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: No quasi-isomorphism between a minimal Sullivan algebra of non-finite type and its realization.



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



§ ABSTRACT 
We prove that the morphisms from a minimal Sullivan algebra of non-finite type to the algebra of polynomial differential forms on its realization cannot be quasi-isomorphic. This provides a positive answer to a question posed by Félix, Halperin and Thomas. Furthermore, we give some discussion about the relationship between the homotopy groups of a topological space and its minimal Sullivan model.




§ INTRODUCTION


The algebraic models formulated by Quillen <cit.> and Sullivan <cit.> serve as cornerstones in rational homotopy theory. Quillen's model is a differential graded Lie algebra constructed from the rational homotopy groups of a simply connected space, while Sullivan's model is a commutative differential graded algebra (CDGA) built upon the rational cohomology ring. Every path connected space possesses a unique CDGA (up to isomorphisms) of special type, which is called its minimal Sullivan model.

The minimal Sullivan model of a space X is a free graded commutative algebra Λ V equipped with a differential that satisfies certain conditions. There need to be a quasi-isomorphism from Λ V to A_PL(X), the CDGA of polynomial differential forms on X. This quasi-isomorphism is sometimes also referred to as the minimal Sullivan model of X.

Moreover, when X is simply connected and H^*(X,ℚ) is of finite type, the rational homotopy group π_n(X)⊗ℚ of X can be readily computed from Λ V. This homotopy group is isomorphic to the dual space of V^n, where V^n denotes the subspace of V in degree n. Such dual spaces are also defined as the homotopy group of Λ V and written as π_n(Λ V).

Conversely, any simply connected CDGA (whose first cohomology is trivial) is quasi-isomorphic to a unique special CDGA Λ V, known as the minimal Sullivan algebra. When Λ V is of finite type, one can construct a topological space |Λ V| such that π_n(|Λ V|)=π_n(|Λ V|)⊗ℚ is isomorphic to the homotopy group of Λ V, and H^n(|Λ V|;ℤ) = H^n(|Λ V|;ℚ) = H^n(Λ V;ℚ). The space |Λ V| is referred to as the realization of Λ V. If Λ V is a minimal Sullivan model of some space X, then |Λ V| and X have the same rational homotopy type.

Therefore, the minimal Sullivan model and the realization give a bijective correspondence between the rational homotopy types of simply connected spaces whose cohomology is of finite type and the isomorphism classes of simply connected minimal Sullivan algebras of finite type.

For the general case, such an identification no longer holds. While we can still construct a minimal Sullivan model based on the cohomology of a space, their rational homotopy groups may differ. Similarly, while we can construct the realization of a minimal Sullivan algebra, their rational cohomologies may not necessarily be isomorphic. These observations lead to the following questions.


  * Given a minimal Sullivan algebra Λ V, under what conditions is the map Λ V→ A_PL(|Λ V|) a quasi-isomorphism?

  * Given a topological space X with a minimal Sullivan model Λ V, when is π_n(X)⊗ℚ→π_n(Λ V) isomorphic for all n≥ 2? Furthermore, under what conditions can we construct certain isomorphisms based on π_1(X) and the homotopy Lie algebra of Λ V?


When H^1(Λ V;ℚ) and V^n   (n≥ 2) are all finite dimensional, these questions have been answered affirmatively. The corresponding path connected space X satisfies H^1(X;ℚ)<∞ and that H^*(X;ℚ) is of finite type, where X is the universal cover of X. Such space a X is called a Sullivan space if its minimal Sullivan model Λ V ≃⟶ A_PL(X) induces isomorphisms π_n(X)⊗ℚ≅⟶π_n(Λ V) for all n. There exist several equivalent conditions for being a Sullivan space <cit.>. On the other hand, if Λ V satisfies the above hypothesis, then the induced map m_|Λ V|: Λ V→ A_PL(|Λ V|) is a quasi-isomorphism if and only if |Λ V| is a Sullivan space <cit.>.

In the case where Λ V is simply connected, m_|Λ V|:Λ V→ A_PL(|Λ V|) is a quasi-isomorphism if and only if V is of finite type <cit.>. Consequently, if the minimal Sullivan model Λ V of a simply connected space X induces isomorphisms on rational homotopy groups, then H^*(X;ℚ) must be of finite type. This is because, under these conditions, X and |Λ V| have the same rational homotopy type.

It remains to consider the case of neither simply connected nor finite type. In <cit.>, Félix, Halperin and Thomas asked of whether there exist minimal Sullivan algebras that are quasi-isomorphic to their realizations, despite not being simply connected or of finite type.


Suppose that Λ V is a minimal Sullivan model of a path connected space X, and the induced map X→ |Λ V| is isomorphic on cohomologies. Is |Λ V| a Sullivan space? In this case must X be a Sullivan space?


When |Λ V| is a Sullivan space, according to Theorem 7.8 of <cit.>, H^1(Λ V;ℚ) and V^n for n≥ 2 are finite dimensional. So the first part of Question <ref> essentially examines whether m_|Λ V|:Λ V → A_PL(|Λ V|) can be a quasi-isomorphism when V^n=∞ for some n≥ 2. We will show that under this condition m_|Λ V| is not quasi-isomorphic, and furthermore, no quasi-isomorphism exists from Λ V to A_PL(|Λ V|). Moreover, the same conclusion holds if V^1=∞.


Let Λ V be a minimal Sullivan algebra. If there exists a quasi-isomorphism Λ V ≃⟶ A_PL(|Λ V|), then V must be of finite type.


Then we can generalize Theorem 7.8 of <cit.> as follows.


Let Λ V be a minimal Sullivan algebra. The following statements are equivalent.

(i) There exists a quasi-isomorphism Λ V ≃⟶ A_PL(|Λ V|).

(ii) m_|Λ V|:Λ V → A_PL(|Λ V|) is a quasi-isomorphism.

(iii) |Λ V| is a Sullivan space.

Moreover, in this case V is of finite type.


We also delve into the homotopy groups of a topological space and its minimal Sullivan model. It is possible that a minimal Sullivan model Λ V ≃⟶ A_PL(X) exists such that it induces isomorphisms π_n(X)⊗ℚ≅⟶π_n(Λ V) for n≥ 2, yet H^1(X; ℚ)=∞. However, this scenario can be excluded if additional constraints are imposed on π_1(X). When H^1(X; ℚ)<∞, π_n(X)⊗ℚ and π_n(Λ V) are isomorphic only if H^*(X; ℚ) is of finite type. It remains an open question whether the same necessity applies to the cohomology of X, i.e. whether H^*(X; ℚ) has to be of finite type.


Let f:Λ V ≃⟶ A_PL(X) be a minimal Sullivan model of a topological space X, and H^1(X) =  H^1(Λ V) < ∞. If for all n≥ 2, the induced map π_n(f):π_n(X)⊗ℚ→π_n(Λ V) is an isomorphism, then H^*(X) is of finite type.


Finally, we present two examples. Firstly, for a Sullivan space, the realization of its minimal Sullivan model may not necessarily be a Sullivan space. Secondly, for a non-Sullivan space, the realization of its minimal Sullivan model can be a Sullivan space. The second example provides a negative answer of the second part of Question <ref>.

This paper is organized as follows. In Section 2, we review the realization of minimal Sullivan algebras, along with the holonomy actions of fibrations and their algebraic representations. These materials serve as the basis for proving Theorem <ref> in Section 3. Moving on, Section 4 delves into the relationship between the homotopy groups of a topological space and its minimal Sullivan model. Throughout these latter sections, we will assume that the ground ring is ℚ, and hence, for brevity, we will write H^*(-;ℚ) simply as H^*(-).

The author would like to thank Jianfeng Lin and Jie Wu for helpful discussions. This research is supported by the National Key Research and Development Program of China No. 2020YFA0713000.



§ PRELIMINARY



 §.§ Realization of minimal Sullivan algebras

A Sullivan algebra is a free graded algebra Λ V generated by a graded vector space V of positive degree, equipped with a differential d such that V has a filtration

    0=V(-1) ⊂ V(0) ⊂ V(1) ⊂…⊂ V(k) ⊂…

and dV(k)⊂Λ V(k-1). For simplicity, we will use Λ V to denote the Sullivan algebra (Λ V,d).

We call Λ V minimal if dV⊂Λ^≥ 2 V. Here Λ^≥ n V denotes the subspace of Λ V spanned by elements with wordlength at least n.

The simplicial commutative cochain algebra, A_PL, is defined as follows. (A_PL)_n is the quotient algebra of the free graded algebra generated by t_0,…,t_n,dt_0,…,dt_n where t_0+…+t_n is identified with 1. Here t_0,…,t_n are all of degree 0. So dt_0,…,dt_n are degree-1 coboundaries and dt_0+…+dt_n=0. The face and degeneracy morphisms of A_PL are defined by 

    ∂_i t_k =
    
    t_k    if  k<i 
    
    0    if  k=i 
    
    t_k-1   if  k>i
      and  
    s_j t_k =
    
    t_k    if  k<j 
    
    t_k+t_k+1   if  k=j 
    
    t_k+1   if  k>j
    
    .


Given a Sullivan algebra Λ V, we can construct a simplical set ⟨Λ V ⟩. Set ⟨Λ V ⟩_n as the morphisms of CDGAs from Λ V to (A_PL)_n. For each σ∈⟨Λ V ⟩_n, set ∂_i(σ)=∂_i∘σ and s_j(σ)=s_j∘σ.

By Milnor realization, ⟨Λ V ⟩ induces a CW complex |Λ V|. More generally, given any simplicial set K, its Milnor realization |K| is defined as follows. Let e_0,e_1,… be the standard basis of ℝ^∞ and Δ^n denote the Euclidean simplex with vertices e_0,e_1,…,e_n. Then

    |K| = _n   (K_n×Δ^n)/∼

where each K_n is equipped with the discrete topology and ∼ is the equivalence relation generated by

    ∂_iσ× x ∼σ×μ_i x,    s_jσ× y ∼σ×π_j y.

Here μ_i:Δ^n-1→Δ^n and π_j:Δ^n+1→Δ^n satisfy

    μ_i e_k =
    
    e_k    if  k<i 
    
    e_k+1   if  k≥ i
    
    ,  π_j e_k =
    
    e_k    if  k≤ j 
    
    e_k-1   if  k>j
    
    .


The homotopy group π_n(|Λ V|) is related to (V^n)^♯, where V^n is the degree n subspace of V and ♯ denotes the dual space.


Let Λ V be a minimal Sullivan algebra. There exists natural bijective maps ι_n:π_n(|Λ V|) → (V^n)^♯ satisfying the following.

(i) ι_n are natural transformations.

(ii) When n≥ 2, ι_n is an isomorphism of abelian groups.

(iii) If H^1(Λ V) is finite dimensional, exp_L∘ι_1 is an isomorphism of groups. Here exp_L is a bijective map on (V^1)^♯, and gives its image a group structure.


For n≥ 2, (V^n)^♯ is called the 𝐧-th homotopy group of Λ V and denoted as π_n(Λ V). To define the fundamental group of Λ V we need to restricted to the case that H^1(Λ V)<∞. It is the image of exp_L and denoted as π_1(Λ V).

On the other hand, for any topological space X, the set of singular simplices Sing   X is a simplicial set. |∙| and Sing are adjoint functors between the categories of the pointed simplicial sets and the pointed topological spaces. Let (K,u_0) be a pointed simplicial set and (X,x_0) be a pointed topological space. Then the based continuous maps (|K|,u_0)→(X,x_0) and the based simplicial maps (K,u_0)→(Sing   X,x_0) are identified.

Thus, the identity map on |K| is adjoint to a simplicial λ_K: K→ Sing   |K|. In particular, λ_K sends σ∈ K_n to the singular simplex Δ^n → |K|,   x↦σ× x.

Let S_n(K) be the ℚ-vector space with basis K_n, and D_n(K) be the subspace spanned by the degenerate simplices. Then the operator ∂=∑_i=0^n (-1)^i∂_i makes C_*(K) a chain complex where C_n(K)=S_n(K)/D_n(K). Its dual, C^*(K)=(C_*(K))^♯=Hom(C^*(K), ℚ), is a cochain complex. λ_K induces isomorphisms H_*(C_*(K))≅⟶ H_*(|K|) and H^*(|K|) ≅⟶ H^*(C^*(K)).

The space of simplicial maps from K to A_PL also has a cochain algebra structure, which is called the cochain algebra of polynomial forms on K and denoted as A_PL(K). An element Φ∈ A_PL^k(K) of degree k sends σ∈ K_n to Φ_σ∈ (A_PL)_n^k and satisfies

    ∂_iΦ_σ = Φ_∂_iσ,    s_jΦ_σ = Φ_s_jσ.

The differential on A_PL(K) is defined as (dΦ)_σ=d(Φ_σ).

There exists a quasi-isomorphism ∮_K:A_PL(K)≃⟶ C^*(K) given by (∮_K Φ)(σ)=∮_n Φ_σ. ∮_n:A_PL^n→ℚ is defined in the section 10 of <cit.>. Its properties, which are needed in this paper, will be described later.

So ∮_Sing   |K| also induces an isomorphism H^*(A_PL(Sing   |K|))≅⟶ H^*(|K|). Note that A_PL(Sing   |K|) is usually written as A_PL(|K|) for simplicity. In conclusion, we have the following commutative square of natural isomorphisms.


    @C=60pt
    H^*(|K|) [r]^H(C^*(λ_K))_≅    H^*(C^*(K)) 
    
    H^*(A_PL(|K|)) [u]^H(∮_Sing   |K|)_≅[r]^H(A_PL(λ_K))_≅    H(A_PL(K)) [u]^H(∮_K)_≅


Consider a minimal Sullivan algebra Λ V again. The identity map on ⟨Λ V ⟩ is adjoint to a map m_⟨Λ V ⟩: Λ V → A_PL(⟨Λ V ⟩) such that (m_⟨Λ V ⟩(x))_σ=σ(x) for x∈Λ V and σ∈⟨Λ V ⟩. It can be shown that the quasi-isomorphism A_PL(λ_⟨Λ V ⟩): A_PL(|Λ V|) → A_PL(⟨Λ V ⟩) is surjective, so m_⟨Λ V ⟩ has a lift m_|Λ V|:Λ V→ A_PL(|Λ V|) satisfying m_⟨Λ V ⟩ = A_PL(λ_⟨Λ V ⟩) ∘ m_|Λ V| <cit.>.

As Λ V is minimal, all the coboundaries are in Λ^≥ 2V. So the map ξ_n:H^n(Λ V) → (Λ^≥ 1V/Λ^≥ 2V)^n ≅ V^n sending the cohomology classes to the classes of representatives is a well defined map.


Let Λ V be a minimal Sullivan algebra. For n≥ 2, suppose α∈π_n(|Λ V|) and β∈ H^n(Λ V). Then

    ⟨ H^n (m_|Λ V|)β, hur α⟩ = ⟨ξ_n(β), ι_n(α) ⟩.


It follows that H^n (m_|Λ V|)^♯∘ hur=ξ_n^♯∘ι_n, i.e. the following diagram commutes.



    @C=60ptπ_n(|Λ V|) [r]^ι_n[d]^hur    (V^n)^♯[d]^ξ_n^♯
    
    H_n(|Λ V|) [r]^H^n (m_|Λ V|)^♯    [H^n(Λ V)]^♯.


When n=1 and H^1(Λ V)<∞, (<ref>) also commutes. To deal with the case that H^1(Λ V)=∞, we consider the map τ_1:(V^1)^♯→π_1(|Λ V|), which is constructed in <cit.> as the inverse of ι_1.

The definition of ι_1 requires the simplicial set Δ[1]/∂Δ[1], which consists of two non-degenerate simplices c_0∈(Δ[1]/∂Δ[1])_0 and c_1∈(Δ[1]/∂Δ[1])_1. So its realization is S^1. There is an element in A_PL(Δ[1]/∂Δ[1]), denoted as dt, satisfying ∮_1 ⟨ dt,c_1⟩ = ∮_1 (dt)_c_1 = -1. This notation arises from the identification of A_PL(Δ[1]) with (A_PL)_1=Λ(t,dt) (see the Example in Section 1.4 of <cit.>).

For f∈ (V^1)^♯, it determines a map φ_f: Λ V→ A_PL(Δ[1]/∂Δ[1]) given by φ_f(v) = -⟨ v,f ⟩ dt. Then the adjoint map ⟨φ_f ⟩: Δ[1]/∂Δ[1] →⟨Λ V ⟩ of φ_f realizes to a continuous map |φ_f|:S^1→|Λ V|. Set τ_1(v)=|φ_f|.


Let Λ V be a minimal Sullivan algebra, f∈ (V^1)^♯, and β∈ H^1(Λ V). Then

    ⟨ H^1 (m_|Λ V|)β, hur∘τ_1(f) ⟩ = ⟨ξ_1(β), f ⟩.

In particular H^1 (m_|Λ V|)^♯∘ hur∘τ_1 = ξ_1^♯.



By the construction of τ_1, hur∘τ_1(f) is the homology class of the cycle |φ_f|∘ q: I→ |Λ V| where q: I → S^1 is the quotient map by identifying the two end points. This homology class can be identified with the class [ψ_f]∈ H_1(C_*(⟨Λ V ⟩)), where ψ_f is defined as ⟨φ_f ⟩(c_1) ∈⟨Λ V ⟩. So

    ∮_1 ψ_f(v) = ∮_1 ⟨φ_f,c_1⟩ = -⟨ v,f ⟩∮_1 ⟨ dt,c_1⟩ = ⟨ v,f ⟩.


