ζ⊣ζ^*)
       =ζ_(x,y)
       ≤ζ(x,y)

for all x∈ X, y∈ Y. Thus ζ=ζ_ is a symmetric -map.    


The aim of this section is to answer the following questions:


  * Is every -map symmetric?

  * Is every -map the graph of a map in ?

In fact, both the answers are negative, and we will provide necessary and sufficient conditions on  for them to be true.

 
Let  be a non-trivial, commutative and unital quantale. We say that:


  *   is lean, if 

    (p∨ q=k and p q=)(p=k or q=k)

and

    p q=k p=q=k

for all p,q∈;

  *   is weakly lean, if

    (_i∈ Ip_i q_i=k and p_i q_j= (i≠ j))(k≤_i∈ I(p_i∧ q_i)(p_i∧ q_i))

for all p_i,q_i∈ (i∈ I).



 


  *  Note that (<ref>) is always true when  is integral. Indeed, p q=k implies that

    k=p q≤ p k=p≤ k,

which forces p=k, and q=k follows from similar calculations. Hence, Definition <ref><ref> generalizes the notion of lean in <cit.>, where the quantale  is required to be integral, and consequently, only (<ref>) is presented there.

  *  The condition (<ref>) is a modification of the condition given in <cit.>. The difference between our following Theorem <ref> and <cit.> is that we deal with the discrete case, i.e., -maps between (crisp) sets (instead of -categories as in <cit.>).



 
If  is lean, then  is weakly lean.



Suppose that p_i,q_i∈ (i∈ I), _i∈ Ip_i q_i=k and p_i q_j= (i≠ j). Since <k, there exists j∈ I such that p_j q_j>. Let

    u:=p_j q_j  and   v:=_i∈ I
     i≠ jp_j q_j.

Then 

    u∨ v=k  and   u v=p_j q_j(_i∈ I
     i≠ jp_i q_i)=_i∈ I
     i≠ jp_j q_j p_i q_i=.

Note that v≠ k. Indeed, if v=k, then u v=u k=u>, contradicting to (<ref>). Thus, since  is lean, we deduce from (<ref>) that u=p_j q_j=k, and consequently p_j=q_j=k. It follows that

    _i∈ I(p_i∧ q_i)(p_i∧ q_i)≥(p_j∧ q_j)(p_j∧ q_j)=(k∧ k)(k∧ k)=k,

showing that  is weakly lean.


 
If  is integral, then  is weakly lean. 



Suppose that p_i,q_i∈ (i∈ I), _i∈ Ip_i q_i=k and p_i q_j= (i≠ j). Note that for each i∈ I,

    p_i   =p_i k
       =p_i(_j∈ Ip_j q_j)
       =_j∈ Ip_i p_j q_j
       =(p_i p_i q_i)∨(_j∈ I
    j≠ ip_i p_j q_j)
       =(p_i p_i q_i)∨
       =p_i p_i q_i.

Since  is integral, we have p q≤ (p k)∧(k q)=p∧ q for all p,q∈, and consequently

    _i∈ I(p_i∧ q_i)(p_i∧ q_i)≥_i∈ Ip_i q_i p_i q_i=_i∈ Ip_i q_i=k,

as desired.



For the examples listed in <ref>:


  * (C_3,,k) is a non-integral lean quantale.

  * Every frame is divisible, and thus integral. Then it follows from Lemma <ref> that every frame is weakly lean. Moreover, F_1={,p,q,⊤} is not lean while F_2={,p,q,r,⊤} is lean. Therefore, a weakly lean quantale need not be lean.

  * The quantale (M_3,,k) is not lean. 

  * For each continuous t-norm * on [0,1], the quantale ([0,1],*,1) is lean. 

  * The quantale ([-∞,∞],+,0) is not weakly lean, while the Lawvere quantale ([0,∞],+,0) is lean.

  * Consider the free quantale (,,{k}) induced by a commutative monoid (M,,k):


  * (,,{k}) is lean if, and only if, k is the only element of M with an inverse; that is, if m∈ M and m≠ k, then there exists no m'∈ M with m m'=k.

  * (,,{k}) is weakly lean if, and only if, there exist no m,m'∈ M such that m≠ m' and m m'=k.

In particular, the free quantale induced by the cyclic group _2=/2={0,1} (under the usual addition modulo 2) is non-integral, weakly lean, but not lean.



 
Let  be a non-trivial, commutative and unital quantale. Then every -map is symmetric if, and only if,  is weakly lean.






For the necessity, suppose that every -map is symmetric. Assume that p_i,q_i∈ (i∈ I), _i∈ Ip_i q_i=k and p_i q_j= (i≠ j). Define a -relation 

    ζ{⋆} I,  ζ(⋆,i)=p_i,

where {⋆} refers to the singleton set. Note that for each i∈ I, from p_i q_i≤ k and q_i p_j= (j≠ i) we have

    q_i   =q_i∧ q_i
       ≤(_j∈ I
    j≠ ip_j)∧(p_i k)
       =(_j∈ I
    j≠ iζ(⋆,j))∧(ζ(⋆,i) k)
       =_j∈ Iζ(⋆,j)𝕀_I(i,j)
       =(ζ𝕀_I)(i,⋆)
       =(_j∈ Ip_j q_j)(_l∈ Iζ(⋆,l)𝕀_I(i,l))
       =[_j∈ I
    j≠ ip_j q_j(_l∈ Ip_l𝕀_I(i,l))]∨[p_i q_i(_l∈ Ip_l𝕀_I(i,l))]
       ≤[_j∈ I
    j≠ ip_j q_j(p_j)]∨[p_i q_i(p_i k)]
       ≤∨ (q_i k)
       =q_i.