On the other hand, H^1 (m_|Λ V|)β is identified with H^1 (m_⟨Λ V⟩)β∈ H^1(A_PL(⟨Λ V ⟩)). Let v_β∈ V^1 be the unique representative of β, then identified H^1 (m_⟨Λ V⟩)β can be identified with the cohomology class of ∮_⟨Λ V⟩ m_⟨Λ V⟩ (v_β) ∈ C^1(⟨Λ V ⟩)). Therefore,

    ⟨ H^1 (m_|Λ V|)β, hur∘τ_1(f) ⟩   = ⟨∮_⟨Λ V⟩ m_⟨Λ V⟩ (v_β), ψ_f ⟩
       = ∮_1 ⟨ m_⟨Λ V⟩ (v_β), ψ_f ⟩
       = ∮_1 ψ_f(v_β) 
       = ⟨ v_β,f ⟩.


By definition ξ(β)=v_β, which completes the proof.




 §.§ Holonomy action


Let F→ E→ B be a fibration where F and B are path-connected. There exists an anti-homomorphism from π_1(B) to the homotopy classes of homotopy equivalences of F, which is called the holonomy action. It induces a right π_1(B)-action on H_*(F) and a left π_1(B)-action on H^*(F). Moreover, when F is simply connected, the holonomy action also induces a right π_1(B)-action on π_*(F).

In certain cases, the fibration can be represented by a relative Sullivan algebra. Moreover, the holonomy action also has an algebraic representation.

Let Λ Z be a Sullivan algebra and W be a graded vector space of positive degree. Suppose that the differential on Λ Z can be extended to Λ Z⊗Λ W, and W has a filtration W(k) satisfies dW(k) ⊂Λ Z⊗Λ W(k-1). Then Λ Z⊗Λ W is called a relative Sullivan algebra. More generally, Λ Z can be replaced by any connected CDGA.

If in addition that dW^n ⊂ dZ⊗Λ W^≤ n, where W^≤ n is the subspace of W spanned by the elements with degree ≤ n, then the relative Sullivan algebra is called minimal. Observe that the quotient algebra (Λ Z⊗Λ W)/(Λ^≥ 1 Z⊗Λ W) is a Sullivan algebra, which is denoted as (Λ W,d̅) or simply Λ W. If Λ Z⊗Λ W is a minimal relative Sullivan algebra, then Λ W is a minimal Sullivan algebra.

Let L_0 denote the space of degree 0 whose suspension sL_0=(Z^1)^♯. It has a Lie algebra structure corresponding to the differential on Z^1, so it is called the fundamental Lie algebra of Λ Z. There exists a left L_0-action on H^*(Λ W) defined as follows. For a cocycle Φ∈Λ Z,

    d(1⊗Φ) = ∑ z_i⊗Θ_i+Ψ,

where z_i∈ Z, Θ_i∈Λ W and Ψ∈Λ^≥ 2 Z⊗Λ W. Set x·[Φ] as the cohomology class of ∑_i sx(z_i)Θ_i. Here sx∈(Z^1)^♯ is recognized as a function on the whole Z vanishing on Z^≥ 2. It has been proved that the above action is indeed a Lie algebra action, and x·[Φ] is independent of the choice of representative Φ <cit.>.

As stated in Theorem <ref> (iii), the bijection exp_L∘ s translates the Lie algebra L_0 into the group π_1(Λ Z) when H^1(Λ Z)<∞. This group is isomorphic to π_1(|Λ Z|). The above L_0-action also induces a group action on H^*(Λ W) <cit.>, given by

    (exp_L(sx))·[Φ] = ∑_i=0^∞1/i! x^i·[Φ].

Here x^i·Φ means x acting on Φ i times, i.e. x·(x·(…(x·Φ)…)). By definition if Φ∈ W(k), a representative of x·[Φ] is spanned by Θ_i∈Λ W(k-1). So x^i·[Φ]=0 when i is sufficiently large. Therefore, ∑_i=0^∞1/i! x^i·[Φ] is a finite sum and the L_0-action is well-defined.

Given a minimal Sullivan algebra Λ V, the subalgebra Λ V^≤ n can be considered as a relative Sullivan algebra Λ V^≤ n-1⊗Λ V^n for n≥ 2. In this case H^n(Λ V^n)=V^n can be acted by the fundamental Lie algebra of Λ V^≤ n-1, and by the group π_1(|Λ V^≤ n-1)| if H^1(Λ V^≤ n-1)<∞. Then it induces a right π_1(|Λ V^≤ n-1)|-action on (V^n)^♯ = π_n(|Λ V^n|) = H_n(|Λ V^n|).

On the other hand, we have a fibration |Λ V^n| → |Λ V^≤ n| → |Λ V^≤ n-1| according to Theorem <ref>. The holonomy action also induce right π_1(|Λ V^≤ n-1)|-actions on π_n(|Λ V^n|) and H_n(|Λ V^n|). These actions actually agree.


Let Λ V be a minimal Sullivan algebra, and H^1(Λ V)<∞. Denote the π_1(|Λ V^≤ n-1|)-action on Λ V^n of the relative Sullivan algebra Λ V^≤ n by ·, and denote the holonomy action of the fibration |Λ V^n| → |Λ V^≤ n| → |Λ V^≤ n-1| by ∙. Then for each α∈π_1(|Λ V^≤ n-1)|, β∈π_n(|Λ V^n|) and v∈ V^n,

    ι_n(β∙α)    = ι_n(β)·α, 
    
    hur(β∙α)    = hur(β)∙α, 
    
    [m_|Λ V^n|(α· v)]    = α∙[m_|Λ V^n|(v)].

Moreover, if m_|Λ V^n| is a quasi-isomorphism, the holonomy action on H^*(|Λ V^n|) is nilpotent.



This theorem is a special case of some statements in <cit.>, including Theorem 4.2, Corollary 4.3, and Section 4.6.


We end this section by giving the relationship between fibrations and relative Sullivan algebras.


Suppose that F→ E→ B is a fibration satisfying the following conditions.


  * F, E and B are path connected.

  * The holonomy action of π_1(B) acts on H^*(F) nilpotently.

  * Either H^*(F) or H^*(B) has finite type.

Let Λ Z be a minimal Sullivan model of B, and Λ Z⊗Λ W be a minimal relative Sullivan algebra making the following diagram commute.

    Λ Z [r] [d]^m_B_≃   Λ Z⊗Λ W [r] [d]^m_E   Λ W [d]^m_F
    
    A_PL(B) [r]     A_PL(E) [r]     A_PL(F)


(i) If m_E is a quasi-isomorphism, then so is m_F.

(ii) If m_F is a quasi-isomorphism, then so is m_E.




§ COHOMOLOGIES OF A MINIMAL SULLIVAN ALGEBRA AND ITS REALIZATION

In this section we prove Theorem <ref>.


Suppose that Λ Z⊗Λ W is a minimal relative Sullivan algebra, and Λ Z itself is a minimal Sullivan algebra with H^1(Λ Z)<∞. The left π_1(|Λ Z|)-action on W induces a right action on W^♯. If there exist c_1,…,c_k∈ℚ, α_1,…,α_k∈π_1(|Λ Z|) and some non-zero f∈ W^♯ such that

    ∑_i=1^k c_i(f·α_i)=0,

then ∑_i=1^k c_i=0.


Equip W with the filtration W(r) as follows. Set W(-1)=0 and

    W(r) = { w∈ W | dw∈Λ Z⊗Λ W(r-1) }

for r≥ 0. Let L denote the homotopy Lie algebra of Λ. Then for each x∈ L, x· W(r)⊂ W(r-1).

Now let r_0 is be smallest number such that there exists some w∈ W(r_0) satisfying f(w)=1. For each α_i, α_i· w=(exp_L   x_i)· w ∈ w+W(r_0-1), where x_i=ι_1(α_i)∈ L <cit.>. By hypothesis f is trivial when restricted to W(r_0-1). It follows that (f·α_i)(w)=f[(exp_L   x_i)· w]=f(w). Therefore,

    0 = ∑_i=1^k c_i(f·α_i)(w) = ∑_i=1^k c_i f(w) = ∑_i=1^k c_i.




Let Λ V be a minimal Sullivan algebra. If V is not of finite type, neither is H^*(|Λ V|).



Let n be the smallest number such that V^n=∞. There are two cases.

Case 1. n≥ 2.

In this case V^1<∞ and V^1=V^1(k) for some finite k. So the fundamental Lie algebra L_0, whose suspension sL_0=(V^1)^♯, satisfies L_0=L_0/L_0^k+2 <cit.>. Let G_L=π_1(|Λ V^1|). There exists a bijective correspondence L_0/L_0^j → G_L/G_L^j for all j≥ 1 <cit.>. It follows that G_L= G_L/G_L^j=G_L/G_L^k+2 <cit.>. Therefore, Theorem 7.3 of <cit.> gives the following quasi-isomorphism.

    m_|Λ V^1|:Λ V^1=Λ V^1(k) ≃⟶ A_PL(|Λ V^1|) ≃ A_PL(BG_L)=A_PL(B(G_L/G_L^k+2)).


Let n≥ 2 be the smallest integer such that V^n=∞. We will use induction to show that m_|Λ V^≤ n-1|:Λ V^≤ n-1→ A_PL(|Λ V^≤ n-1|) is a quasi-isomorphism.

Suppose that m_|Λ V^≤ i|: Λ V^≤ i→ A_PL(|Λ V^≤ i|) is quasi-isomorphic for some 1≤ i<n-1. Then we have the following commutative squares for the fibration |Λ V^i+1| → |Λ V^≤ i+1| → |Λ V^≤ i|.


    Λ V^≤ i[r] [d]^m_|Λ V^≤ i|_≃   Λ V^≤ i+1[r] [d]^m_|Λ V^≤ i+1|   Λ V^i+1[d]^m_|Λ V^i+1|
    
    A_PL(|Λ V^≤ i|) [r]     A_PL(|Λ V^≤ i+1|) [r]     A_PL(|Λ V^i+1|)


Λ V^i+1 is concentrated in degree i+1>1 and of finite type, so m_|Λ V^i+1|: Λ V^i+1→ A_PL(|Λ V^i+1|) is a quasi-isomorphism <cit.>. Then by Theorem <ref>, π_1(|Λ V^≤ i|) acts nilpotently in H^*(|Λ V^i+1|). Therefore, the hypotheses of Theorem <ref> (ii) are all satisfied, and m_|Λ V^≤ i+1|: Λ V^≤ i+1→ A_PL(|Λ V^≤ i+1|) is a quasi-isomorphism.

By induction, we have a quasi-isomorphism m_|Λ V^≤ n-1|: Λ V^≤ n-1→ A_PL(|Λ V^≤ n-1|). In particular, H^*(|Λ V^≤ n-1|) has finite type.

Next consider the fibration |Λ V^n| → |Λ V^≤ n| → |Λ V^≤ n-1|. Its Serre spectral sequence satisfies

    E_2^0,n = H^0(|Λ V^≤ n-1|; ℋ^n(|Λ V^n|))

consists of all φ∈ H^n(|Λ V^n|) which are invariant under the π_1(|Λ V^≤ n-1|)-action <cit.>. Here ℋ^n(|Λ V^n|) is a local coefficient system.

As V^1<∞, (V^1)^♯, which is bijective to π_1(|Λ V^≤ n-1|), is a countable set. Then for each f∈ H_n(|Λ V^n|), the subspace spanned by f·π_1(|Λ V^≤ n-1|) is at most countable dimensional. Note that this space is the minimal π_1(|Λ V^≤ n-1|)-invariant subspace containing f. On the other hand, V^n=∞, so the dimension of H_n(|Λ V^n|)=π_n(|Λ V^n|)=(V^n)^♯ is uncountable. Therefore, H_n(|Λ V^n|) can be decomposed as a direct sum ⊕_λ H_λ of infinite (actually uncountable) components, and each component H_λ is a minimal π_1(|Λ V^≤ n-1|)-invariant subspace spanned by f_λ·π_1(|Λ V^≤ n-1|) for some f_λ∈ H_n(|Λ V^n|).

For each H_λ, set φ_λ∈ H^n(|Λ V^n|) = [H_n(|Λ V^n|)]^♯ such that φ_λ(f_λ·α)=1 for all α∈π_1(|Λ V^≤ n-1|) and φ_λ vanishes on the other components. It is well-defined because if ∑_i=1^k c_i(f_λ·α_i)=0 with c_1,…,c_k∈ℚ and α_1,…,α_k∈π_1(|Λ V^≤ n-1|), then by Lemma <ref>,

    ∑_i=1^k c_i φ_λ(f_λ·α_i) = ∑_i=1^k c_i = 0.

By definition, φ_λ is π_1(|Λ V^≤ n-1|)-invariant and all φ_λ are linearly independent. Thus, E_2^0,n = ∞.

In the Serre spectral sequence, all d_r: E_r^-r,n+r-1→ E_r^0,n is trivial, and the only non-trivial d_r   (r≥ 2) from E_r^0,n is d_n+1:E_n+1^0,n→ E_n+1^n+1,0. Since E_2^n+1,0 = H^n+1(|Λ V^≤ n-1|; ℋ^0(|Λ V^n|)) = H^n+1(|Λ V^≤ n-1|; ℚ) is finite dimensional, so is E_n+1^n+1,0 (actually E_n+1^n+1,0=E_2^n+1,0). Then the kernel of d_n+1:E_n+1^0,n→ E_n+1^n+1,0, which is E_n+2^0,n, is infinite dimensional. It follows that E_∞^0,n=∞ and H^n(|Λ V^≤ n|)=∞.

Finally consider the fibration |Λ V^>n| → |Λ V| → |Λ V^≤ n|. As shown above, E_2^n,0 = H^n(|Λ V^≤ n|; ℚ) of its Serre spectral sequence is infinite dimensional. Since |Λ V^>n| is n-connected, all d_r   (r≥ 2) whose domain or codomain is E_r^n,0 are trivial. Therefore, E_∞^n,0 = E_2^n,0 is infinite dimensional. Then so is H^n(|Λ V|).

Case 2. n=1.

When V^1=∞, the dimension of its dual (V^1)^♯ is uncountable and so is the cardinality. By Theorem <ref>, π_1(|Λ V^1|) is a uncountable set, and |Λ V^1| is homotopy equivalent to Bπ_1(|Λ V^1|) where π_1(|Λ V^1|) is equipped with the discrete topology.

On the other hand, as mentioned in <cit.>, H^2(Bπ_1(|Λ V^1|)) is H^2(π_1(|Λ V^1|); ℚ), the second group cohomology of π_1(|Λ V^1|). Moreover, Hu <cit.> (and independently Heller <cit.>) showed that H^2(π_1(|Λ V^1|); ℚ) is an extension of π_1(|Λ V^1|) by ℚ. Therefore, H^2(π_1(|Λ V^1|); ℚ) is a uncountable set, then its dimension is also uncountable.

Consider the classifying space fibration |Λ V^≥ 2|→|Λ V|→|Λ V^1|. In its Serre spectral sequence, E_2^2,0=H^2(|Λ V^1|; ℚ)=H^2(π_1(|Λ V^1|); ℚ) is of uncountable dimensional. Since |Λ V^≥ 2| is simply connected, E_2^0,1=0. It follows that E_∞^2,0=E_3^2,0=E_2^2,0. So H^2(|Λ V|) must be uncountable.



When |Λ V| is a Sullivan space, its cohomology is of finite type. Theorem <ref> implies that that V^1<∞. This also follows from the proof of Theorem 7.8 of <cit.>, which shows that Λ V^1 is a Sullivan model of |Λ V^1| and V^1 is at most countable. So if V^1=∞, H^2(|Λ V^1|)=H^2(Λ V^1) is at most countable. But Theorem <ref> shows that H^2(|Λ V^1|) is uncountable in this case, which is a contradiction.



Let Λ V be a minimal Sullivan algebra. If there exists a quasi-isomorphism Λ V ≃⟶ A_PL(|Λ V|), then H^*(Λ V)=H^*(|Λ V|) is of finite type.


Suppose that such quasi-isomorphism exists and H^*(Λ V) is not of finite type. Then there exists a smallest n such that H^n(Λ V) =  H^n(|Λ V|) = ∞. Since H^n(|Λ V|) = [H_n(|Λ V|)]^♯, its dimension must be uncountable. We denote this dimension by the cardinal number ℵ.

Claim. n is also the smallest number such that V^n is uncountable. Moreover, the image of ξ_n: H^n(Λ V) → V^n is of ℵ-dimensional.

Assume that i<n is the smallest number such that V^i is uncountable. Then (Λ V^≤ i-1)^i+1 is at most countable. Hence, the kernel of d:V^i→(Λ V^≤ i-1)^i+1 is of uncountable dimensional. So is H^i(Λ V) because Λ V is minimal. This contradicts the hypothesis. Therefore, V^≤ n-1 must be of at most countable dimensional, and so are the cycles in (Λ V^≤ n-1)^n+1. Since H^n(Λ V)=ℵ, which is uncountable, there must be an ℵ-dimensional subspace of V^n containing all cycles. In particular, this subspace is ξ_n, which proves the claim.

As the restriction of ξ_n^♯ to (ξ_n)^♯ is injective, the dimension of ξ_n^♯ is at least (ξ_n)^♯, which is strictly greater than ℵ.

When n≥ 2, by Theorem <ref>, (ξ_n^♯∘ι_n)=ξ_n^♯, so its degree is strictly greater than ℵ. On the other hand, H^n(|Λ V|) is an infinite dimensional dual space of H_n(|Λ V|). Thus, H_n(|Λ V|) is strict smaller than ℵ, and so is (H^n(m_|Λ V|)^♯∘ hur). But ξ_n^♯∘ι_n = H^n(m_|Λ V|)^♯∘ hur, which is a contradiction.

When n=1, by Lemma <ref>, the dimension of (H^1 (m_|Λ V|)^♯∘ hur∘τ_1) = ξ_1^♯ is strictly greater than ℵ. But (hur∘τ_1) ⊂ H_1(|Λ V|). It follows that

    (H^1 (m_|Λ V|)^♯∘ hur∘τ_1) ≤ (hur∘τ_1) ≤ H_1(|Λ V|) < ℵ,

which is a contradiction.


Combining Theorem <ref> and Theorem <ref> together, we have proved Theorem <ref>.



§ HOMOTOPY GROUPS OF A TOPOLOGICAL SPACE AND ITS MINIMAL SULLIVAN MODEL


We may also examine the relationship between the homotopy group of a topological space X and that of its minimal Sullivan model Λ V. Suppose that H^1(X)<∞ and that the cohomology H^*(X) of the universal cover of X, is of finite type. The spaces that induce isomorphisms π_n(X) ≅⟶π_n(Λ V) for all n≥ 2 are defined as Sullivan spaces. Without the hypotheses on cohomologies, there are other spaces which share the same higher homotopy groups with their minimal Sullivan models.


Let X=⋁_i=1^∞ S^1_i be an infinite wedge of circles. Then

    H^n(X)=
    
    1,    if  n=0, 
    ∞,    if  n=1, 
    
    0,    if  n≥ 2.

So X is not a Sullivan space. On the other hand, by Lemma 7.7 of <cit.>, the minimal Sullivan model of X is Λ V^1 where V^1 is concentrated in degree 1. Then both π_n(X) and π_n(Λ V) are trivial for n≥ 2.


Given that the fundamental Lie algebra of a minimal Sullivan algebra Λ V is related to π_1(|Λ V|), we may inquire about the conditions under which a similar relationship exists between exists between π_1(X) of a space X and the fundamental Lie algebra of its minimal Sullivan model. Using Malcev ℚ-completions, we can formulate a criterion that applies to spaces encompassing Sullivan spaces, while excluding those X with H^1(X)=∞.

A group Γ is called Malcev ℚ-complete if each Γ^n/Γ^n+1 is a ℚ-vector space and Γ = Γ^n/Γ^n+1. The Malcev ℚ-completion of a group G is a homomorphism ϕ:G→Γ where Γ is Malcev ℚ-complete and ϕ induces isomorphisms

    (G^n/G^n+1) ⊗_ℤℚ≅⟶Γ^n/Γ^n+1.


When X is a Sullivan space, H^1(X)<∞. Apply Corollary 7.4 and Corollary 2.4 of <cit.> we have


Let Λ V be a minimal Sullivan model of a topological space X. The induced morphism π_1(X)→π_1(Λ V) is a Malcev ℚ-completion. Moreover, Let L_0 denote the fundamental Lie algebra of Λ V.

    (π_1(X)^n/π_1(X)^n+1)⊗_ℤℚ≅ L_0^n/L_0^n+1

for all n as isomorphisms of abelian groups.


On the other hand, the isomorphisms between (π_1(X)^n/π_1(X)^n+1)⊗_ℤℚ and L_0^n/L_0^n+1 do not exist if H^1(X)=∞.


Let Λ V be a minimal Sullivan model of a topological space X, and L_0 be the fundamental Lie algebra of Λ V. If H^1(X)=∞, then H_1(X) = (π_1(X)/[π_1(X),π_1(X)])⊗_ℤℚ and L_0/[L_0,L_0] are not isomorphic.


Equip V with the filtration V(r) by setting V(-1)=0 and

    V(r) = { v∈ V | dv∈Λ V(r-1) }.

Then [L_0,L_0] vanishes on V(0)=H^1(Λ V) <cit.>. As sL_0=(V^1)^♯, this induces a surjective morphism L_0/[L_0,L_0] → [H^1(Λ V)]^♯. Therefore, the cardinal number of L_0/[L_0,L_0] is greater than or equal to the cardinal number of [H^1(Λ V)]^♯=[H^1(X)]^♯, which is strict larger than H_1(X) by the hypothesis that H^1(X)=∞. Therefore, there is no isomorphisms between H_1(X) and L_0/[L_0,L_0].


It is unknown whether only Sullivan spaces induce isomorphisms on the higher homotopy groups and a Malcev ℚ-completion on the fundamental group to their minimal Sullivan algebras. Nevertheless, applying the idea from Theorem <ref>, we can prove that the cohomologies of such spaces must be of finite type. This is stated in Proposition <ref>.


Let f:Λ V ≃⟶ A_PL(X) be a minimal Sullivan model of a topological space X, and H^1(X) =  H^1(Λ V) < ∞. If for all n≥ 2, the induced map π_n(f):π_n(X)⊗ℚ→π_n(Λ V) is an isomorphism, then H^*(X) is of finite type.


Assume that H^*(X) is not of finite type. Then there exists a smallest n such that H^n(Λ V) =  H^n(X)=∞. This dimension must be uncountable since H^n(X)=[H_n(X)]^♯ and denote it by ℵ. As shown in the claim of Theorem <ref>, n is also the smallest number such that V^1 uncountable, and the image of ξ_n: H^n(Λ V) → V^n is of ℵ-dimensional. Then the restriction of ξ_n^♯ to (ξ_n)^♯ is injective, and the cardinality number of ξ_n^♯ is strict larger than ξ_n=ℵ.

f induces a map |f|:X → |Λ V| and satisfies ι_n∘π_n(|f|) = π_n(f) for n≥ 2 <cit.>. Then π_n(|f|): π_n(X)
→π_n(|Λ V|) is an isomorphism. By the naturality of hur and (<ref>), we have the following commutative diagram.

    @C=60ptπ_n(X) [r]^π_n(|f|)_≅[d]^hur   π_n(|Λ V|) [r]^ι_n_≅[d]^hur    (V^n)^♯[d]^ξ_n^♯
    
    H_n(X) [r]^H_n(|f|)    H_n(|Λ V|) [r]^H^n (m_|Λ V|)^♯    [H^n(Λ V)]^♯.


So the image of ξ_n^♯∘ι_n∘π_n(|f|) is ξ_n^♯, and the cardinality of its dimension is strictly larger than ℵ. But on the other hand,

    ξ_n^♯∘ι_n∘π_n(|f|)    =  H^n (m_|Λ V|)^♯∘ H_n(|Λ V|) ∘ hur 
       ≤ H^n (m_|Λ V|)^♯∘ H_n(|Λ V|) 
       ≤ H_n(X) 
        < ℵ,

which is a contradiction.


It is still possible that V^n=∞ under the hypothesis of Proposition <ref>. Actually, there exists a minimal Sullivan model Λ V of a Sullivan space that satisfy V^1=∞.


Let X=S^1∨ S^1, then X is homotopy equivalent to B(ℤ♯ℤ), where ℤ♯ℤ is the free product of two ℤ. Since Bℤ≃ S^1 is a Sullivan space, so is X <cit.>.

Let Λ V be the minimal Sullivan model of X, then V^1=∞. Indeed, let v_1,v_2∈ V^1 be the two cocycles whose cohomology classes span H^1(Λ V)=H^1(X). Since [v_1v_2] ∈ H^2(Λ V)=H^2(X)=0, there must be some v_3∈ V^1 such that dv_3=v_1v_2. v_1v_3 is also a cocycle in Λ V, so we need some v_4∈ V^1 such that dv_4=v_1v_3. Continue this process, we can find an infinite sequence {v_n} such that dv_n∈ v_1v_n-1.

By Theorem <ref>, |Λ V| is not a Sullivan space. So it is possible that a Sullivan space taking minimal Sullivan model then realizing is no longer a Sullivan space.


On the other hand, there are also non-Sullivan spaces whose minimal Sullivan model can be realized to Sullivan spaces. This gives a negative answer of the second part of Question <ref>.


Let X=ℝP^2. Then H^*(X)=ℚ. So the minimal Sullivan model of X is ℚ trivially, and its realization is a point, which is a Sullivan space. However, π_2(X)=ℤ≠ 0=π_2(ℚ). Thus, X is not a Sullivan space. 


plain


1 cm
Beijing Institute of Mathematical Sciences and Applications, Huairou District, Beijing, China 101408

Email address: jiaweizhou@bimsa.cn


 Paper Title: Hamiltonian elements in algebraic K-theory.



 
Supported by CONAHCYT 
research grant CF-2023-I
yasha.savelyev@gmail.com
University of Colima and IAS

§ ABSTRACT   
Recall that topological complex K-theory associates to
an isomorphism class of a complex vector bundle E over
a space X an element of the complex K-theory group of X. Or
from algebraic K-theory perspective, one assigns
a homotopy class [X → K (𝒦)], where
𝒦 is the ring of compact operators on the Hilbert space. We show
that there is an analogous story for algebraic K-theory of
a general commutative ring k, replacing complex vector
bundles by certain Hamiltonian fiber bundles. 




The construction actually first assigns elements in
a certain categorified algebraic K-theory, analogous to
Toën's secondary K-theory of k. And there is a natural
map from this categorified algebraic K-theory to the
classical variant. 























The construction in
 <cit.> and the setup
here produce a natural group map g: ℤ→ K _4 ^Cat, ℤ _2
(ℤ), with the latter group, the group of maps S ^4→ K ^Cat, ℤ _2 (ℤ), i.e. the
fourth “categorified algebraic K-theory group” of
ℤ. Based on a geometric analysis calculation in
 <cit.> we show that
g is an injection provided a certain independent
conjecture of algebraic topological nature holds.
Rognes has shown that K _4 (ℤ )=0, with the
latter the classical fourth algebraic K-theory group of
ℤ. We may thus (conditionally) conclude that K ^Cat,
ℤ _2 (ℤ ) has finer arithmetic
information than K (ℤ ).

This naturally leads us to formulate a generalization of the homological mirror symmetry phenomenon to the algebraic K-theory context.


































Hamiltonian elements in algebraic K-theory
    Yasha Savelyev
    
==========================================



§ INTRODUCTION

Our main goal is to tie up a pair of at a glance
unrelated threads in geometry and in algebraic
topology/algebra. The first thread is the existence of
a homotopy coherent action, in the sense of 
Porter-Stasheff <cit.>, of the group of
Hamiltonian symplectomorphisms on the Fukaya category,
particularly as developed in the author's
 <cit.> and
 <cit.>, see also
Oh-Tanaka <cit.> for a different
perspective. The second thread is ideas of Toën
 <cit.> on
secondary K-theory of commutative rings k, roughly speaking replacing
k-modules by dg-categories over k, and using Waldhausen
K-theory.  This culminates in

a connection of symplectic geometry and algebraic
K-theory, and in particular we get a certain geometric
insight into what we call categorified algebraic K-theory
of ℤ. We are then naturally lead to speculate on generalizations
of the homological mirror symmetry phenomenon to algebraic
K-theory context. We now describe some of the ingredients.

Although in the main examples here the principal actors are
compact groups and PU (n) in particular, the framework is
most naturally understood from the perspective of the group
of Hamiltonian symplectomorphisms of some symplectic
manifolds, so we quickly introduce this.
Let (M, ω ) be a symplectic manifold, throughout
assumed to be compact, with ω a closed non-degenerated 2-form on M. The group of
diffeomorphisms ϕ of M,
satisfying ϕ ^*ω = ω, is called the
symplectomorphism group of (M, ω ). When M is
simply connected, this
is also the group of Hamiltonian symplectomorphisms that we
denote by ℋ  = Ham (M,  ω). In general,
a symplectomorphism is Hamiltonian if it is the time one map
of an isotopy generated by a family of vector fields {X
_t}, t ∈ [0,1], satisfying: 
    ω (X _t, ·) = dH _t
    (·),
 for
some family of smooth functions {H _t} on M.

The group ℋ is one of the
more enigmatic objects of mathematics. It is always (unless
M=pt) an
infinite dimensional Fréchet Lie group with its C
^∞ topology. But it has a natural, bi-invariant
Finsler metric called the Hofer metric. And so in some ways
it behaves like a compact group. It's topological and algebraic
structure is tied to the main problems of Hamiltonian dynamics, see for instance Polterovich
 <cit.>.

Much of the algebraic/topological structure of ℋ is reflected in the
properties of Hamiltonian fibrations, which are the main
geometric ingredients in what follows. A Hamiltonian fibration M ↪ P → X, is
a fiber bundle over X, with fiber a symplectic
manifold (M,ω), whose
structure group is contained in Ham (M,
ω).  For example, take (M, ω
) = (ℂℙ ^n, ω _FS) where ω _FS
is the Fubini-Study symplectic form, and take the structure group G= PU (n+1) ⊂Ham (ℂℙ ^n,
ω _FS).


The other ingredient is algebraic K-theory, which is a construction associating
an infinite loop space K (k) [In this paper, the term infinite loop space is meant to be synonymous with the term
Ω-spectrum (not necessarily connective), i.e. a sequence of spaces {E _n} _n ∈ℕ with E _n≃ΩE _n+1.  Then
K (k) is the notation for the 0-space E _0 of the corresponding spectrum.]  to any ring k, and so abelian groups:

    K _m (k) = π _m  (K (k)),    m ∈ℕ,

called K-theory groups of k.
This is a powerful tool in
algebraic topology and ring theory, with applications in number theory, geometry and
geometric topology.

Taking k to be the ring  𝒦 of compact operators on a separable complex Hilbert space,  the
algebraic K-theory space K (𝒦) [Rather one must take a certain non-connective variant of the algebraic K
theory construction.] is
identified with the topological K-theory space KU = BU
×ℤ, see Karoubi  <cit.>,
Suslin-Wodzicki  <cit.>.   

In particular, a complex vector bundle E over a space X
determines a homotopy class [f _E: X →
K (𝒦)], which we might call a geometric
element of the corresponding abelian group.
One goal is to extend this to general k.




There are two versions of the construction, corresponding to
the pair of geometric cases of monotone and Calabi-Yau
symplectic manifolds. In the monotone case, for each
commutative ring k, we construct an 
infinite loop space K ^Cat, ℤ _2 (k).  This is
obtained by Waldhausen K-theory construction on a certain
category, whose objects are pretriangulated A
_∞ categories over k, with a certain algebraic
finiteness condition that we call finite monotone type, Definition <ref>. This
categorified algebraic K-theory of k is a mostly minor
variation of the secondary K-theory of k introduced by Toën.
K ^Cat, ℤ _2 (k) admits a natural map to the
2-periodic variant K ^ℤ _2 (k) of algebraic K-theory of k.
Technically, K ^ℤ _2 (k) is the algebraic
K-theory built using the exact category of 2-periodic complexes of k-modules. 
 
It was
pointed out to me by Bertrand Toën that for k a regular
ring, for example k= ℤ, K ^ℤ _2 _m (k) ≃ K _m (k) ⊕
K _m-1 (k). This can be proved with A _1 homotopy
theory. 



To obtain “geometric elements”, we start with a Hamiltonian
fibration M ↪ P → X, 
with fiber a finite monotone type 
symplectic manifold. We show that for each k, the data
of the isomorphism class of P, determines a homotopy class [f _P: X → K ^Cat, ℤ _2 (k)]. 
As one consequence of this, there are group homomorphisms:

    π _m (BPU (n)) K ^Cat, ℤ
    _2 _m (ℤ) → K _m ^ℤ _2
    (ℤ ),   ∀ m,n ∈ℕ,

with the first map g geometrically constructed. We will
discuss some partial results towards computation of g. In particular, when m=4, and n=2 g is injective conditionally on a certain independent algebraic topological conjecture. 





















A similar construction also exists for Hamiltonian
fibrations with finite Calabi-Yau type fiber. In this case
we must restrict k, necessitated by the condition that the
Fukaya category is suitably well defined with k
coefficients. Usually one takes k to be a certain
Novikov field Λ over ℂ. In particular, 
a Hamiltonian fibration with such a finite Calabi-Yau type fiber
over S ^n, induces an element of the standard algebraic
K-theory group K _n (Λ).














§ PRELIMINARIES
 


Let k be a commutative ring. Denote by A _∞Cat _k the category of strictly unital, ℤ graded A _∞ categories with
morphisms A _∞ functors. We set A
_∞Cat _k ^ℤ _n to be like
A _∞Cat _k  but with chain complexes
ℤ _n graded instead of ℤ graded.
This means that all structure maps, functors, etc., are with 
respect the ℤ _n grading.
We will assume homological grading conventions on all chain
complexes. Various subscripts k may be omitted.

There is an endofunctor 
    Perf: A _∞Cat   →A _∞Cat,
 s.t. Perf (B) is
idempotent complete and pretriangulated
for each B.  One reference for this material is
<cit.>. We may denote
by C the category Perf (C).