Thus q_i=(ζ↘𝕀_I)(i,⋆)=ζ^*(i,⋆). It follows that 

    (ζ^*∘ζ)(⋆,⋆)=_i∈ Iζ^*(i,⋆)ζ(⋆,i)=_i∈ Iq_i p_i=k=𝕀_{⋆}(⋆,⋆),

which in conjunction with Lemma <ref> shows that ζ is a -map. Since every -map is symmetric, in particular we have

    q_i=ζ^*(i,⋆)=ζ^(i,⋆)=ζ(⋆,i)=p_i

for all i∈ I, and consequently

    _i∈ I(p_i∧ q_i) (p_i∧ q_i)=_i∈ I(p_i∧ p_i) (q_i∧ q_i)=_i∈ Ip_i q_i=k.

Hence  is weakly lean.

Conversely, for the sufficiency, suppose that  is 
weakly lean. In order to show that every -map ζ X Y is symmetric, by Lemma <ref> it suffices to show that ζ_ is always a symmetric -map. To this end, let x∈ X. First,

    k=𝕀_X(x,x)≤(ζ^*∘ζ)(x,x)=_y∈ Yζ^*(y,x)ζ(x,y)≤_y∈ Y𝕀_Y(y,y)=k,

Second, for any y,z∈ Y with y≠ z,

    ζ(x,z)ζ^*(y,x)≤(ζ∘ζ^*)(y,z)≤𝕀_Y(y,z)=.

Thus

    _y∈ Yζ^*(y,x)ζ(x,y)=k  and  ζ(x,z)ζ^*(y,x)= (y≠ z)

Since  is weakly lean, (<ref>) forces

    ζ_^∘ζ_(x,x)=_y∈ Yζ_^(y,x)ζ_(x,y)=_y∈ Y(ζ(x,y)∧ζ^*(y,x))(ζ(x,y)∧ζ^*(y,x))≥ k=𝕀_X(x,x).

Moreover,

    ζ_∘ζ_^(y,z)=_x∈ Xζ_(x,z)ζ_^(y,x)≤_x∈ Xζ(x,z)ζ^*(y,x)≤𝕀_Y(y,z)

for all y,z∈ Y. Thus ζ_ζ_^; that is, ζ_ is a symmetric -map.


 
If  is weakly lean, then for any -maps ζ,η X Y, ζ=η whenever ζ≤η. 



First, ζ≤η implies that η^*≤ζ^*. Second, since  is weakly lean, Theorem <ref> indicates that η^=η^*≤ζ^*=ζ^. Thus η≤ζ, as desired.



































Following the terminology of <cit.>, Corollary <ref> actually says that  is map-discrete when  is weakly lean.

We would also like to point out that every -map is symmetric if  is modular (cf. <cit.>) and localic (cf. <cit.>), which is an immediate consequence of <cit.>. However, this is only a sufficient condition, since a weakly lean quantale need not be localic; for example, the quantale M_3 given in Example <ref><ref> is lean but not localic. 


 
Let  be a non-trivial, commutative and unital quantale. Then every -map is the graph of a map in  if, and only if,  is lean. In this case,  and  are isomorphic categories.



For the necessity, suppose that every -map is the graph of a map in . 

First, let p,q∈ with p∨ q=k and p q=. Then both p and q are idempotent, since 

    p   =p k=p (p∨ q)=(p p)∨(p q)=(p p)∨=p p  and
    
    q   =k q=(p∨ q) q=(p q)∨(q q)=∨(q q)=q q.

It follows that

    p≤ p k,   q≤ q k,   p≤ q  and   q≤ p.

Define a -relation

    ζ{⋆}{p,q},  ζ(⋆,p)=p,  ζ(⋆,q)=q.

Then, from (<ref>) we deduce that

    ζ^*(p,⋆)=(ζ𝕀_Y)(p,⋆)=(ζ(⋆,p)𝕀_Y(p,p))∧(ζ(⋆,q)𝕀_Y(p,q))=(p k)∧(q)≥ p∧ p=p,

and similarly ζ^*(q,⋆)≥ q. Thus

    (ζ^*∘ζ)(⋆,⋆)=(ζ^*(p,⋆)ζ(⋆,p))∨(ζ^*(q,⋆)ζ(⋆,q))≥(p p)∨(q q)=p∨ q=k.

By Lemma <ref>, ζ is a -map, and hence the graph of a map f{⋆}→{p,q} in , which forces p=k or q=k. 

Second, let p,q∈ with p q=k. Define a -relation 

    η{⋆}{⋆},  η(⋆,⋆)=p.

Since

    (η^*∘η)(⋆,⋆)=(η(⋆,⋆)𝕀_{⋆}(⋆,⋆))η(⋆,⋆)=(p k) p≥ q p=k,

η is a -map by Lemma <ref>. Thus η is a graph of the unique (identity) map g{⋆}→{⋆}, which forces p=k, and consequently q=q k=q p=k.

Conversely, for the sufficiency, suppose that  is lean. Let ζ X Y be a -map. For each x∈ X, since

    k=𝕀_X(x,x)≤(ζ^*∘ζ)(x,x)=_y∈ Yζ^*(y,x)ζ(x,y)≤_y∈ Y(ζ∘ζ^*)(y,y)≤_y∈ Y𝕀_Y(y,y)=k,

we have

    _y∈ Yζ^*(y,x)ζ(x,y)=k.