Let Cat _k denote the category of small k-linear
categories.
For C ∈𝒜, 𝒜 ^ℤ _n, we
set hC ∈ Cat _k to be the category with the same
objects and with hom _hC (a,b) = H _0 (hom _C (a,b)). 

Given a morphism f: A → B in 𝒜,
𝒜 ^ℤ _n, we denote by f _*: hA → hB the induced morphism in Cat _k.
We say that that







f is quasi fully-faithful if f _* is
fully-faithful, and f is a quasi equivalence if f
_* is an equivalence.  We will say that f is
a Morita equivalence  if Perf (f): Perf (A) → Perf
(B) is a quasi equivalence.










§ BUILDING A  WALDHAUSEN CATEGORY FROM
A _∞ CATEGORIES 
 
To build an algebraic K-theory from the category of A _∞
categories, we need
some analogue of “exact sequences” of A _∞
categories.  
For dg categories there are already very good answers to
this, and this results in a definition of “secondary
K-theory”, Toën  <cit.>.
The main difficulty in the case of A _∞ categories
(with non-strict functors) is the existence of suitable colimits. 
In fact, strict colimits of diagrams in A _∞Cat do
not exist. This is not surprising because the “correct”
(in some geometric and algebraic sense, for instance
Keller <cit.>)
notion of functors of A _∞ categories, preserves
A _∞ structures only in homotopy coherent sense,
the latter is also discussed in Porter-Stasheff
 <cit.>.  
Despite this, there is a suitable theory of homotopy colimits, 
which will be sufficient for our purposes.















We also need to put some algebraic finiteness conditions on our
A _∞ categories, otherwise any K-theory
we construct will be trivial due to the Eilenberg swindle,
the same reason that K-theory of infinite dimensional
vector bundles is trivial.
One natural idea is to require our A _∞
categories be smooth and proper or saturated in the sense of
Kontsevich-Soibelman  <cit.> , as this is a dualizability
condition, see  <cit.>. We will work with
a somewhat less restrictive notion, which naturally generalizes
to the ℤ _n graded case,  and is not too hard
to check for geometric examples.







Correspondingly, we will get
a pair of categories 𝒜 _k and
𝒜 ^ℤ _n _k, with the latter
corresponding to the n-periodic case.

Let 𝒜 _k⊂A _∞Cat _k be the
full-subcategory  whose objects are pretriangulated,
A _∞ categories B s.t.  the following holds:
 
B is Morita equivalent to an A _∞ algebra, whose
underlying chain complex is perfect, and whose Hochschild
homology complex is perfect.

  Let C be smooth and
proper dg
category, then it satisfies Condition
<ref>, Toën-Vaquié
 <cit.>. 







The category 𝒜
^ℤ _n_k⊂A _∞Cat ^ℤ
_n _k is defined to
be  the full-subcategory whose objects are pretriangulated,
A _∞ categories B s.t. the following holds:
 
B is Morita equivalent to an A _∞ algebra, whose
underlying chain complex B _∙, in each degree is
projective and finitely generated over k.
The same holds for the Hochschild homology complex of B.






















It is shown in  <cit.> that 𝒜,
𝒜 ^ℤ _n have
certain homotopy colimits,  in particular for the pushout diagram:

    ∙   ∙ [l]  [r]      ∙.

 
Strictly speaking only distinguished homotopy cocones are
constructed in  <cit.>, although they are expected to be homotopy colimits.
Moreover, these distinguished cocones are suitably
functorial, that is maps of diagrams induces natural maps of
these distinguished cocones. The structure of distinguished
functorial homotopy cocones is enough for the
S-construction underlying the infinite loop space
(<ref>). This is a very minor generalization
of the standard Waldhausen S-construction. To simplify the
discussion, we will just call them homotopy colimits.











To define a Waldhausen structure on 𝒜,
𝒜 ^ℤ _n  we 
need to specify a class of morphisms 𝒞 called cofibrations and
a class of morphisms 𝒲 called weak equivalences. We set these
to be quasi fully-faithful maps and quasi isomorphisms
respectively.







  With respect to 𝒞,
𝒲 as above,  both 𝒜 and
𝒜 ^ℤ _n are Waldhausen categories.
More specifically, we have: 





















  * 𝒜, 𝒜 ^ℤ _n have a zero object. 
	
  * The canonical morphism 0 → A is a cofibration for
	all A ∈𝒜, 𝒜 ^ℤ
	_n. 
	
  * All isomorphisms are weak equivalences, and all weak
	equivalences are cofibrations. 
	
  * Weak equivalences are closed under composition.
	
	
  * All diagrams of the form (<ref>), where the right pointing arrow is a cofibration, have homotopy colimits.
	
  * Given a homotopy push-out diagram  
	
    A [r, "f"]  [d, ""]      B  [d,""] 
    
    	C  [r, "g"]        D,

if f is a cofibration then so is g.	

  * Given a diagram: 

    A [r, "f"]  [d, ""]      B  [d,""]  [r,"e"]
    	     C  [d, ""] 
    
    	D  [r, "g"]        E  [r,"e'"]      F,

where e,e' are cofibrations, and vertical arrows are weak
equivalences, the induced map of homotopy push-outs is a weak
equivalence.


 The zero object is defined to be the
zero category (the category with unique object and morphism).
Property <ref> then holds vacuously. Property
<ref> and <ref> hold by
definitions of 𝒞, 𝒲.
To check property <ref> we can work directly with the
construction in  <cit.>.

Specifically, if 
    F: J → A _∞Cat
 is 
the functor representing the diagram 

    A [r, "f"]  [d, ""]      B 
    
    	C        ,

then first we form a type of Grothendieck construction
Groth  F, which is an A
_∞ category, which we briefly describe.  There are
fully-faithful embeddings A,B,C →Groth
F, with images denoted by just A,B,C, s.t. 
    Groth  F =  A ⊔ B ⊔
    C.
 There is an identification 
    hom _Groth  F
    (a, b) = hom _B (f (a), b),
 for a ∈ A, b ∈ B.
And a similar identification for hom _Groth  F
(a, c). We call morphisms corresponding to the identity
morphisms in this identification, adjacent identities.

These are all the non-zero hom sets,  i.e. there is
a non-zero morphism from o _1∈
F (i _1) to   o _2∈ F (i _2) only if there is
a morphism from i _1 to  i _2 in J.

The homotopy colimit D is then defined to be the
localization (an A _∞ analogue of the usual
localization, as defined in  <cit.>) of Groth  F at the set of adjacent identity morphisms.
There is then a homotopy commutative diagram:

    A [r, "f"]  [d, ""]      B  [d,""] 
    
    	C  [r, "g"]        D.

By construction,  the induced commutative diagram:

    hA [r, "f _*"]  [d, ""]      hB  [d,""] 
    
    	hC  [r, "g _*"]        hD,

is a push out diagram in Cat_k. 
As f _* is
fully-faithful, g _* is fully-faithful by 
Stanculescu <cit.>.

The final property is verified in  <cit.>. 
 
















§ CATEGORIFIED ALGEBRAIC K-THEORY
 
We define K ^Cat (k) to be the infinite loop space given
by the Waldhausen S-construction  <cit.>, on the
Waldhausen category (𝒜 _k, 𝒲, 𝒞).

More specifically,

    K ^Cat (k) = Ω |w S _∙ (𝒜 _k)|,

where (very briefly)

	
  * S _∙ (𝒜 _k) is a simplicial
category. For each d, objects of S _∙ (𝒜 _k) (d) are certain length d sequences of
cofibrations in 𝒜 _k (with an additional
structure.)

  * wS _∙ (𝒜 _k) is the
simplicial category with wS _∙ (𝒜 _k)
(d) = w (S _∙ (𝒜 _k) (d)), where the
right-hand side is the category of weak equivalences of the
category S _∙ (𝒜 _k) (d).

  * |·| is the geometric realization of
a simplicial category, with the latter understood as a bi-simplicial set, taking nerve of each w S _∙ (𝒜 _k) (d).

Likewise, K ^Cat, ℤ _n (k) is defined
by the S-construction on (𝒜 ^ℤ _n
_k, 𝒲, 𝒞). 



 §.§ Some computation, and relation with Toën's secondary K-theory
 
Toën  <cit.>, constructs what he calls the secondary
K-theory spectrum K ^(2) (k) of k.
Our construction above is an essentially minor modification
of this construction. 
More specifically, Toën works with smooth and proper ℤ-graded, pretriangulated dg categories.  
As mentioned,
for dg categories the smooth and proper condition in particular implies  the finiteness Condition (<ref>), but can only be expected
to hold for Fukaya categories of compact Calabi-Yau manifolds.  
Nevertheless, the spectrum K ^(2) (k) should be closely
related to the spectra K ^Cat, ℤ _n (k) and
K ^Cat (k).


















Define: 

    K ^Cat, ℤ ^n _m (k) = π _m (K ^Cat,
    ℤ ^n (k)).

And define:

    K ^Cat _m (k) = π _m (K ^Cat (k)).


To partially demystify this, we note that K ^Cat _0 (k)
has the following description (similarly with K ^Cat,
ℤ _n _0 (k)). By a cofiber sequence of
A _∞ functors we mean a diagram in 𝒜
_k:

    A  B → C,

with f quasi fully-faithful, which completes to
a homotopy pushout diagram in 𝒜 _k. 

    A [r, "f"]  [d, ""]      B  [d,""] 
    
    0  [r, ""]        C.

Then K ^Cat _0 (k) is the quotient of the free abelian
group, generated by the set of isomorphism classes of objects in
𝒜 _k,
by the
subgroup generated by

    { [B] - [A] - [C]  | there is a cofiber sequence
    (<ref>). }


Let us also recall the definition of K _0 (k). This is
the quotient of the free abelian
group, generated by the set of isomorphism classes of
projective finitely generated k modules, by  the
subgroup generated by

    { [B] - [A] - [C]  | exists an exact sequence 0
    → B → A → C → 0. }



More generally if ℰ is an exact category,  
one defines K _0 (ℰ )  to be the quotient of
the free abelian group, generated by the set of isomorphism
classes in ℰ, by the
subgroup generated by

    { [B] - [A] - [C]  | exists an exact sequence B → A → C. }

It is well known that K
_0 (D ^perf (Mod _k)) is isomorphic to K _0 (k),
(together with higher K-groups), where 
Mod _k denotes the category of projective finitely
generated k modules and 
D ^perf (Mod _k)
denotes the derived category of perfect complexes.
  
Denote by Hoch _∙ (C) the Hochschild homology
complex of C ∈𝒜 _k. There is a natural map 

    K ^Cat _0 (k) → K _0
    (D ^perf(Mod _k)) ≃ K _0 (k)  
        [C] ↦ [Hoch _∙ (C)],

and it is non-trivial, where [· ] denote the classes
in the associated Grothendieck K-groups. 

Toën's  proof of this proceeds as follows. One shows that images of cofiber
sequences of A _∞ functors, by the functor C
↦ Hoch
_∙ (C), give exact sequences in D ^perf (Mod
_k),
of the associated Hochschild complexes. This property of the
Hochschild functor is discussed
for example in Keller
 <cit.>.
Other interesting computations in degree 0 have been made
by Tabuada <cit.>.


There is an analogous map in the case of K ^Cat,
ℤ _n _0 (k) but in general we no longer have that
K _0 (k) ≃ K _0 (D _n (Mod _k)), where D
_n (Mod _k) denotes the n-periodic derived category, as
there are certain corrections, see
Saito <cit.>. We do have:
 
For n even, the natural homomorphism:

    ψ: K _0 (D _n (Mod _k)) → K _0 (k) 
       ψ ([V]) = ∑_i = 0^i=n	(-1) ^i[H _i (V)],

is an isomorphism.

 
The composition map

    K ^Cat, ℤ _2 _0 (ℤ)
     K _0 (D
    _2 (Mod _ℤ))  K _0
    (ℤ),

where ϕ is the map induced by [C] ↦ [Hoch _∙
(C)],
is non-trivial.



Take C to be a ℤ _2 graded A
_∞ algebra over ℤ such that
HH_0 (C) = ℤ, HH _1 (C) = 0, HH_2 (C)
= ℤ, where HH _∙ (C) is the homology of
Hoch _∙ (C). It is not hard to construct an example. As a geometric example take C= FH _∙ (L _0, L _0) where the
latter is the Floer homology of the equator L _0 in S
^2, then the Hochschild homology HH _∙ (C) is 
isomorphic to the singular homology H _∙ (S ^2, ℤ
),  <cit.>. 

So we get 
    ψ ([Hoch _∙ (C)]) = 2 [ℤ] ≠ 0 ∈
    K _0 (ℤ) ≃ℤ.


One natural question is how to produce interesting elements
in K ^Cat _m (k) for higher m. We will show
that there are somewhat exotic elements coming by way of
symplectic geometry. 







 




§ |𝒜 _K|, |𝒜 _K
^ℤ_N| AS REPRESENTING SPACES OF FUKAYA FIELD
THEORIES.
 
We briefly review simplicial sets. 
We denote by  the simplex category:

	
  * The set of objects of Δ^ is ℕ.
	
  * _
 (n, m) is the set of non-decreasing maps [n] → [m], where [n]= {0,1, …, n}, with its natural order.
 
A simplicial set X is a functor                                 

    X:  ^op→ Set.
 
The set X (n) is called the set of n-simplices of X.
^d will denote a particular
simplicial set: the standard representable
d-simplex, with 
    ^d (n) = hom _ (n, d).
 
  For X a simplicial set, Δ (X) will denote the simplex category of X. This is the
category s.t.: 

	
  * The set of objects Δ (X)  is the set
of simplicial maps:

    Σ: Δ ^d→ X,    d ≥
       0.

	 
  * Morphisms f: Σ _1→Σ
      _2 are commutative diagrams in s-Set:

    ^d[r, ""]  [dr,
       "Σ _1"]     ^n
       [d,"Σ _2"] 
        X.



The main construction of
 <cit.>  has as input 
a Hamiltonian fibration M ↪ P → X [Since the
construction in  <cit.>
is carried out on the universal level, we do not need any
smoothness assumptions on P here.], with
(M, ω ) a compact monotone symplectic manifold, e.g.
(ℂℙ ^n, ω _FS). 
And the output is a functor

    F _P: Δ^ (X _∙) →A _∞Cat
    ^ℤ _2 _k, [In  <cit.> we
    work with k=ℚ but this is only needed for
    inverting quasi equivalences. As we don't need to do this
    here, the construction works perfectly well over
    any commutative ring.]

well defined up to concordance of functors as defined below.
We may understand F _P as giving a kind of topological
field theory - the Fukaya field theory  associated to P. 

Let Y be a simplicial set and C any category. Let F_i:
Δ^ (Y) → C, i=0,1	be functors.
A concordance of F _0 to F _1  is
a functor F: Δ^ (Y × I) → C s.t.
F| _Δ^ (Y ×{0}) = F _0 and F| _Δ^
(Y ×{1}) = F _1.


Then we obtain a functor:

    (F  _P = Perf ∘ F _P):
    Δ(X _∙)   →A _∞Cat
    ^ℤ _2,

again well defined up to concordance of functors, with image
now consisting of pretriangulated idempotent complete A
_∞ categories.


We then produce an ∞-functor as follows.
If W is the collection of all morphisms of Δ(X
_∙), then it is well known that the
∞-localization [By ∞-localization we
mean simplicial localization followed by
the simplicial nerve.] NΔ(X _∙) [W ^-1] is an ∞-groupoid equivalent
to X _∙.


By construction F  _P takes all morphisms to quasi equivalences in A _∞Cat
^ℤ _2. So we get an induced map:

    ℱ _P:  |Δ^ (X
    _∙)|  → |w A _∞Cat
    ^ℤ _2|,

where |C| is shorthand for the geometric realization of
the nerve of a category C.



The same construction applies to a concordance of functors,
yielding a homotopy:

    |Δ^ (X
    _∙)| × [0,1]  → |w A _∞Cat
    ^ℤ _2|.

Since X ≃ |Δ^ (X _∙)|, we conclude
that ℱ _P induces a well defined
class 
    [X → |w A _∞Cat ^ℤ _2|]
 in π _0 (Maps  (X,
|w A _∞Cat ^ℤ _2|)), where Maps is the
space of continuous maps. By slight abuse we denote this
class by [ℱ _P].

To summarize, we obtain the following variation of the main
theorem of  <cit.>. 
  Let M ↪
P → X be a Hamiltonian fibration with fiber (M, ω
) a compact monotone symplectic manifold. 
Suppose that Fuk (M,  ω) satisfies the finiteness Condition
(<ref>).
Then the assignment:

    [P] ↦  [ℱ _P] ∈π _0
    (Maps  (X,  |w𝒜 _k
    ^ℤ _2|)),

is well defined, where [P] is the Hamiltonian isomorphism
class of the fibration P.

 
The finiteness condition is readily satisfied for 
Fuk (ℂℙ ^n,  ω _FS),
where the latter denotes the monotone Fukaya category,
 <cit.>. [More concretely, we may define a simplified
category Fuk
(ℂℙ ^n,  ω _FS) to have objects oriented
Lagrangians Hamiltonian isotopic to the Clifford torus L
_0. In this case it is automatic that Fuk
(ℂℙ ^n,  ω _FS) is quasi-isomorphic to
the A _∞ algebra HF (L _0, L _0), and
finiteness follows.]  

 
For each n ≥ 1, there is a natural map:

    cl _n: B Ham (ℂℙ ^n,  ω
    _FS)   → |w𝒜 ^ℤ _2 _k|,

well defined up to homotopy.