Since k>, there exists f(x)∈ Y with p:=ζ^*(f(x),x)ζ(x,f(x))>. Let

    q:=_y∈ Y
     y≠ f(x)ζ^*(y,x)ζ(x,y).

Then p∨ q=k, and

    p q   =ζ^*(f(x),x)ζ(x,f(x))(_y∈ Y
     y≠ f(x)ζ^*(y,x)ζ(x,y))
       =_y∈ Y
     y≠ f(x)ζ^*(f(x),x)ζ(x,f(x))ζ^*(y,x)ζ(x,y)
       =_y∈ Y
     y≠ f(x)ζ(x,f(x))ζ^*(f(x),x))
       =,

where the third equality follows from

    ζ(x,z)ζ^*(y,x)≤(ζ∘ζ^*)(y,z)≤𝕀_Y(y,z)=

for all y,z∈ Y with y≠ z. 

Note that q≠ k, because q=k would lead to p q=p k=p>, contradicting to p q=. Therefore, given that  is lean, we conclude that p=ζ^*(f(x),x)ζ(x,f(x))=k, and consequently 

    ζ^*(f(x),x)=ζ(x,f(x))=k.

We claim that ζ is the graph of the map f X→ Y. Indeed, it is obvious that f_∘≤ζ, and hence the conclusion follows immediately from Lemma <ref> and Corollary <ref>.




As mentioned in Remark <ref><ref>, Theorem <ref> reduces to <cit.> when  is integral:


 (See <cit.>.)
Let  be a non-trivial, commutative and integral quantale. Then every -map is the graph of a map in  if, and only if,

    (p∨ q=⊤ and p q=)(p=⊤ or q=⊤)

for p,q∈.




§ QUANTALE-VALUED PARTIAL MAPS
 

Note that coproducts in the category  are given by disjoint unions of sets as in :

 
 has all coproducts.



Let X=∐_i∈ IX_i be the disjoint union of a family of sets {X_i}_i∈ I, and let ι_i X_i X (i∈ I) be the graphs of the inclusion maps. For any family of -maps {ζ_i X_i Y}_i∈ I, define a -relation

    ζ X Y,  ζ(x,y)=ζ_i(x,y)  if x∈ X_i, y∈ Y.

Then ζ is clearly a -map, and ζ∘ι_i=ζ_i (i∈ I) since









    (ζ∘ι_i)(x,y)   =_w∈ Xζ(w,y)ι_i(x,w)
       =(_w∈ X∖ X_iζ(w,y))∨(_w∈ X_iζ(w,y)𝕀_X_i(x,w))
       =_w∈ X_iζ_i(w,y)𝕀_X_i(x,w)
       =ζ_i(x,y)

for all x∈ X_i, y∈ Y. Finally, for the uniqueness of ζ, let η X Z be a -map satisfying η∘ι_i=ζ_i for all i∈ I. Then

    ζ_i(x,y)=η∘ι_i(x,y)=_w∈ Xη(w,y)𝕀_X(x,w)=η(x,y)

for all x∈ X_i, y∈ Y, which completes the proof.


 
If ζ X Y, η X' Y' are -maps, then 

    ζ⨿η X⨿ X' Y⨿ Y'
 
is a -map. 



It is straightforward to check that ζ^*⨿η^* Y⨿ Y' X⨿ X' is the right adjoint of ζ⨿η in .


Let

    X_+:=X⨿{⋆}

denote the disjoint union of every (crisp) set X and the singleton set {⋆}.


Let X, Y be (crisp) sets. A partial -map ζ from X to Y is a -map ζ X Y_+.



The singleton set {⋆} may not be a terminal object of . For example, if =M_3 (see Example <ref><ref>), then there are two different -maps from {⋆} to {⋆}:


  * the graph of the identity map on {⋆};

  * the -map

    ζ{⋆}{⋆},  ζ(⋆,⋆)=a,

whose right adjoint is given by

    η{⋆}{⋆},  η(⋆,⋆)=b.





When  is an integral quantale, every partial -map is a fuzzy partial function defined in <cit.>, but not vice versa.


In order to construct the category of sets and partial -maps, note that by setting 

    =  and   A={⋆}

in (<ref>) and (<ref>) we obtain a pair of adjoint functors

    |a|/@->@<6pt>/<1000,0>[`;]
    (1000,0)|b|/@->@<6pt>/<-1000,0>[`;]
    (450,0)[]

given by the following data:


  *  is the forgetful functor;

  * the functor  sends each set X to the graph of the inclusion map {⋆} →/^(->/X_+, and sends each -map ζ X Y to the -map 

    ζ_+:=ζ⨿𝕀_{⋆} X_+ Y_+;


  * for each set X, the component of the unit ι_X  X X_+ is the graph of the inclusion map X →/^(->/X_+; 

  * for each object μ{⋆} X in , the component of the counit _μ X μ is given by 

    _μ(x,y)=𝕀_X(x,y)    if x,y∈ X,
    μ(⋆,y)    if x=⋆ and y∈ X.


The adjunction ⊣ induces the maybe monad (,,ι) on , whose endofunctor = carries each set X to X_+, and whose multiplication is given by

    _X (X_+)_+ X_+,  _X(x,y)=𝕀_X(x,y)    if x,y∈ X,
    
    k    if x=y=⋆,
       if x=⋆, y∈ X or x∈ X, y=⋆.