Let ℂℙ ^n↪  P → B Ham (ℂℙ ^n,  ω
_FS) be the 
tautological Hamiltonian fibration, that is the fiber bundle
associated to the universal principal Ham (ℂℙ ^n,  ω _FS) bundle over B Ham (ℂℙ ^n,  ω
_FS). By the Example
<ref> we may apply Theorem
<ref> to get a well defined up
to homotopy map: 

    B Ham (ℂℙ ^n,  ω
    _FS) → |w𝒜 ^k
    _ℤ|.










































In  <cit.> we deal
with monotone symplectic manifolds, but all constructions can also be performed for compact Calabi-Yau manifolds. In this case, we can get
ℤ grading, but we must work with a more
specific type of coefficient ring. The standard type of choice is
the universal Novikov field Λ, cf.
Sheridan <cit.>, 
whose elements are formal sums

    ∑_j=0 ^∞ c_j r^λ _j,

where c_j ∈ℂ, and λ _j∈ℝ ^  is an increasing sequence of real numbers
such that 
    lim _j →∞λ _j = ∞.


In this case,   we similarly obtain:
  Let M ↪
P → X be a Hamiltonian fibration with fiber (M, ω
) a compact Calabi-Yau symplectic manifold, s.t. Fuk (M,
ω ) satisfies the finiteness Condition
<ref>.  Then 
the assignment:

    [P] ↦  [ℱ _P] ∈π _0
    (Maps  (X,  |w 𝒜 _Λ |)),

is well defined.


For brevity we call the symplectic manifolds in Theorem
<ref> respectively Theorem <ref> finite monotone
type and finite Calabi-Yau type respectively. 




§ HAMILTONIAN ELEMENTS OF CATEGORIFIED ALGEBRAIC K-THEORY
 
Now recall the definition of the infinite loop space
(<ref>). There is a natural map, see
 <cit.>,  

    |w 𝒜 _k | →Ω |w S _∙ (𝒜 _k)| = K ^Cat (k),

Similarly, there is a natural map 
    |w 𝒜
    ^ℤ _n _k | → K ^Cat, ℤ _n
    (k).


Combining with Corollary <ref> we get:
 
For each commutative ring k, there is a natural homotopy class of a map:

    𝒰: BHam (ℂℙ
    ^n,  ω _FS)) → K ^Cat, ℤ _2  (k), ∀ n ∈ℕ _≥ 1,

and so a natural group homomorphism:

    𝒰 _m, *: π _m (BHam
    (ℂℙ ^n,  ω _FS)) → K ^Cat,
    ℤ _2 _m (k).




























The image of 𝒰 _m, * is part of what we call  
Hamiltonian algebraic K-theory elements. Are these
elements known to be non-trivial? Not
yet, but we do know that the map 

    (ℤ≃π
    _4(BPU (2)) ≃π _4  (BHam
    (S ^2,  ω _FS)) )
    →π _4 (|w 𝒜 ^ℤ _2
    _ℤ|, Fuk (S ^2,
    ω _FS)  
    ),

induced by cl _2 is injective,
 <cit.>.
[ <cit.> does
not work with pretriangulations,   however pretriangulating
does not change the relevant computation.]









 










Thus, if we knew that 

    π _4 (| w𝒜
    ^ℤ _2 _ℤ|, Fuk (S ^2, ω _FS)) ⊗ℚ K ^Cat, ℤ
    _2 _4 (ℤ) ⊗ℚ

was injective,  we would obtain
non-triviality of (some) Hamiltonian algebraic K-theory
elements. And in particular we would have: 

    K ^Cat, ℤ _2 _4 (ℤ) 
    conditionally admits
    a ℤ injection.

In general, there is no reason to expect 𝒰
_m,* to be injective, but m=4 is in the stable
range of homotopy groups of BPU (2). Here `stable' means the
range in which BPU (n) → BPU injects on homotopy
groups. It is thus tempting to conjecture that there
is a similar stabilization phenomenon for the more abstract
mapping:
 

    π _2k (|w𝒜 ^ℤ _2
    _ℤ|,  Fuk (ℂℙ ^n-1, ω _FS))) ⊗ℚ K ^Cat,
    ℤ _2 _2k (ℤ) ⊗ℚ is injective, for 2k ≤ 2n.[We tensor
    with ℚ because all the geometric evidence, e.g.  <cit.>,
    is over ℚ but it is not obviously of essence.]


The above is party motivated by
 <cit.>,
which says that the topological stable range of BPU (n) is
related to algebraic stable range of certain quantum
characteristic classes, related to the Fukaya category
theory underlying the construction of the Hamiltonian
algebraic K-theory elements. 

It should be emphasized that the latter conjecture is of
algebraic topological nature, there is no analysis involved
in its formulation. On the other hand the content and
the proof of the injectivity of (<ref>) does involve geometric analysis.


















§ HAMILTONIAN ELEMENTS IN ALGEBRAIC K-THEORY
 
It was our initial goal to obtain geometric elements of
“classical” algebraic K-theory. First we have the
“Calabi-Yau” case. There is a natural functor:

    ϕ: 𝒜 _k→Perf  (k),

where Perf  (k) is the Waldhausen category
of perfect complexes over k, ϕ (C) = Hoch _∙ (C) as in Section <ref>. Keller's theorem
 <cit.>
readily implies that this is a Waldhausen functor, that is
it induces a map 
    ϕ _*: K ^Cat (k) → K (k).


In particular, combining with Theorem
<ref> we get that an isomorphism
class of a Hamiltonian
fibration over S ^k with compact Calabi-Yau type fiber
determines an element of K _k (Λ ), as discussed in
the introduction.

Now we discuss the  monotone case. We have a Waldhausen category Ch
_k ^ℤ _2 of 2-periodic complexes of
projective finite rank k-modules. It's Waldhausen
K-theory infinite loop space will be denoted by K ^ℤ _2 (k).
We again have a map:

    ϕ _*: K ^Cat, ℤ _m (k) → K ^ℤ
    _m (k).

In particular, we obtain homomorphisms:

    π _m (BHam
    (ℂℙ ^n,  ω _FS)) → K ^Cat,
    ℤ _2 _m (k)) → K ^ℤ _m (k)   ∀ m,n ∈ℕ.

And this specializes to give (<ref>) in the
introduction, using the natural map 
    BPU (n) →BHam (ℂℙ ^n-1 ,  ω).



§ MIRROR HAMILTONIAN ALGEBRAIC K-THEORY ELEMENTS.
 
We may reformulate the famed homological mirror symmetry
conjecture of Kontsevich in algebraic K-theory terms. In the case of Calabi-Yau
manifolds, we propose that
there is an equality of classes in K  ^Cat_0 (Λ): 
    [Fuk (M)] = [D ^b (M ^mirror)]
 and 
    [Fuk (M ^mirror)] = [D
    ^b (M)],
  where M,
M ^mirror are mirror dual, finite type Calabi-Yau manifolds.
We have a construction of elements in K  ^Cat_d
(Λ) for all d ≥ 0, which we may call
“A-model
Hamiltonian algebraic K-theory elements”. That is the
elements in K ^Cat _d (Λ ), associated to Hamiltonian fibrations M
↪ P → S ^d, via Theorem
<ref>. 

We might then try to extend homological mirror symmetry to the context
of general Hamiltonian algebraic K-theory elements. 
For concreteness let us suppose that G is a compact Lie
group acting by Hamiltonian symplectomorphisms on
a finite type Calabi-Yau symplectic manifold (M, ω).
In this context, there  is a proposal of Teleman
 <cit.>, (see also his 2014 ICM address) 
on extension of the homological mirror symmetry. 
He expects that the Langlands dual of ^LG
^ℂ plays a role in the B-model, where G
^ℂ is the complexification. This has now been
partially confirmed in some cases,
 <cit.>.

In our setup we obtain a natural “A-model” homotopy class:

    [f_M: BG → K ^Cat (Λ)],

as the composition BG →BHam (M, ω)
→ K ^Cat (Λ) .
To mimic Teleman's idea we may propose that there is also
a natural “B-model” homotopy class 

    [f _(G  M) ^mirror: B(^LG ^ℂ)  →
    K ^Cat (Λ)],

whenever G   M has a Teleman mirror (G  M)
^mirror.
We may then conjecture that the induced maps:

    π _∙ (BG) →π _∙ (K ^Cat
    (Λ)) = K ^Cat _∙ (Λ), 
    π _∙ (B(^LG ^ℂ)) →π _∙ (K ^Cat
    (Λ)) = K ^Cat _∙ (Λ)

have the same image. 
What is curious at the moment is not the possible correspondence
itself but how (<ref>) might be constructed. 








 





§ ACKNOWLEDGMENTS
 This work was initiated during my
stay as a member at IAS in Spring 2024, and I am very
grateful for all that was provided. Many thanks to Bertrand
Toën, Jacob Lurie and Nick Sheridan for answering many
background questions. I have also benefited from a discussion
with Mohammed Abouzaid.














































































































































































For a k-simplex Σ: Δ^ (k) →  X _∙, let C ⊂Δ^ (X































































siam

 Paper Title: On Hopkins' Picard group.



§ ABSTRACT 
    We compute the algebraic Picard group of the category of K(n)-local spectra, for all heights n and all primes p. In particular, we show that it is always finitely generated over _p and, whenever n ≥ 2, is of rank 2, thereby confirming a prediction made by Hopkins in the early 1990s. In fact, with the exception of the anomalous case n=p=2, we provide a full set of topological generators for these groups. Our arguments rely on recent advances in p-adic geometry to translate the problem to a computation on Drinfeld's symmetric space, which can then be solved using results of Colmez–Dospinescu–Niziol.


[
    [
    
=====




  

    < g r a p h i c s >
 

René Magritte, The Voice of Space, 1931








§ INTRODUCTION




 §.§ Results

This article settles a question about the K(n)-local Picard group _n,p, relative to a prime p and an integer n≥ 1, formulated by Hopkins in the early 1990s (<cit.>). Here, K(n) denotes Morava K-theory at height n, the prime p being implicit.  Also known as Hopkins' Picard group, _n,p is the group of isomorphism classes of invertible K(n)-local spectra, under the operation of K(n)-local smash product.  The study of _n,p was initiated by Hopkins–Mahowald–Sadovsky <cit.>, who defined an approximation to _n,p in the form of a comparison map
    
    ε_n,p→_n,p^.

The target _n,p^ of this map is defined in a purely algebraic manner in terms of equivariant line bundles on Lubin–Tate space (reviewed below).  The map ε is an isomorphism for p large enough with respect to n, but generally it has a kernel κ_n,p which is a (possibly infinite) product of cyclic p-groups with bounded exponent.  Our main theorem gives a complete description of _n,p^ together with the image of ε (up to a /2 ambiguity in the case (n,p)=(2,2)). 

[<Ref>]  Let p be a prime, assume n≥ 2, and write 𝒵_n,p for the pro-cyclic group[This notation allows us to deal with all primes uniformly. When p is odd, 𝒵_n,p≅_p ⊕/(2p^n-2), while 𝒵_n,2≅_2 ⊕/(2^n-1).] lim_k /p^k(2p^n-2). 
There is an isomorphism

    _n,p^𝒵_n,p⊕_p  ⊕ (/2)^⊕ e(n,p),

where

    e(n,p)=
        
            0,     p≠ 2; 
    
            2,     p=2 and  n≥ 3; 
    
            3,     p=2 and  n=2.

In all cases except possibly (n,p)=(2,2), the map ε is surjective. 
In the case (n,p)=(2,2), the map ε is split with cokernel of order at most 2.


The case n=1 has long been understood, so we have excluded it from the statement of the theorem.  The computation for n=2 has also been known by work of Hopkins and Lader (p ≥ 5), Karamanov as well as Goerss–Henn–Mahowald–Rezk (<cit.> p=3), and Henn (p=2, unpublished). These computations rely on explicit resolutions such as those constructed in <cit.> in the case n=2 and p=3, and thus seem very difficult to extend to larger heights. Indeed, prior to the present work, there was not a single pair (n,p) with n≥ 3 for which _n,p^ had been computed. 

A precise version of <ref> appears as  <ref>, which in fact provides explicit topological generators for _n,p^. The case p=2 being rather delicate, for the moment we focus on what happens when p is odd.  As a profinite group, _n,p^𝒵_n,p⊕_p is generated by the images under ε of the following two classes in _n,p:  the suspended K(n)-local sphere Σ𝕊_K(n), and the (p-1)-fold power of the determinant sphere 𝕊_K(n)[], defined in <ref>.  (In the case n=1, 𝕊_K(1)[] and Σ^2𝕊_K(1) have the same image in _1,p^, and _1,p^𝒵_1,p is generated by the image of Σ𝕊_K(1).) Combined with the known isomorphism range of the comparison map ε, for all heights and generic primes p, we obtain a full computation of the topological K(n)-local Picard group, stated here only for the previously unknown cases:

[<Ref>]
     Let n > 2 and p>(n^2+1)/2. There is an isomorphism 
        
    _n,p≅_p ⊕_p ⊕/(2p^n-2).

   As a profinite group, _n,p is freely generated by Σ𝕊_K(n) and 𝕊_K(n)[]^⊗ (p-1).


<Ref> can be reduced to a purely algebraic result concerning deformations of formal groups, as we now explain.  Let H_n be a 1-dimensional formal group of height n over _p, and define the Lubin–Tate ring A_n as the deformation ring of H_n.  Then A_n is non-canonically isomorphic to a power series ring in n-1 variables over the ring of Witt vectors of _p.  Finally, define the Morava stabilizer group _n as the group of automorphisms of the pair (H_n,_p).  Then _n is a profinite group acting continuously on the topological ring A_n.  

The ring A_n admits a spectral incarnation E=E(H_n,_p), known as Morava E-theory or the Lubin–Tate spectrum.  This is a K(n)-local commutative ring spectrum with homotopy ring E_* = A_n[u^± 1] for a class u of degree -2.  The spectrum E is constructed functorially from the pair (H_n,_p), so that it admits an action of _n.  The completed E-homology of a spectrum is a graded E_*-module which is derived complete with respect to the maximal ideal of π_0 E=A_n, and which admits a continuous action of _n lying over the action on E_*.  We call such a structure a Morava module.

The algebraic Picard group _n,p^ is defined as the group of isomorphism classes of invertible Morava modules. The comparison map ε appearing above is given by applying completed E-homology to invertible K(n)-local spectra. The algebraic Picard group sits in an exact sequence

    0 → H^1_(_n,A_n^∗)→_n,p^→/2→ 0.

This is due to the fact that E_* is 2-periodic, so an invertible Morava module is supported in either odd or even degrees, giving the map to /2;  since the shift E_*[1] is supported in odd degrees, the map to /2 is surjective.  The kernel of this map is isomorphic to the group of isomorphism classes of invertible A_n-modules carrying an equivariant continuous action of _n.  Since A_n is a local ring, its Picard group is trivial.  Equivariant continuous actions of _n on a free A_n-module of rank 1 correspond exactly to classes in the continuous cohomology H^1_(_n,A_n^∗), where A_n^∗ is the group of units in A_n.  

The exact sequence (<ref>) allows us to reduce <ref> to a computation of H^1_(_n,A_n^∗).  
Once again we assume n≥ 2, the case n=1 being already well-understood.
 

    
    Assume n≥ 2.  There is an isomorphism
    
    H^1_(_n,A_n^∗)_p^⊕ 2⊕/(p^n-1) ⊕ (/2)^⊕ e(n,p),

    where e(n,p) is as in <ref>.    

<Ref> implies the description of _n,p^, at least up to the question of which extension of /2 by H^1_(_n,A_n^∗) is represented by (<ref>).  
What is more, we show in the course of proving <ref> that every element of H^1_(_n,A_n^∗) (or rather an index 2 subgroup of it, in the case n=p=2) lifts to an element of _n,p of even degree;  this implies the properties of ε claimed by <ref>.  

Let A_n^⊂ A_n^∗ be the subgroup of principal units (i.e., topologically unipotent elements).  
It is convenient to divide <ref> into pro-p and prime-to-p parts using the _n-equivariant decomposition A_n^∗ A_n^⊕_p^∗.  The prime-to-p part poses no difficulty (<Ref>), so <ref> is reduced to the claim H^1_(_n,A_n^)_p^⊕ 2⊕ (/2)^⊕ e(n,p).  We prove a precise statement (<Ref>) about H^1_(_n,A_n^), and show that it implies <Ref>.   




 §.§ Main ideas of the proof


An even invertible Morava module (meaning a class in _n,p^ which maps to the identity in /2 in (<ref>)) corresponds to a free A_n-module with a continuous equivariant _n-action.  We may view such modules geometrically, as line bundles on the stack [_n/_n], where _n= A_n is the Lubin–Tate deformation space, i.e., the affine formal scheme parameterizing deformations of the formal group H_n.  
The stack [_n/_n] arises naturally;  it is isomorphic to the completion of the moduli space of formal groups at the height n point corresponding to H_n (see <ref>).  

To access the Picard group of [_n/_n], we rely heavily on Faltings' isomorphism between the Lubin–Tate and Drinfeld towers, as interpreted as a relation between perfectoid spaces <cit.>.  This isomorphism is only available after passage to the generic fiber.  Therefore let K be the fraction field of the ring of Witt vectors of _p, and let _n,K be the adic generic fiber of _n, so that _n,K is isomorphic to a rigid-analytic (n-1)-dimensional open ball over K.  