The isomorphism between  and the Eilenberg-Moore category ^ is not difficult to be observed. Indeed, a -algebra (X,μ) consists of a set X and a -map μ X_+ X such that 

    μ∘ι_X=𝕀_X,

which is clearly uniquely determined by an object in . Moreover, it is easy to see that a morphism ζ (X,μ)(Y,λ) of -algebras is a -map ζ X Y such that 

    (ζ∘μ)(⋆,y)=λ(⋆,y)

for all y∈ Y, which is simply a morphism in . Therefore, in what follows we identify ^ with .

















Furthermore, objects in the Kleisli category _ are sets, and a morphism from X to Y in _ is exactly a -map X Y_+; that is, a partial -map from X to Y. Thus, we denote by

    :=_

the category of sets and partial -maps. By Lemma <ref>,  is embedded in  as a full subcategory.

 
The composite of partial -maps ζ X Y_+ and η Y Z_+ is given by 

    (η♢ζ)(x,z)=_y∈ Yη(y,z) ζ(x,y)    if x∈ X and z∈ Z,
    (_y∈ Yη(y,⋆)ζ(x,y))∨ζ(x,⋆)    if x∈ X and z=⋆.




(<ref>) follows from η♢ζ=_Z∘(η)∘ζ and

    (_Z∘(η)∘ζ)(x,z)   =_w∈ (Z_+)_+_y∈ Y_+_Z(w,z) (η)(y,w)ζ(x,y)=_y∈ Yη(y,z)ζ(x,y),
    
    (_Z∘(η)∘ζ)(x,⋆)   =_z'∈ (Z_+)_+_y∈ Y_+_Z(z',⋆) (η)(y,z')ζ(x,y)=(_y∈ Yη(y,⋆)ζ(x,y) k)∨(ζ(x,⋆) k)

for all x∈ X, z∈ Z.



































 
Let ζ X→ Y_+ be a partial -map from X to Y. The value

    ζ(x,⋆)

is interpreted as the degree that x has no image under the map ζ, and so is the value

    ζ^*(⋆,x)=_z∈ Y_+ζ(x,z)𝕀_Y_+(⋆,z),

because the above expression means that


  * For each z∈ Y_+, if z is the image of x under ζ, then z is equal to ⋆; in other words, x has no image under ζ within Y.
  
The adjunction ζ⊣ζ^* can be translated as follows:


  * For every x∈ X, there may not be y∈ Y such that y is the image of x under ζ; because 𝕀_X≤ζ^*∘ζ means that

    k≤_y∈ Y_+ζ^*(y,x)ζ(x,y)=(ζ^*(⋆,x)ζ(x,⋆))∨(_y∈ Yζ^*(y,x)ζ(x,y))

for all x∈ X.

  * If y,z∈ Y are both the images of x under ζ, then y is equal to z; because ζ∘ζ^*≤𝕀_Y_+ means that 

    _x∈ Xζ(x,z)ζ^*(y,x)≤𝕀_Y_+(y,z)=𝕀_Y(y,z)

for all y,z∈ Y. Moreover, setting y∈ Y and z=⋆ in the above inequality we obtain that

    ζ(x,⋆)ζ^*(y,x)=𝕀_Y_+(y,⋆)=

for all x∈ X; that is, “x has no image under ζ” and “y is the image of x under ζ” cannot happen simultaneously. 

Given another partial -map η Y Z_+ from Y to Z, the formula (<ref>) given in Lemma <ref> is understood as follows:


  * z∈ Z is the image of x under η♢ζ if, and only if, there exists y∈ Y such that y is the image of x under ζ, and z is the image of y under η.

  * x has no image under η♢ζ if, and only if, either of the following cases holds:
    
    
  * x has no image under ζ;
    
  * x has an image y under ζ, but y has no image under η.
    




Let  be a non-trivial, commutative and unital quantale. Then every partial -map is the graph of a partial map in  if, and only if,  is lean. In this case, ^∂ and  are isomorphic categories.



Since  is embedded in  as a full subcategory, the necessity is an immediate consequence of Theorem <ref>. For the sufficiency, suppose that  is lean. Since every partial -map is a -map ζ X Y_+, there is a map f X→ Y_+ in  such that ζ=f_, which is exactly a partial map in .


Let (X_+,τ_X)∈ denote the free -algebra on a set X. Then

    τ_X{⋆} X_+,  τ_X(⋆,x)=
    k    if x=⋆,
       if x∈ X

is the restriction of _X (X_+)_+ X_+ (see (<ref>)), which is exactly the graph of the inclusion map {⋆} →/^(->/X_+. It is well known that the canonincal functor

    (=_)→(≅^)

that sends each -map ζ X Y_+ (i.e., partial -map from X to Y) to

    ζ (X_+,τ_X) (Y_+,τ_Y),   (ζ)(x,y)=ζ(x,y)    if x∈ X, y∈ Y,
    
    k    if x=⋆=y,
       else

is fully faithful (see, e.g., <cit.>).






























 
Assuming the axiom of choice, every object of  is isomorphic to a free -algebra, hence the functor  is essentially surjective.



Let (X,μ)∈. Since X is the codomain of the -map μ{⋆} X, it is clear that X≠∅. By the axiom of choice and <cit.>, there exists a binary operation · on X such that (X,·) is a group, with e∈ X being its unit. Then


  *  x· z≠ y· z and z· x≠ z· y for all x,y,z∈ X with x≠ y, and

  *  x· y≠ x for all x,y∈ X with y≠ e.