In <ref> we interpret Faltings' isomorphism as a relation between stacks on Scholze's category of diamonds:

    [_n,K^♢/_n]  [^n-1,♢/_n(_p)].

On the left side of the isomorphism is the “diamond generic fiber” of the stack [_n/_n].  Appearing on the right side is (the diamond version of) Drinfeld's symmetric space ^n-1, defined as the complement in rigid-analytic ^n-1__p of all _p-rational hyperplanes.  

The isomorphism (<ref>) is indispensable for the proof of <ref>, because it trades the opaque action of _n on A_n for the transparent (indeed linear) action of _n(_p) on ^n-1.  We offer a summary of this proof.  Recall we have already reduced 
<ref> to the calculation of H^1_(_n,A_n^).  In particular, we want to show that for n≥ 2, the torsion-free part of H^1_(_n,A_n^) is _p^⊕ 2, generated by the pro-p parts of the even Morava modules ε(Σ^2𝕊_K(n)[]) and ε(𝕊_K(n)[]).  The proof brings together two important constructions, accounting for each of these two generators.

The first construction involves determinants of formal groups.  The idea is that in certain contexts, it is possible to take the determinant of a height n formal group to obtain a height 1 formal group.  See <ref> for the precise statement.  One manifestation of this phenomenon is that
there is a determinant homomorphism _n→_1 which is compatible with the natural inclusion map A_1→ A_n;  as a result there is an induced map  

    _^* H^1_(_1,A_1^)→ H^1_(_n,A_n^).

The source of this map can be calculated directly,

    H^1_(_1,A_1^) _p,     p≠ 2; 
    _2⊕ (/2)^⊕ 2,     p= 2,

and the image of the generator of the free part of this group under _^* is the pro-p part of ε(𝕊_K(n)[]), the Morava module of the determinant sphere.

The second construction is the one that leverages (<ref>).  In <ref> we define a sheaf  of principal units on an adic space, and we observe that A_n^∗∗ H^0(_n,K,^∗∗).  Let C be the completion of an algebraic closure _p of _p, and let _C^n-1 be the base change of ^n-1 to C.  Another version of (<ref>) is 

    [_n,K^♢/_n]  [^n-1,♢_C/((_p/_p)×_n(_p))].

As a formal consequence of this isomorphism, we obtain a map 

    b  H^1_(_n,A_n^)→ H^1_(_C^n-1,)^(_p/_p)×_n(_p).

The following theorem appears as <ref> combined with <ref>.


     Assume (n,p)≠ (2,2).  The maps _^* and b fit into an exact sequence
    
    0→ H^1_(_1,A_1^)  H^1_(_n,A_n^)b→ H^1_(_C^n-1,)^(_p/_p)×_n(_p)→ 0.

If (n,p)=(2,2), there is a similar exact sequence, except the first nonzero term in the sequence is H^1_(_1,A_1^)⊕/2.

Interestingly, the reason that (n,p)=(2,2) is treated differently in <ref> traces back to the fact that it is the only case where there are nontrivial continuous homomorphisms _n(_p)→_p^ (see <ref>).  

 
At this point we apply the recent advances of Colmez–Dospinescu–Niziol <cit.> on the cohomology of _C^n-1, which seem to be tailor-made for our purposes.  From their work we deduce (see <ref>) that if n≥ 2, then 
H^1_(_C^n-1,)^(_p/_p)×_n(_p) is canonically isomorphic to _p .  The final challenge (<ref>) is to show that the map b carries the pro-p part of ε(Σ^2𝕊_K(n)) onto a generator of this _p, so that b is surjective.  

Combining <ref> with (<ref>) shows that H^1_(_n,A_n^)_p^⊕ 2⊕ (/2)^⊕ e(n,p), which completes the proof of <ref>.  





 §.§ Acknowledgements


We thank Agnès Beaudry, Pierre Colmez, Ehud de Shalit, Shai Evra, Paul Goerss, Drew Heard, Hans-Werner Henn, Mike Hopkins, Wiesława Nizioł, Ori Parzanchevski, and Peter Scholze for many helpful discussions. 

TB is supported by the European Research Council (ERC) under Horizon Europe (grant No. 101042990) and would like to thank the Max Planck Institute for its hospitality. NS and TMS were supported by the US-Israel Binational
Science Foundation under grant 2018389. NS was supported by a Sloan research fellowship, NSF Grants DMS-2304781, and a grant from the Simons Foundation (MP-TSM-00002836, NS).  TMS was supported by ISF1588/18 and the
ERC under the European Union’s Horizon 2020 research and innovation program (grant agreement
No. 101125896)









§ THE K(N)-LOCAL PICARD GROUP


The purpose of this section is two-fold. First, it reviews some background material on Picard groups in chromatic homotopy theory. We then explain how to reduce the determination of Hopkins' Picard group to a cohomological computation (<ref>), which will then be tackled in the remainder of this paper. 



 §.§ Motivation: invertible spectra and degree


Let = (,⊗,) be a symmetric monoidal ∞-category. The Picard group () of  is defined as the collection of isomorphism classes of ⊗-invertible objects in , i.e., those objects X ∈ for which there exists y ∈ with x ⊗ y ≃. If  is presentably symmetric monoidal, then () is a set <cit.>. The symmetric monoidal product on  then equips () with the structure of a group, with unit .

If = (,⊗,𝕊) is the symmetric monoidal ∞-category of spectra, then we can classify invertible objects via their degree. Indeed, for any X ∈(), the Künneth isomorphism implies that H_*(X;) is concentrated in a single degree, which we define as (X). A Postnikov tower argument <cit.> then shows that the induced map
    
    () [r]

is a bijection, with inverse n ↦𝕊^n. In particular, () is generated by Σ𝕊.

Hopkins, Mahowald, and Sadofsky observed that the situation becomes more interesting when we turn to local stable homotopy theory. This article concerns the invertible K(n)-local spectra.  Here K(n) denotes Morava K-theory of height n at the (fixed) prime p. This is an associative ring spectrum with coefficients 
    
    π_*K(n) ≅   if  n = 0; 
    _p[v_n^± 1]    if  0 < n < ∞; 
    _p    if   n = ∞,

with v_n of degree 2(p^n-1). By convention, K(0) = H independently of p, while K(∞) = H_p. Our focus in this paper will be on the case of intermediate characteristic 0 < n < ∞, and we will often keep the ambient prime p implicit in the notation. The Morava K-theories are fields in , i.e., they obey a Künneth formula 
    
    K(n)_*(X ⊗ Y) ≅ K(n)_*(X) ⊗_K(n)_* K(n)_*(Y)

for any X, Y ∈, so that every module spectrum over K(n) is free. In fact, as a consequence of the nilpotence theorem due to Devinatz, Hopkins, and Smith <cit.>, this provides a full classification of all prime fields in . For more background information on chromatic homotopy theory, we refer the reader to <cit.>.

Bousfield localization constructs the category _K(n) of K(n)-local spectra, i.e., the universal localization (-)_K(n)→_K(n) which inverts all maps f of spectra for which K(n) ⊗ f is an equivalence. Localization admits a fully faithful right adjoint _K(n)⊂, so we may view a K(n)-local spectrum as a spectrum. The smash product on  descends to a symmetric monoidal structure on _K(n); explicitly, the product of two K(n)-local spectra X,Y is given by 
    
    X ⊗̂ Y = (X⊗ Y)_K(n).

Equipped with this structure, _K(n) = (_K(n), ⊗̂,𝕊_K(n)) forms a presentably symmetric monoidal stable ∞-category.


    Fix a height n≥ 1 and an implicit prime p. We define the K(n)-local Picard group as 
        
    _n,p(_K(n)).



By (<ref>), the Morava K-homology K(n)_*(X) of any invertible X ∈_K(n) is concentrated in degrees congruent to d modulo 2(p^n-1) for some 0 ≤ d < 2(p^n-1). Following the global story above, we define this number as the local degree of X: _n,p(X)  d. The next result is proven in <cit.>, relying on <cit.>.


    Let n≥ 1. The K(n)-local degree provides a surjective map 
        
    _n,p_n,p@->>[r]    /2(p^n-1)

    with kernel an infinite abelian pro-p-group. 


In particular, in contrast to the case of (), the naive notion of degree in the K(n)-local setting is only a rather coarse approximation to the Picard group. In the next subsection, we will review a refined invariant which provides a much closer algebraic approximation to _n,p.




 §.§ Morava modules


As the starting point for the construction of a good algebraic approximation to _n,p, we take the following fundamental result of Devinatz–Hopkins <cit.>. A general account of the Galois theory of commutative ring spectra was developed by Rognes <cit.>.  

Let H_n be a 1-dimensional formal group of height n over _p.  Let E_n be Morava E-theory of height n associated with H_n. This a K(n)-local commutative ring spectrum with
    
    π_*E_n = A_n[u^± 1], u=-2.

Here π_0E_n=A_n is the deformation ring of H_n, known as the Lubin–Tate deformation ring.  There is a (non-canonical) isomorphism A_n  W(_p) u_1,…, u_n-1, with W(_p) the ring of Witt vectors of _p.  Let H_n^ be the universal deformation of H_n;  then π_-2E_n= H_n^=A_nu is the Lie algebra of H_n^. The spectrum E_n admits a continuous action of the Morava stabilizer group _n=(H_n,_p).  


    The unit map 𝕊_K(n)→ E_n witnesses the target as a K(n)-local Galois extension with Galois group _n. In particular, there is a canonical equivalence 𝕊_K(n)≃ E_n^h_n.


We can twist the equivalence of this theorem by any X ∈_K(n), resulting in a new equivalence X ≃ (E_n ⊗̂ X)^h_n, where X is equipped with trivial _n-action. This in turn gives rise to a (Bousfield–Kan) descent spectral sequence of signature (see <cit.>)
    
    E_2^s,t(X):= H^s(_n,E_n,t(X))π_t-sX

that has excellent convergence properties. Indeed, since 𝕊_K(n)→ E_n is a descendable morphism of commutative ring spectra by the smash product theorem <cit.> (see also <cit.>), the spectral sequence converges strongly and collapses on a finite page with a horizontal vanishing line. When p-1 does not divide n, the stabilizer group _n is p-torsion free and thus has finite cohomological dimension (as opposed to merely finite virtual cohomological dimension), in which case (<ref>) collapses already on the E_2-page. 

The input to the spectral sequence (<ref>) is the continuous _n-cohomology of E_*(X) = π_*L_K(n)(E_n ⊗ X), which by a mild abuse of notation here refers to the completed E-homology of X. This structure is captured algebraically in the category of Morava modules, _E_*^↺_n, defined as follows: Let m= (p,u_1,…, u_n-1) ⊂π_0E_n be the maximal ideal of π_0E_n. A Morava module is a derived 𝔪-complete graded E_*-module equipped with a continuous semi-linear action of _n. Morphisms of Morava modules are continuous equivariant maps. The category of Morava modules can be modeled as (π_*E_n,E_*E)-comodules whose underlying E_*-module is derived m-complete. Completing the usual tensor product induces a symmetric monoidal structure on _E_*^↺_n, denoted ⊗̂. With these definitions, we obtain a canonical lift 
    
    _E_*^↺_n[d]^U 
    _K(n)[r]_-E_*(-)[ru]^-E_*(-)   _E_*,

where U denotes the evident forgetful functor, which forgets the action by _n on the given Morava module. Note that E_*(-) is in general not symmetric monoidal. However, when restricted to spectra X with E_*(X) projective (as E_*-module), it is symmetric monoidal. In particular, this applies to invertible K(n)-local spectra, in light of the following characterization:


    For X ∈_K(n), the following conditions are equivalent:
        
            
  * X is invertible;
            
  * E_*(X) is free of rank 1 as a E_*-module;
            
  * K(n)_*(X) is free of rank 1 as a K(n)_*-module.
        



    The algebraic Picard group _n,p^ is defined as the group of isomorphism classes of ⊗̂-invertible Morava modules. The Morava module functor (<ref>) then induces a comparison map
        
    ε_n,p⟶_n,p^(_E_*^↺_n),    X ↦ E_*(X).



The comparison map ε turns out to be a close approximation to the topological Picard group, essentially due to the excellent properties of the descent spectral sequence (<ref>). The next result summarizes the situation:



    Let p be prime such that 2p-2 > max{n^2-1,2n}, then the comparison map is bijective:
        
    ε_n,p_n,p^.
 


The injectivity part of this result was established in <cit.>, while surjectivity was proven under slightly stronger hypothesis in <cit.>; the version given here is due to Goerss and Hopkins, <cit.>.

In general, however, the map ε is not injective; non-trivial elements in the kernel are called exotic. This can only occur in the non-algebraic realm, i.e., when the descent spectral sequence (<ref>) for X ∈_K(n) does not collapse for cohomological reasons. 


    The kernel of ε is the exotic Picard group κ_n,p, fitting into an exact sequence
        
    0 [r]    κ_n,p[r]    _n,p[r]^-ε   _n,p^.




    For any n and p, the group κ_p,n is an abelian pro-p group of finite exponent.  It is pro-p as a consequence of <ref>.  The finiteness of its exponent can be seen as follows: Consider the descent spectral sequence (<ref>). Since it collapses at a finite page with a horizontal vanishing line, uniformly for any X ∈_n,p, there is an induced descent filtration on κ_n,p with subquotients E_r^r,r-1(𝕊), see for example <cit.>. By <cit.>, the latter groups are all p-torsion of finite exponent, hence so is κ_n,p. It follows from the Prüfer–Baer theorem that κ_p,n is then a (possibly infinite) product of finite cyclic p-groups. For odd p, this has already been observed by Heard <cit.>.  





 §.§ Two distinguished spheres


We discuss here the two distinguished classes in the local Picard group _n,p which make an appearance in <ref>.

The first of these classes is simply the suspension Σ𝕊_K(n).   The Morava module E_n,*(Σ𝕊_K(n)) is odd, but its tensor square E_n,*(Σ^2𝕊_K(n)) is
    
    ε(Σ^2𝕊_K(n))=π_-2E_n ≅ H_n^,

where H_n^ is the Lie algebra of the the universal deformation H_n^ of the height n formal group H_n. Since K(n)_*(Σ^2p^n-2𝕊_K(n)) ≅ K(n)_*, the local degree of  Σ^2p^n-2𝕊_K(n) is zero. In light of <ref>, this implies that the assignment k ↦Σ^(2p^n-2)k𝕊_K(n) extends to a homomorphism
    
    ι__p [r]    _n,p[r]    _n,p^.

In fact, we can do better, extending to a larger procyclic group generated by Σ^1𝕊_K(n). To do so, we will introduce an auxiliary piece of notation (inspired by <cit.>) that allows us to treat the cases of odd and even primes uniformly:


    For each prime p and height n, we set _n,plim_k /(p^k(2p^n-2)).



    For any n ≥ 1 and prime p, the group _n,p is procyclic, isomorphic to
        
    _n,p≅_p ⊕/2(p^n-1)    if  p>2; 
    _2 ⊕/(2^n-1)    if  p=2.

    For example, if p is odd, a generator for the torsion summand in this presentation is given by the element lim_k p^nk. 


Based on the periodicity theorem <cit.>, Hopkins, Mahowald, and Sadofsky <cit.>, one can use “v_n-adic interpolation” to construct an extension of ι_, which we denote by the same symbol:
    
    4pcι__n,p[r]^-1 ↦Σ^1𝕊_K(n)   _n,p^.

Here, the map _p →_n,p is induced by maps /p^k →/(p^k(2p^n-2)) determined by 1 ↦ 2p^n-2. In particular, for p odd, this splits the presentation of <ref>. However, for p=2, this embedding includes _2 as an index 2 subgroup of the _2 appearing in <ref>.

The second important class is the determinant sphere.  Let 𝕊(1) be the p-complete sphere with its natural continuous _p^∗-action. The stabilizer group then acts on 𝕊(1) through the determinant map _n →_p^∗. The determinant sphere is defined as 
    
    𝕊_K(n)[]  (E_n ⊗𝕊(1))^h_n,

where _n acts diagonally on the p-completed smash product E_n ⊗𝕊(1). As shown in <cit.>, the determinant sphere topologically realizes the Morava module A_n[], meaning the free A_n-module of rank 1 where the _n-action has been twisted by .  Thus:
    
    ε(𝕊_K(n)[]) = A_n[].



    The significance of having a concrete topological model for the determinant sphere comes from Gross–Hopkins duality, which describes the Morava module of the dualizing module for the K(n)-local category. The latter is by definition the Brown–Comenetz dual of the n-th monochromatic layer of the sphere M_n𝕊 = (L_n𝕊→ L_n-1𝕊), denoted I_n  IM_nS^0. Here, Brown–Comenetz duality I is a lift of Pontryagin duality to the category of spectra. Gross and Hopkins <cit.> prove that I_n ∈_n,p with Morava module E_*(I_n) ≃π_n-n^2E_n[], see also <cit.>. We may then restate this algebraic identification as a K(n)-local equivalence 
        
    I_n ≃Σ^n^2-n𝕊_K(n)[] ⊗̂ P_n,p,

    where P_n,p∈κ_n,p. This formula is a source for the construction of exotic elements in the Picard group, see for example <cit.>. 