Our strategy is to show that (X,μ) is isomorphic to

    (X∖{e})=((X∖{e})_+,τ_X∖{e}).

Explicitly, we define -relations 

    ζ X (X∖{e})_+,  ζ(x,a)=μ^*(x,⋆)    if x∈ X, a=⋆
    μ^*(x· a,⋆)    if x,a∈ X

and 

    η (X∖{e})_+ X,  η(a,x)=μ(⋆,x)    if a=⋆, x∈ X
    μ(⋆,x· a)    if a,x∈ X,
 
and we show that ζ and η establish an isomorphism between (X,μ) and ((X∖{e})_+,τ_X∖{e}) in .

First, η∘ζ=𝕀_X and ζ∘η=𝕀_(X∖{e})_+, so that ζ and η are both -maps, and they are inverses of each other. Indeed, it follows from <ref> and Lemma <ref> that

    (η∘ζ)(x,y)=_a∈(X∖{e})_+η(a,y)ζ(x,a)=(μ(⋆,y)μ^*(x,⋆))∨(_a∈ X∖{e}μ(⋆,y· a)μ^*(x· a,⋆))=

for all x,y∈ X with x≠ y, and

    (η∘ζ)(x,x)   =(μ(⋆,x)μ^*(x,⋆))∨(_a∈ X∖{e}μ(⋆,x· a)μ^*(x· a,⋆))
       ≤𝕀_X(x,x)∨(_a∈ X∖{e}𝕀_X(x· a,x· a))   (μμ^*)
       =k
       =(μ^*∘μ)(⋆,⋆)     (Lemma <ref>)
       =(μ(⋆,x)μ^*(x,⋆))∨(_y∈ X
     y≠ xμ(⋆,y)μ^*(y,⋆))
       =(μ(⋆,x)μ^*(x,⋆))∨(_y∈ X
     y≠ xμ(⋆,x· x^-1· y)μ^*(x· x^-1· y,⋆))
       ≤(μ(⋆,x)μ^*(x,⋆))∨(_a∈ X∖{e}μ(⋆,x· a)μ^*(x· a,⋆))   (x≠ y x^-1· y≠ e)
       =(η∘ζ)(x,x)

for all x∈ X. Thus η∘ζ=𝕀_X. In order to show that ζ∘η=𝕀_(X∖{e})_+, from <ref>, <ref> and Lemma <ref> we deduce that

    (ζ∘η)(a,b)   =_x∈ Xζ(x,b)η(a,x)=_x∈ Xμ^*(x· b,⋆)μ(⋆,x· a)=,
    
    (ζ∘η)(a,⋆)   =_x∈ Xζ(x,⋆)η(a,x)=_x∈ Xμ^*(x,⋆)μ(⋆,x· a)=,
    
    (ζ∘η)(⋆,b)   =_x∈ Xζ(x,b)η(⋆,x)=_x∈ Xμ^*(x· b,⋆)μ(⋆,x)=

for all a,b∈ X∖{e} with a≠ b. Moreover, it follows from Lemma <ref> that

    (ζ∘η)(⋆,⋆)=_x∈ Xζ(x,⋆)η(⋆,x)=_x∈ Xμ^*(x,⋆)μ(⋆,x)=(μ^*∘μ)(⋆,⋆)=k,

and

    (ζ∘η)(a,a)   =_x∈ Xμ(⋆,x· a)μ^*(x· a,⋆)
       ≤_x∈ X𝕀_X(x· a,x· a)   (μμ^*)
       =k 
       =(μ^*∘μ)(⋆,⋆)   (Lemma <ref>)
       =_x∈ Xμ^*(x,⋆)μ(⋆,x)
       =_x∈ Xμ^*(x· a^-1· a,⋆)μ(⋆,x· a^-1· a)
       ≤_y∈ Xμ^*(y· a,⋆)μ(⋆,y· a)
       =(ζ∘η)(a,a)

for all a∈ X∖{e}, as desired.

Second, ζ∘μ=τ_X∖{e} and η∘τ_X∖{e}=μ, so that ζ(X,μ)→((X∖{e})_+,τ_X∖{e}) and η((X∖{e})_+,τ_X∖{e})→(X,μ) are isomorphisms in . Indeed, ζ∘μ=τ_X∖{e} because

    (ζ∘μ)(⋆,⋆)   =_x∈ Xζ(x,⋆)μ(⋆,x)
       =_x∈ Xμ^*(x,⋆) μ(⋆,x)
       =(μ^*∘μ)(⋆,⋆)
       =k   (Lemma <ref>)
       =τ_X∖{e}(⋆,⋆)

and

    (ζ∘μ)(⋆,a)   =_x∈ Xζ(x,a)μ(⋆,x)
       =_x∈ Xμ(⋆,x)μ^*(x· a,⋆)
       =   (<ref> and Lemma <ref>)
       =τ_X∖{e}(⋆,a)

for all a∈ X∖{e}. Meanwhile, η∘τ_X∖{e}=μ because

    (η∘τ_X∖{e})(⋆,x)=_a∈ (X∖{e})_+η(a,x)τ_X∖{e}(⋆,a)=η(⋆,x) k=η(⋆,x)=μ(⋆,x)

for all x∈ X, where the second equality follows from (<ref>).
































Lemma <ref> guarantees that → is an equivalence of categories:

 
Assuming the axiom of choice,  is equivalent to  and, therefore, is monadic over .





§ ACKNOWLEDGEMENT


The authors acknowledge the support of National Natural Science Foundation of China (No. 12071319) and the Fundamental Research Funds for the Central Universities (No. 2021SCUNL202).













abbrv





 Paper Title: The least subtopos containing the discrete skeleton of $Ω$.







Predicting nonlinear-flow regions in highly heterogeneous porous media using adaptive constitutive laws
 and neural networks
    Chiara Giovannini^1 ^2 ^3 Alessio Fumagalli^1 Francesco Saverio Patacchini^2
 August 5, 2024
    ^1 Department of Mathematics, Politecnico di Milano, p.za Leonardo da Vinci 32, 20133 Milano, Italy

^2 IFP Energies nouvelles, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison, France

^3 Department of Engineering Mechanics, KTH Royal Institute of Technology, Brinellvägen 8, 114 28 Stockholm, Sweden

====================================================================================================================================================================================================================================================================================================================







§ ABSTRACT 
Let p : → be  a pre-cohesive geometric morphism.
We show that the least subtopos of  containing both the subcategories p^* : → and p^! : → exists, and that it coincides with the least subtopos containing p^*, where   denotes the subobject classifier of .














§ INTRODUCTION


At least since his PhD  <cit.>, and throughout his career until <cit.>, Topos Theory was one of Pieter's many interests.
Each time we met we spoke about it. Certainly a lot when Pino Rosolini invited him to Genova while I was there in 2002, and then also a little each time we met at CT meetings.  He was a very witty conversationalist so it was also a pleasure to discuss other things. Still,  I would have liked to tell him about the result in the present paper because it concerns toposes `of spaces' which, as explained in <cit.>, have a canonical Homotopy Theory, another one of Pieter's interests <cit.>.



More specifically, this paper is about the Dimension Theory outlined in <cit.>.  
We recall some of the basic definitions and refer to  <cit.> for  additional information.
A level of a topos   is just an essential subtopos of .
If l : → is a level of , then the left adjoint to the inverse image functor  l^* : → is denoted by l_!. Following <cit.> we picture a level of  as a `dimension' and l_! : → as the full subcategory  consisting of the objects X of  such that `X ≤ l'. For any X in , the counit l_! (l^* X) → X is called the  (l)-skeleton of X. 
Abusing the terminology a bit, we sometimes call the object l_! (l^* X) the  (l)-skeleton of X. 
In other words, we sometimes confuse a skeleton with its domain.


Levels  may be partially ordered as subtoposes so, in this way, each topos determines its poset `of dimensions'.
Also, for levels l, m of a given topos, we say that m is way above l if m is above l as subtoposes and, moreover,  l_! factors through m_*. The Aufhebung of a level l is the least level that is way above l. 

 
Notice that it also makes sense to say that a subtopos is way above some  level l. Hence, we may consider the problem of finding the least subtopos that is way above  l. Of course, if this least subtopos is essential then it is the Aufhebung of l. The resulting `weak Aufhebung' problem seems interesting because, on the one hand, little is known about  conditions guaranteeing the existence of the Aufhebung of an arbitrary level of an elementary topos and, on the other hand, for several important toposes `of combinatorial spaces', such as simplicial sets, every subtopos is essential.



Remark <ref> and other well-known results in Topos Theory suggest the following. 


For a topos     and a functor F : →, the envelope of F is, if it exists, the  least subtopos f : → such that F factors through the direct image functor f_* : →.


Roughly speaking, the envelope of F is the  least subtopos of  through which F factors.
As suggested above, given a level l : →, the `weak Aufhebung' of l is the envelope of  the join of the subcategories l_! : → and l_* : →. 
The main result of the paper shows that it exists in certain cases. 
Before stating the result precisely it is relevant to mention other examples of envelopes.

It makes perfect sense to consider the envelope of an object in a topos.
Indeed, we may define it as the envelope of the obvious functor from the terminal category to the  topos.
The envelope of any object in a topos exists by  <cit.>.


[The least dense subtopos.]
The envelope of the initial object coincides with the subtopos for the double-negation topology <cit.>.



[Weak generation.]
A topos is said to be weakly generated by an object if the envelope of that  object is the whole topos. For example,  <cit.> shows that every topos is weakly generated by its subobject classifier; but toposes may be weakly generated by smaller objects. The topological topos is weakly generated by 2 <cit.>. Sufficiently Cohesive toposes over Boolean bases are weakly generated by objects with a unique point <cit.>.


If p : → is a geometric morphism and   is the subobject classifier of , then we may consider the envelope of  p^*.


[The least pure subtopos.]
If p : → is locally connected and bounded, then <cit.>  shows that the envelope of  p^* is the smallest pure subtopos of  and it is also locally connected over .






For a level l : →  we may consider the envelope of  l_! where  is the subobject classifier of , and we may also consider the envelope of  the (domain of the) l-skeleton of the subobject classifier of . Occasionally, these two subtoposes coincide; in particular, this is the case if the inverse image functor l^* preserves the subobject classifier <cit.>.


[The `weak Aufhebung' of level -∞.]
Level -∞ is the essential subtopos  0→ where 0 is the topos with exactly one object. 
The inverse image functor trivially preserves the subobject classifier and the skeleton of any object in  is initial. So, by Example <ref>,  the envelope of  the -∞-skeleton of the subobject classifier coincides with the subtopos of sheaves for the double-negation topology. 