 §.§ Conjectures and results


The following conjecture, dating from the early 1990s and due to Hopkins, summarizes the expected behaviour of the sequence (<ref>):


    For all heights n ≥ 1 and primes p:
        
            
  * κ_n,p is a finite p-group;
            
  * _n,p^ is finitely generated over _p.
        


Taken together, the two parts of the conjecture imply that _n,p is finitely generated over _p. In fact, this is almost an `if and only if' statement, up to the following


    Is the map ε in (<ref>) surjective?


<Ref> has been confirmed at heights 1 and 2 and all primes p through explicit computations; the following table summarizes the state of the art before the present work:


The main result of the present paper is a complete calculation of the algebraic Picard group of _K(n), for all heights n and all primes p; in particular, we recover the the computations of _n,p^ for n≤ 2 summarized in the table above. In addition, we provide explicit topological elements which under the comparison map ε generate the algebraic Picard group with the possible exception of the case n=p=2. The following is <ref> from the introduction:


    For any prime p and any height n≥ 2, the algebraic Picard group of the category of K(n)-local spectra is given by:
    
    _n,p^≅
                    
                    _n,p⊕_p    if   p>2  and  n ≥ 2;
    _n,2⊕_2 ⊕ (/2)^⊕ 2   if   p=2   and  n>2; 
    _n,2⊕_2 ⊕ (/2)^⊕ 2⊕/2    if   p=2   and  n=2,

    with _n,p as in <ref>. For all primes p, the summands _n,p and _p are generated by ε(Σ^1 𝕊_K(n)) and ε(𝕊_K(n)[]^⊗ (p-1)), respectively. For p=2, generators for the additional (/2)^⊕ 2 summand are described in <ref>. 


In the range where the comparison map ε is an isomorphism (<ref>), we thus obtain a complete computation of the Picard group of _K(n); this is <ref>.


    When 2p-2 > max(n^2-1,2n), there is an isomorphism 
        
    _n,p≅_p ⊕_p ⊕/(2p^n-2),

    with generators given by Σ𝕊_K(n) and 𝕊_K(n)[]^⊗ (p-1).


Another direct consequence of <ref> is the surjectivity of the comparison map in all cases except for possibly the anomalous one when n=p=2; see also <ref>.


    For all heights n ≥ 1 and all primes p with the possible exception of n=p=2, the comparison map ε_n,p→_n,p^ is surjective.


The goal of the remainder of this section is to reduce <ref> to a computation in p-adic geometry, stated as <ref> below, which we then carry out in the rest of this paper. 






 §.§ The even algebraic local Picard group


We now begin our reduction of <ref> to a purely cohomological statement. Observe that the K(n)-local Picard group and its algebraic counterpart are naturally /2-graded, since E_* is concentrated in even degrees and is 2-periodic. Let _n,p^,0 be the even algebraic Picard group, consisting of those invertible Morava modules whose underlying E_*-module is even, i.e., concentrated in even degrees.  Thus there is a short exact sequence
    
    0 [r]    _n,p^,0[r]    _n,p^[r]    /2 [r]     0.



The even algebraic Picard group has a simple description in terms of continuous group cohomology. 
Let M ∈_n,p^,0 be an even invertible Morava module.  We can think of M as an invertible A_n-module with continuous _n-action.  Since A_n is a local ring, the underlying A_n-module of M is free of rank 1, say with generator m.  Define a continuous 1-cocycle ϕ_M_n→ A_n^∗ by
    
    g · m = ϕ_M(g)m

for any g ∈_n. The class of ϕ_M in H^1_(_n,A_n^∗) does not depend on the chosen generator.  Conversely, a continuous cocycle in H^1_(_n,A_n^∗) determines an even Morava module by the same formula, and we obtain:


    The assignment M ↦ϕ_M induces an isomorphism
        
    _n,p^,0≅ H^1_(_n,A_n^∗).




    Let _n,p^0 be the even part of the local Picard group, i.e., those invertible K(n)-local spectra X with even Morava module E_*(X). We can translate the Morava modules of the two distinguished spheres of <ref> into a 1-cocycle description. Indeed, for each prime p and height n, under the composite 
        
    _n,p^0 [r]^-ε   _n,p^,0[r]^-ϕ_-≅    H^1_(_n,A_n^∗)

    we have:
        
            
  * since ε(Σ^2𝕊_K(n)) ≅ H, it maps under ϕ to the class of the cocycle t_0_n  → A_n^∗, where t_0 is determined by the formula g· u=t_0(g)u for g∈_n and u a generator of π_-2E_n= H.  

            
  * by construction, 𝕊_K(n)[] maps to the composition of the determinant map _n →_p^∗ with the inclusion map _p^∗→ A_n^∗.  
        


In order to better understand these elements and in particular their relation, it will be useful to separate the prime-to-p part of _n,p^,0 from its principal part. Recall that we consider Morava E-theory based on _p. As a topological ring, 
    
    A_n Wu_1,…,u_n-1,

where W=W(_p) is the ring of Witt vectors. The ring A_n is endowed with the m-adic topology, where m=(p,u_1,…,u_n-1) is the maximal ideal. The quotient map A_n→ A_n/m_p admits a multiplicative splitting via the Teichmüller lift _p→ W.  Thus there is a _n-equivariant isomorphism of topological groups:
    
    A_n^∗ A_n^∗∗⊕_p^∗,

where A_n^∗∗=1+m.  We call A_n^∗∗ the group of principal units; principal units will appear in various contexts throughout the article;  we will always use the double-star notation to refer to them. 

Accordingly, the even algebraic Picard group decomposes into prime-to-p and principal parts:
    
    _n,p^,0 H^1_(_n,_p^∗)⊕ H^1_(_n,A_n^∗∗).

With respect to this decomposition, let us write 
    
    ε = ε^p ⊕ε_p,

where ε^p, resp. ε_p, refers to the composition of ε with the projection onto the prime-to-p part, resp. the principal part. 



 §.§ The prime-to-p part and a relation


We begin by determining the prime-to-p part of _n,p^,0. 


    There are isomorphisms 
        
    H^1_(_n,_p^∗)_(_p/_p)(_p^n^∗,_p^∗) ≅/(p^n-1).



    For the proof, let us abbreviate Γ=(_p/_p).  We write σ for the Frobenius automorphism, so that σ generates Γ as a profinite group.  We recall here some facts about the structure of the Morava stabilizer group _n. It is an extension
        
    0 →_D^∗→_n→Γ→ 0,

    where _D= H_n is the ring of integers in a central simple algebra D of invariant 1/n over _p.  The ring _D has a unique two-sided maximal ideal J, with quotient ring _p^n. Let _D^∗∗=1+J be the group of principal units;  then _D^∗∗ is a normal pro-p subgroup of _n, with quotient
        
    _n/_D^_p^n^∗⋊Γ.

    The pro-p subgroup _D^∗∗⊂_n acts trivially on _p^∗, on which p acts invertibly, so we have an isomorphism for all i≥ 0:
        
    H^i_(_n,_p^∗) H^i_(_p^n^∗⋊Γ, _p^∗).

    Consider the Lyndon–Hochschild–Serre spectral sequence
        
    H^i_(Γ,H^j(_p^n^∗,_p^∗))  H^i+j_(_n,_p^∗).

    Since Γ is pro-cyclic, its cohomology vanishes in degrees ≥ 2, so we have a short exact sequence
        
    0→ H^1_(Γ,_p^∗)→ H^1_(_n,_p^∗) → H^1(_p^n^∗,_p^∗)^Γ→ 0.

    The term  H^1_(Γ,_p^∗) is the cokernel of σ-1 on _p^∗, which is 0.  The term H^1(_p^n^∗,_p^∗)^Γ is the group of Γ-equivariant homomorphisms _p^n^∗→_p^∗.  This is a cyclic group of order p^n-1, generated by the inclusion map _p^n^∗↪_p^∗.


Next, we can determine the fate of Σ^2𝕊_K(n) and 𝕊_K(n)[] in the prime-to-p part.

 

As a cyclic group of order p^n-1, H^1_(_n,_p^∗) is generated by ε^p(Σ^2𝕊_K(n)). Furthermore, the following relation holds in H^1_(_n,_p^∗):
    
    ε^p(𝕊_K(n)[])=p^n-1/p-1ε^p(Σ^2𝕊_K(n)).


  
We use the identification of <ref> and claim that this group is generated by the prime-to-p part of ε(Σ^2𝕊_K(n)).  Recall from <ref> that ε(Σ^2𝕊_K(n)) is the Morava module H_n^, where H_n^ is the universal deformation of the (unique up to isomorphism) formal group H_n  H_n^⊗_A_n_p of height n over _p.  The prime-to-p part of ε(Σ^2𝕊_K(n)) is the class in H^1_(_n,_p^∗) represented by the _n-equivariant _p-vector space H_n^⊗_A_n_p H_n.  All we need to know about the _n-action on H_n is that the subgroup _D^∗ acts on it through the composition _D^∗→_p^n^∗↪_p^∗.  Tracing through the calculation of H^1_(_n,_p^∗) in the proof of <ref>, we see that it is generated by the class of H_n^.  

With respect to the isomorphism H^1_(_n,_p^∗)_Γ(_p^n^∗,_p^∗), the 
class ε^p(𝕊_K(n)[]) corresponds to the norm map _p^n^∗→_p^∗↪_p^∗.  The norm map is (p^n-1)/(p-1) times the inclusion map, giving the relation claimed in the second part of the proposition. 


The proposition also allows us to compute the local degree of 𝕊_K(n)[], as we shall explain next. First, we will denote the even Picard group of the K(n)-local category by _n,p^0; by definition, this is the subgroup of _n,p consisting of those elements that have even Morava module. Restricting the local degree from <ref> to the even part, it factors through the inclusion /(p^n-1) ⊆/2(p^n-1) determined by 1 ↦ 2. Therefore, we obtain the even local degree homomorphism
    
    _n,p^0_n,p^,0[r]    /(p^n-1).

We observe that the target is normalized so that _n,p^0ε(Σ^2𝕊_K(n)) = 1. Moreover, the quotient map A_n^∗→ (A_n/ m)^∗≅_p^∗ is _n-equivariant, and hence induces a reduction map on continuous cohomology, q H^1_(_n,A_n^∗) → H^1_(_n,_p^∗). The next result provides an alternative perspective on the even local degree:


    There is a commutative diagram:
        
    _n,p^0[r]^-_n,p^0[d]_ϕ_(-)∘ε   /(p^n-1) [d]^≅
    
                H^1_(_n,A_n^∗) [r]^-q    H^1_(_n,_p^∗).



    Recall that the kernel of the even local degree map is a pro-p-group by <ref>, while H^1_(_n,_p^∗) is a group of order prime to p, see <ref>. It follows that the composite q ∘ϕ_(-)∘ε     factors through _n,p^0, i.e., there is a commutative square
        
    _n,p^0[r]^-_n,p^0[d]_ϕ_(-)∘ε   /(p^n-1) [d] 
    
                H^1_(_n,A_n^∗) [r]^-q    H^1_(_n,_p^∗).

    We claim that the induced right vertical map is an isomorphism. Indeed, it suffices to verify that on a generator of /(p^n-1), which we may take to be _n,p^0(Σ^2𝕊_K(n)) = 1. As already observed in the proof of <ref>, we have that  q ∘ϕ_(-)∘ε(Σ^2𝕊_K(n)) = ε^p(Σ^2𝕊_K(n)) is the canonical inclusion _p^n^∗↪_p^∗. This is our preferred generator of the cyclic group H^1_(_n,_p^∗), which verifies the claim.



    The even local degree of 𝕊_K(n)[] is p^n-1/p-1. In particular, we have 
        
    ε^p(𝕊_K(n)[]^⊗ (p-1)) = 0.



    This follows from <ref> and <ref>.





 §.§ A reduction of <ref>


The next result, whose proof will occupy the remaining sections of this paper, describes the principal part of the even algebraic Picard group. Assuming it, we can then deduce <ref> up to some special features for the prime 2 that are treated separately in <ref>.


    Let p be a prime n ≥ 2 some height. There are isomorphisms
        
    H^1_(_n,A_n^∗∗) ≅_p^2     if  p>2; 
    _2^2 ⊕ (/2)^⊕ 2   if  p=2  and  n>2; 
    _2^2 ⊕ (/2)^⊕ 3   if  p=2  and  n=2.

    For all primes p, the free summands are generated by the 1-cocycles t_0 and  of <ref>, respectively. For p=2, generators for the (/2)^⊕ 2-summand are described in <ref>. 



    By <ref>, <ref>, <ref>, and (<ref>), we have isomorphisms
        
    ψ_n,p^0_n,p^,0≅ H^1_(_n,A_n^∗) ≅_p^2 ⊕/(p^n-1)    if  p>2; 
    _2^2 ⊕/(p^n-1) ⊕ (/2)^⊕ 2   if  p=2  and  n>2; 
    _2^2 ⊕/(p^n-1) ⊕ (/2)^⊕ 3   if  p=2  and  n=2.

    Since the 2-torsion part for p=2 is dealt with separately in <ref> below, we will ignore these summands for the remainder of this proof. Keeping in mind <ref>, it follows from the identifications of the generators in <ref> together with <ref> that the group _p ⊕/(p^n-1) is generated by ε(Σ^2𝕊_K(n)), while the other _p summand is generated by ε(𝕊_K(n)[]). 

    In order to deduce <ref> from this description, it remains to solve the extension problem of (<ref>). To this end, recall from <ref> that we have constructed a map
        
    8pcψ_n,p_n,p⊕_p [r]^-(Σ^1𝕊_K(n),𝕊_K(n)[]^⊗ (p-1))   _n,p^,
    b
 
    whose two components send the distinguished generators of _n,p and _p to ε(Σ^1𝕊_K(n)) and ε(𝕊_K(n)[]^⊗ (p-1)), respectively. This map fits into a commutative diagram of short exact sequences
        
    0 [r]     (_p ⊕/(p^n-1)) ⊕_p [r]^-(2,1)[d]_ψ_n,p^0   _n,p⊕_p [r] [d]^ψ_n,p   /2 [r] [d]     0 
     
                0 [r]    _n,p^,0[r]    _n,p^[r]    /2 [r]     0,

    where the top left map is induced by multiplication by 2 resp. the identity map on the two cyclic summands. The left vertical map is injective with cokernel (/2)^⊕ e(n,p) for 
        
    e(n,p)=
                    
                        0,     p≠ 2;
    
                        2,     p=2 and  n≥ 3;
    
                        3,     p=2 and  n=2.

    Tracing ε(Σ^1𝕊_K(n)) in the right square, we deduce that the right vertical map is an isomorphism. Therefore, the snake lemma shows that ψ_n,p is an isomorphism  with cokernel (/2)^⊕ e(n,p), as desired. 






 §.§ Addenda and anomalies for p=2


The case of the even prime has some oddities, which we discuss separately in this subsection. The description of _n,p^,0 in <ref> implies that, at least if p≠ 2, the comparison map ε_n,p→_n,p^ is surjective, since _n,p^ is generated as a profinite group by the images of Σ𝕊_K(n) and 𝕊_K(n)[] under ε.  For completeness' sake we discuss the surjectivity of ε when p=2, referring to <ref> for the anomalous case n=p=2.


    The comparison map ε_n,2→_n,2^ is surjective if n>2.


    Let n>2. In light of <ref>, it remains to find K(n)-local spectra which realize the 2-torsion in _n,2^, i.e., to find topological lifts for the non-trivial elements in /2 ⊕/2  = _n,2^[2]. This has already been achieved in <cit.>; for the convenience of the reader, we sketch the construction. There is a cyclotomic (/8)^∗-Galois extension 𝕊_K(n)→𝕊_K(n)[ω_n^8], obtained from 𝕊_K(n)[B^n/8] by splitting off a certain idempotent. By <cit.>, the cofibers of the unit maps of the three /2-Galois subextensions of 𝕊_K(n)[ω_n^8] then give the three non-zero elements in _n,2^[2].     



    Combining <ref> and <ref> show that the comparison map ε is surjective in all cases except potentially when (n,p) = (2,2). We currently do not know if the generator of the third, “anomalous” /2-summand that arises in the algebraic Picard group _2,2^ can be realized topologically. This question has also been studied in unpublished work of Hans-Werner Henn. 





 §.§ A toy case: height 1


For completeness and because we will make use of the following lemma later in the proof of <ref>, we briefly include a discussion of the case n=1.


There are isomorphisms 
    
    H^1_(_1,A_1^) ≅_(_p^∗,_p^) ≅_p    if  p>2; 
    _2 ⊕ (/2)^⊕ 2   if  p=2.



Recall that A_1≅ W(_p), equipped with its canonical action by the Galois group (_p/_p) ≅. We first claim that the inclusion _p^→ A_1^ induces an equivalence _p^≅ (A_1^)^h. Indeed, taking ϕ to be a topological generator of , what we need to prove is that there is a short exact sequence
    
    0→_p^→ W(_p)^  W(_p)^→ 0.

Using the compatible p-adic filtrations on _p and A_1, it suffices to show that the corresponding sequence of  associated graded pieces 
    
    0→_p →_p _p → 0

is short exact, which is indeed the case. It follows that there are isomorphisms
    
    H^1_(_1,A_1^) H^1((A_1^)^h(_p^∗×))  H^1((_p^)^h_p^∗) _(_p^∗,_p^)

The calculation of _(_p^∗,_p^∗∗) is routine.