Let p : → be a geometric morphism. The map p is hyperconnected if  both the unit and counit of p^* ⊣ p_* are monic. The map p is local if its direct image functor has a fully faithful right adjoint, usually denoted by p^!. In this case, the string p^* ⊣ p_* ⊣ p^! : → is a level of  called the centre of p, and, for any X in , the counit p^* (p_* X ) → X is the skeleton of X (determined by the centre of p).
 Also, p is pre-cohesive if it is local, hyperconnected and the inverse image functor has a finite-product preserving left adjoint, usually denoted by p_!  <cit.>. In this case, we picture the objects of  as `spaces' and p^* : → as the full subcategory of `discrete spaces'. Also,  p_!  is thought of as sending a space X to the discrete space p_! X of connected components. Then, for every space X, the (monic) skeleton p^* (p_* X) → X is called the discrete skeleton of X.
Allowing ourselves to identifying `discrete' with `0-dimensional', it makes sense to say that  the centre of the pre-cohesive p is level 0.
 


We prove that, for a pre-cohesive p : →, the `weak Aufhebung' of level 0 exists and, moreover, it coincides with the least subtopos containing the 0-skeleton of Ω. Notice that this is analogous to  the fact that, if  is Boolean, then level 0 is the Aufhebung of level -∞, as already observed in  <cit.> and Example <ref>.



We stress that for an arbitrary geometric morphism p : → towards a topos  with subobject classifier 2, the envelope of p^* 2 need not be a level of , even if p is locally connected. 



[The smallest pure subtopos of Sh(ℝ^2)  is  not essential.]
The present example is due to Jon Funk who patiently explained it to me in a private communication.
Let U ⊆ℝ^2 be a pure open subet and r ∈ U.
Then U - {r}⊆ℝ^2 is also pure and open (and strictly smaller than U, of course).
Hence, the smallest pure subtopos of Sh(ℝ^2) is not open.

Also,  <cit.> implies that the levels of a topos of sheaves on a Hausdorff space are precisely its open subtoposes.
Therefore, the smallest pure subtopos of Sh(ℝ^2)  cannot be essential, for otherwise it would be open.







In contrast, if  is a topos such that every subtopos is a level, then, trivially, every envelope is a level. In such toposes there is an endofunction on the  poset of levels of  that sends a level l to the envelope of  the l-skeleton of the subobject classifier of . As a source of examples consider the topos of presheaves on any small category with split-epi/split-mono factorizations and such that every object has a finite set of subobjects <cit.>. In particular, we may consider the topos of simplicial sets, or the classifier of non-trivial Boolean algebras. We don't know if any of these functions coincides with the Aufhebung of the respective topos.




§ INTERNAL AND STABLE ORTHOGONALITY



Let  be a topos with subobject classifier Ω and let f : W → X be a morphism in .

We say that f is orthogonal to an object Z if for every g : W → Z there exists a unique g' : X → Z such that g' f = g. Alternatively, f is orthogonal to Z if and only if (f, X) : (X , Z)  →(W, Z) is an isomorphism. 
We say that f is internally orthogonal to Z if Z^f : Z^X → Z^W is an isomorphism. 




 It is well-known that Ω^(_) : →  reflects isomorphisms. In other words, a map f is internally orthogonal to Ω if and only if f is an isomorphism.







The map f is stably orthogonal to Z if for every pullback as below


    U [d]_-f'[r]     W [d]^-f 
    
    V [r]     X


the map f' is orthogonal to Z. 


 If f is stably orthogonal to Z, then it is internally so.



Simply observe that  the map f is internally orthogonal to Z if and only if,  for every object Y,
f × Y : W × Y → X × Y is orthogonal to Z .



The proof that the least subtopos containing an object exists  <cit.>
 involves  the following.

 
If _j → is  the least subtopos containing the object Z in  then, 
a subobject u : U → X in  is j-dense   if and only if u is stably orthogonal to Z.



Combining Lemmas <ref> and <ref> we obtain the next auxiliary fact.


 If _j → is the least subtopos containing the object Z in , then every j-dense subobject is internally orthogonal to Z. 







Let  be another topos with subobject classifier denoted by 2.



  If  R : → is a full and faithful functor with a finite-product preserving left adjoint  L : →, then   the following are equivalent:



  * The map f is internally orthogonal to R 2.

  * The map L f: L W → L X is an isomorphism  in .

  * For every A in , f is internally orthogonal to R A.




As L preserves finite products by hypothesis, the canonical  R(A^L Y)  → (R A)^Y is an isomorphism natural in both A and Y.
In particular we have


    (R 2)^X [d]_-≅[rr]^-(R 2)^f       (R 2)^W [d]^-≅
    
    R( 2^L X) [rr]_-R(2^L f)       R (2^L W)


so, if the top map is an isomorphism, then the bottom map is an isomorphism.
As R is fully faithful, ^L f is an isomorphism. That is, L f is internally orthogonal to .
So L f is an isomorphism by Remark <ref>.



To prove that the second item implies the third consider the diagram


    (R A)^X [d]_-≅[rr]^-(R A)^f       (R A)^W [d]^-≅
    
    R( A^L X) [rr]_-R(A^L f)       R (A^L W)


where the vertical maps are iso by adjointness and the bottom map is iso by hypothesis.
It follows that the top map is also an iso, showing that f is internally orthogonal to R A.

Finally, the third item trivially implies the first.









§ THE `WEAK AUFHEBUNG' OF LEVEL 0  OF A PRE-COHESIVE TOPOS


Let p : → be a pre-cohesive geometric morphism.
The centre of the local p is called Level 0.
 Let  be the subobject classifier  of  but we stress  that we are not assuming that  is Boolean. 


 For every f : W → X in , the following are equivalent:



  * The map f is internally orthogonal to p^* 2.

  * The map p_! f : p_! W → p_! X is an isomorphism in .

  * For every A in , f is internally orthogonal to p^* A.




Follows from Lemma <ref> applied to the adjunction p_! ⊣ p^*.



In other words, f is internally orthogonal to discrete spaces if and only if f induces an isomorphism at the level of connected components.
Notice that the equivalence between the first and third items of Lemma <ref> appears to be a variant of <cit.>.
We stress  that  monomorphisms satisfying the equivalent conditions of Lemma <ref> need not be stable under pullback.

 
Consider the topos Δ_1 of reflexive graphs and the pre-cohesive geometric morphism p : Δ_1→. Let  X = ·⇉· be the  reflexive graph with two nodes and two (non-identity) parallel edges. Pick one of the non-identity edges and consider the associated subobject u of X. Its intersection with the subobject determined by the other edge is the discrete set of nodes of X. So u is a monomorphism such that p_! u is an isomorphism, but it is not stably orthogonal to p^* 2. Notice also that in this example, p_* u is  an isomorphism. 





 Let _j → be least subtopos containing   p^* 2. A monic u : U → X in  is j-dense if and only if, for every pullback square 


    V  [d]_-v  [r]     U [d]^-u 
    
    Y [r]     X



in ,  p_! v is an isomorphism in . Hence, if u is j-dense then p_* u is an isomorphism.



Lemma <ref> implies that the pullback condition in the statement holds if and only if u is stably orthogonal to p^* 2.
So the first part of the result follows from Lemma <ref>. To prove the second part of the statement observe that the square below 


    p^* (p_* U) [d]_-p^* (p_* u)[r]^-β    U [d]^-u 
    
    p^* (p_* X) [r]_-β    X


is a pullback because p is hyperconnected so p_! (p^* (p_* u)) : p_! (p^* (p_* U)) → p_!(p^* (p_* X)) is an isomorphism in .
That is, p_* u is an isomorphism.






Let  p : → be the canonical geometric morphism from the topos of presheaves on the small category .
The map p is not necessarily pre-cohesive but it is locally connected so p^* has a left adjoint p_! and <cit.> implies that a sieve R on an object C of  is covering for the least subtopos containing p^*  if and only if f^* R is connected for every map f with codomain C in . This  is  essentially the pullback condition of Lemma <ref> but restricted to the site. So this example illustrates that lemma and suggests a possible generalization to locally connected, but not necessarily pre-cohesive, maps.






 
 Let  be a topos with subobject classifier . If p : → is pre-cohesive, then  the  least subtopos of  containing the subcategories p^* , p^! : → exists and it coincides with the least subtopos of  containing p^*.



 Let _j → be  the least subtopos containing  p^* and let A be an object in .
 Lemma <ref> easily implies that every j-dense mono is orthogonal to p^! A. 
 That is, p^! A is a j-sheaf and therefore, _j → is above the centre of p.
 
 

It remains to prove that the inclusion p^* : → factors through the inclusion _j →. 
By Lemma <ref>, j-dense subobjects are internally orthogonal to p^* 2  and therefore orthogonal to p^* A by 
Lemma <ref>. In other words, p^* A is a j-sheaf.

We have proved that the subtopos _j → is way-above level 0.
It must be the least one because any subtopos of  that is way-above level 0 must contain p^* 2.






10

BungeFunkBook
M. Bunge and J. Funk.
Singular coverings of toposes.
Berlin: Springer, 2006.

FunkHofstra2022
J. Funk and P. Hofstra.
Pseudogroups and their torsors.
Semigroup Forum, 104(2):281–319, 2022.

Hemelaer2022
J. Hemelaer.
Some toposes over which essential implies locally connected.
Cah. Topol. Géom. Différ. Catég., 63(4):425–451,
  2022.

Hofstra2003
P. J. W. Hofstra.
Completions in realizability.
PhD thesis, Universiteit Utrecht, September 2003.

HofstraWarren2013
P. J. W. Hofstra and M. A. Warren.
Combinatorial realizability models of type theory.
Ann. Pure Appl. Logic, 164(10):957–988, 2013.

elephant
P. T. Johnstone.
Sketches of an elephant: a topos theory compendium, volume
  43-44 of Oxford Logic Guides.
The Clarendon Press Oxford University Press, New York, 2002.

Lawvere91
F. W. Lawvere.
Some thoughts on the future of category theory.
In Proceedings of Category Theory 1990, Como, Italy, volume
  1488 of Lecture notes in mathematics, pages 1–13. Springer-Verlag,
  1991.

Lawvere07
F. W. Lawvere.
Axiomatic cohesion.
Theory Appl. Categ., 19:41–49, 2007.

LawvereMenni2015
F. W. Lawvere and M. Menni.
Internal choice holds in the discrete part of any cohesive topos
  satisfying stable connected codiscreteness.
Theory Appl. Categ., 30:909–932, 2015.

MMlevelEpsilon
F. Marmolejo and M. Menni.
Level ϵ.
Cah. Topol. Géom. Différ. Catég., 60(4):450–477, 2019.

Menni2019
M. Menni.
Every sufficiently cohesive topos is infinitesimally generated.
Cah. Topol. Géom. Différ. Catég., 60(1):3–31, 2019.

Menni2019a
M. Menni.
Monic skeleta, boundaries, Aufhebung, and the meaning of
  `one-dimensionality'.
Theory Appl. Categ., 34:714–735, 2019.