    The calculation in the proof of <ref> recovers the computation of the algebraic Picard group at height 1 due to Hopkins, Mahowald, and Sadofsky <cit.>. Indeed, combined with <ref> and <ref>, we get the even algebraic Picard group _1,p^0,. The extension problem (<ref>) is then resolved as in the proof of <ref>. We obtain isomorphisms
        
    _1,p^≅_p ⊕/(2p-2)    if  p >2; 
    _2 ⊕ (/2)^⊕ 2   if  p = 2.

    For odd primes, the algebraic Picard group is generated by ε(Σ^1 𝕊_K(n)); for the even prime, the same class generates the _2-summand. Explicit generators for the 2-torsion part of _1,2^ can again be constructed as in the proof of <ref>; the original reference is <cit.>. For p>2, the comparison map gives an isomorphism _1,p≅_1,p^, while the exotic part for p=2 is a /2, see <cit.>. 






§ MODULI OF FORMAL GROUPS


We have reduced our main topological result to an algebraic one, namely <ref>, which describes the continuous cohomology group H^1_(_n,A_n^).  Its proof will require a dive into the theory of moduli of formal groups, especially those modern aspects of the theory which meet p-adic geometry and especially perfectoid spaces and diamonds.  The star player in the proof is _^n, the completion of the moduli stack of formal groups along the height n locus in characteristic p.  It enters the picture via <ref>, which states that _^n is the quotient of the formal scheme A_n by _n.  

In this section we offer the reader an exposition of the theory of moduli of formal groups and present two results required for the proof of <ref>.   Neither result is entirely original;  our work here is more or less a direct adaptation of material appearing in <cit.> and <cit.>.  Both results concern the “diamond generic fiber” of _^n, which we call 𝔐_n.  In a nutshell, 𝔐_n classifies formal groups of height n over perfectoid spaces of characteristic 0.

The first result is <ref>, which gives an alternative description of 𝔐_n in terms of linear algebra objects (“modifications of vector bundles on the Fargues–Fontaine curve”).  Faltings' isomorphism between the Lubin–Tate and Drinfeld towers appears within <ref>.

The second result is <ref>, which leverages the linear-algebraic description of 𝔐_n to construct a determinant morphism 𝔐_n→𝔐_1.  This result states that if H is a 1-dimensional formal group of height n over a perfectoid space S, there exists a functorially associated 1-dimensional formal group ⋀^n H of height 1 over S, whose associated linear algebra objects (the Tate and Dieudonné modules) are the top exterior powers of those of H.  Ultimately this result will help us understand the class of the determinant sphere in H^1_(_n,A_n^).




 §.§ The stack _^n of height n formal groups


For an introduction to the theory of moduli of formal groups aimed at the algebraic topologist, see <cit.>.  Unless specified otherwise, all formal groups in this discussion have dimension 1.

Let _ be the moduli stack of formal groups on the category of schemes with the fpqc topology.  Then _ admits a uniformization by an affine scheme L, where L=[t_1,t_2,…] is the Lazard ring.  The scheme L parameterizes formal group laws, which carry a choice of coordinate T.  For each n∈, let [n](T)∈ LT be the multiplication-by-n series in the universal formal group law over L.

Fix a prime number p.  The base change _⊗_(p) 
admits a well-known height stratification by closed subsets _FG^≥ n for 0≤ n≤∞, characterized as follows.  For n≥ 0, let v_n be the coefficient of T^p^n in [p](T).  In particular v_0=p.  For 1≤ n≤∞, the vanishing locus of (v_0,v_1,…,v_n-1) in L is coordinate-invariant, meaning that it descends to a closed substack of _⊗_(p), to wit _^≥ n.   We refer to the locally closed substack _^n_FG^≥ n+1∖_FG^≥ n as the moduli stack of formal groups of height n.  It is characterized this way:  a formal group H of height n over a characteristic p ring R is one for which the p-torsion H[p] is a locally free group scheme of rank p^n over R.  

Formal groups of height n obey an important rigidity condition with respect to isomorphisms. This is proven by Goerss in <cit.>, who attributes it to Lazard; see also <cit.>.

 
Let R be a ring of characteristic p, and let H,H' be formal groups over R of height n, where 1≤ n < ∞.  Let (H,H') denote the functor sending an R-algebra R' to the set of isomorphisms H⊗_R R'→ H'⊗_R R'.  Then (H,H') is represented by an affine scheme which is pro-finite étale over R.


For each 1≤ n≤∞, there exists a formal group H_n of height n over _p.  By  <ref> (or by choice of p-typical coordinate when n=∞), H_n is unique up to isomorphism.  H_∞ is the formal additive group, while for 1≤ n<∞, H_n is p-divisible.  Assume henceforth that 1≤ n<∞.  Recalling that _n=(H_n,_p), we have an isomorphism of stacks over _p:

    _^n  [_p/ _n].


For 1≤ n<∞, let _^n be the completion of _ along the locally closed substack _^n.  This is a formal stack over _p.  We may describe it by giving its values on rings in which p is nilpotent.  Equivalently, we may describe it by giving its values on admissible _p-algebras R, meaning that R is complete with respect to the topology defined by an ideal containing p.  For a ring R in which p is nilpotent (respectively, an admissible _p-algebra R), the R-points of _^n classify formal groups H over R such that H⊗_R R/I has height n for some nilpotent ideal (respectively, an open ideal) I⊂ R containing p.  

The stack _^n has a standard presentation as a pro-étale quotient of a finite-type formal scheme.  Let 

    _n=(H_n)

be the deformation space of H_n.  This means that for an Artinian local ring R with residue field _p, the R-points of _n classify 
pairs (H,ρ), where H is a formal group over R, and ρ H_n→ H⊗_R _p is an isomorphism. 
Note that such an R is necessarily an algebra over W=W(_p).  
Then _n is pro-representable by an affine formal scheme:  _n= A_n, where A_n is an admissible _p-algebra, isomorphic to Wu_1,…,u_n-1 with its (p,u_1,…,u_n-1)-adic topology.  The formal scheme _n admits an action of _n, which is compatible with the action of (_p/_p) on W.

    We have an isomorphism of stacks of formal schemes in the pro-étale topology over _p:

    _^n [_n/_n].



<Ref> is standard, but we discuss its proof in some detail as a means of building up the necessary machinery for what follows.  To do so, it will be important to have a moduli interpretation for the R-points of _n, where R is a general W-algebra (not Artinian or even Noetherian) in which p is nilpotent.  This interpretation, appearing in the moduli spaces studied by Rapoport–Zink <cit.>, shifts the focus from formal groups of finite height to the larger category of p-divisible groups, which we review in the next subsection.




 §.§ p-divisible groups


These were introduced by Tate <cit.>.  

 Let R be a ring.  A p-divisible group (or Barsotti–Tate group) of height n over R is an inductive system G=_ν≥ 1 G^ν of commutative group schemes G^ν over R whose underlying scheme is locally free of rank p^nν over R.  It is required that G^ν→ G^ν+1 induces an isomorphism of G^ν with the p^ν-torsion in G^ν+1.  Let _R denote the category of p-divisible groups over R, with the evident notion of morphism; then _R is a _(p)-linear category.  


The same axioms applied to an abstract group would force G=(_p/_p)^⊕ n.  Indeed, for whatever ring R, the constant group (_p/_p)_R=_ν≥ 1  (p^-ν/)_R is an example of a p-divisible group.  


There is a close link between formal groups and p-divisible groups.  Let R be an admissible _p-algebra, and let H/R be a formal group representing an R-point of _^n:  thus there is an open ideal I⊂ R containing p such that H⊗_R R/I has height n.  Locally on R we may choose a coordinate T for H;  then in (R/I)T, the multiplication-by-p series is a unit multiple of T^p^n.  It follows that the torsion H[p^ν]= RT/[p^ν]_H(T) is a connected free group scheme over R of rank p^ν h, and that (globally on R) H[p^∞]:=_ν≥ 1 H[p^ν] is a connected p-divisible group over R.  

As an example, if H is the formal multiplicative group over R, then H[p^∞]=μ_p^∞ is the group of pth power roots of unity over R.

A theorem of Tate <cit.> states that if R is a Noetherian local ring of residue characteristic p, then the functor H↦ H[p^∞] is an equivalence between the category of formal groups over R (of whatever dimension) which are p-divisible (meaning that the p-torsion is a locally free group scheme) and the full subcategory of _R whose objects are connected p-divisible groups.  

A p-divisible group G=_ν G^ν over an admissible _p-algebra R determines a functor of points valued in abelian groups:  if S is an admissible R-algebra with ideal of definition I, define G^ν(S)=_m G^ν(S/I^m) and G(S)=_ν G^ν(S).  The Lie algebra of G is defined via the formula 
    
    G=(G(R[ε]/ε^2)→ G(R)).

It is a locally free R-module, whose rank is by definition the dimension of G.  If G=H[p^∞] for a finite-height formal group H over R, then the functor of points and Lie algebra for G agree with the analogous constructions for H.  Via the functor of points it is possible to embed both formal groups and p-divisible groups as full subcategories of the category of fppf sheaves of abelian groups.  From that point of view a p-divisible formal group H is identified with its p-divisible group H[p^∞];  nonetheless we shall continue to notationally distinguish the two sorts of objects.

  Let G and G' be p-divisible groups over a ring R.  An isogeny f G→ G' between p-divisible groups over R is a morphism whose kernel is a finite locally free group scheme over R.  If f is an isogeny, its kernel f has rank p^n, where n is a locally constant function on R;  we call n=(f) the height of f.  Let _R^0=_R⊗ and call this the category of p-divisible groups up to isogeny. An isomorphism in _R^0 is called a quasi-isogeny.


Thus the multiplication-by-p map on G is an isogeny whose height equals the height of G.  Note that if f G→ G' is an isogeny of height n, then f factors through multiplication by p^n, so that f is also a quasi-isogeny.  Conversely, a quasi-isogeny becomes an isogeny when multiplied by a sufficient power of p.  There is an extension of the notion of height to quasi-isogenies, so that if f is a quasi-isogeny, its height (f) is a (possibly negative) integer, and if the composition f'∘ f of isogenies is defined, then (f'∘ f)=(f')+(f).  

Quasi-isogenies satisfy a rigidity property similar to what appears in <ref>.  For p-divisible groups G and G' over a ring R, let (G,G') be the functor which associates to an R-algebra S the set (G_S,G'_S), meaning the set of quasi-isogenies G⊗_R S→ G'⊗_R S.  Then (G,G') is formally étale over R:


     Let R be a ring in which p is nilpotent, and let I⊂ R be a nilpotent ideal.  Let G and G' be p-divisible groups over R.  The natural map (G,G')→(G_R/I,G'_R/I) is a bijection. 


Consequently, given p-divisible groups G and G' over R, an isogeny G⊗_R R/I→ G'⊗_R R/I will lift to an isogeny G→ G' once multiplied by a sufficiently high power of p. 



 §.§ Lubin–Tate space as a moduli space of p-divisible groups


Recall that H_n is the (unique up to isomorphism) 1-dimensional formal group of height n over _p.  Let

    G_n = H_n[p^∞],

so that G_n is a p-divisible group over _p of height n.  Let _D_n= H_n G_n, and let D_n=_D_n[1/p].  Then D_n is a division algebra over _p of invariant 1/n, and _D_n is its ring of integers.  Therefore the group of automorphisms of G_n (as an object in __p) is _D_n^∗, and the group of quasi-isogenies of G_n (that is, automorphisms in ^0__p) is D_n^∗.  By <ref>, the functor (G_n,G_n) is formally étale over an algebraically closed field; it is therefore the constant group scheme associated to D_n^∗.  

We can view the Morava stabilizer group _n as the group of automorphisms of the pair (G_n,_p).  Let _n^0 denote the group of automorphisms of the pair (G_n,_p) in the isogeny category.  Explicitly, an element of _n^0 is a pair (τ,u), where τ∈Γ__p, and u G_n→τ^*G_n is a quasi-isogeny.

The groups discussed here fit into the diagram

    0 [d]     0 [d]       
    
    0 [r]    _D_n^∗[r] [d]     D_n^∗[r]^[d]    [r]     0 
    
    0 [r]    _n [r] [d]    _n^0 [r]^[d]    [r]     0
       Γ__p[d] [r]_=   Γ__p[d]       
        0     0

in which rows and columns are exact, and Γ__p=(_p/_p).  

The following definition is a special case of the constructions appearing in <cit.>. 

 Let _n be the functor which inputs a ring R in which p is nilpotent and outputs the set of triples (G,ι,ρ), where G/R is a p-divisible group, ι is a ring homomorphism _p→ R/p, and 

    ρ G_n ⊗__p,ι R/p→ G⊗_R R/p

is a quasi-isogeny. 


The homomorphism ι determines a homomorphism W→ R, so that there is a natural transformation _n→ W.  
There is a decomposition of functors over W:

    _n=∐_h∈_n^(h),
 where _n^(h) classifies those (H,ι,ρ) where ρ has constant height h.  

There is an action of the group _n on _n lying over the action of Γ__p on W:    An element g=(τ,u)∈ D_n^∗ sends (G,ι,ρ) to (G,ι∘τ^-1,ρ∘ u^-1).

Under this action, an element g∈ D_n^∗ induces an isomorphism _n^(h)_n^(h-(g)).  Therefore all components _n^(h) are isomorphic, and in order to describe _n with its _n^0-action, it suffices to describe _n^(0) with its _n-action.  

Recall that A_n is the universal deformation of H_n.  Let 𝔪 be the maximal ideal of A_n, and let H_n^ be the universal deformation of H_n, so that there is an isomorphism H_n H_n^⊗_A_n A_n/𝔪.  Let G_n^=H_n^[p^∞], so that we have an isomorphism 

    G_n G_n^⊗_A_n A_n/𝔪.

For each k≥ 1, we apply <ref> over the ring A_n/(p,𝔪^k);  the above isomorphism lifts to a quasi-isogeny

    ρ_k G_n⊗_A_n A_n/(p,𝔪^k) → G_n^⊗_A_n A_n/(p,𝔪^k)

of height 0.  Thus one obtains a morphism A_n/𝔪^k→_n^(0) for all k≥ 1, or all the same, a morphism _n= A_n→_n^(0).  This morphism is _n-equivariant.  In fact:


  
 
    The morphism _n→_n^(0) is an isomorphism.
 


We can now complete the proof of  <ref>, which claims that _^n [_n/_n]. The universal deformation of H_n determines a _n-equivariant morphism _n→_^n.  Composing with the isomorphism in <ref>, we obtain a _n-equivariant morphism _n^(0)→_^n, which extends to a _n^0-equivariant morphism _n→_^n.  To complete the proof, we need to show that this morphism is a pro-étale _n^0-torsor.


Suppose R is a ring in which p is nilpotent, and that we are given a morphism R→_^n, corresponding to a formal group H over R
such that H⊗_R R/I has height n for a nilpotent ideal I containing p. The fiber F:= R×__^n_n is the functor which sends an R-algebra S to the set of pairs (ι,ρ), where ι_p→ S/p is a ring homomorphism, and 

    ρ G_n⊗__p,ι S/p → H[p^∞]⊗_R S/p

is a quasi-isogeny.  Our objective is to show that F is a pro-étale _n^0-torsor over R.

First we show that F→ R has pro-étale local sections.  Let R'=R⊗__p W, so that R'→ R is pro-étale.  By <ref>, 

    (H_n⊗__p R'/I, H⊗_R R'/I)

is represented by an affine scheme which is pro-finite étale and surjective over R'/I;  let S be the lift of this affine scheme to R'.    
Let H_n⊗__p S/I→ H⊗_R S/I be the universal isomorphism. By <ref> applied to the ring S/p, the associated isomorphism of p-divisible groups lifts to a quasi-isogeny G_n⊗__p S/p→ H[p^∞]⊗_R S/p of height 0, determining a morphism S→ F.  Therefore F→ R admits pro-étale local sections.

Any two sections of F over an R-algebra S differ by a pair (τ,u), where τ∈Γ__p and u G_n⊗__p S→τ^*G_n⊗__p S is a quasi-isogeny.  The quasi-isogeny u represents an S-point of the _p-scheme Q(G_n,τ^*G_n).  By <ref>, Q(G_n,τ^*G_n) is formally étale over _p, which (since the latter is algebraically closed) must be the constant scheme associated to the set of quasi-isogenies G_n→τ^*G_n.  Thus any two sections of F differ by an S-point of the constant scheme associated with _n^0. We have shown that F→ R is a _n^0-torsor, which completes the proof.




 §.§ Dieudonné modules


The theory of Dieudonné modules <cit.> provides a functor from p-divisible groups to linear algebra objects.  It is roughly analogous to the functor that sends an abelian variety to its first de Rham cohomology group.  We give here a brief summary.  

Let R be a p-adically complete ring, and let G be a p-divisible group of height n over R.   We can think of G as defining an fppf sheaf on the category of R-algebras.  Let R'→ R be a p-adically complete PD thickening, meaning that R' is p-adically complete and R'→ R is a surjection of rings whose kernel is equipped with a divided power structure.  Let G' be any lift of G to R', and let EG'→ G' be the universal vector extension of G'.  (A vector extension of G' is a surjective morphism of sheaves of abelian groups whose kernel is isomorphic to a direct sum of copies of the additive group.)  The (covariant) Dieudonné crystal is defined as
    
    M(G)(R'→ R) =  EG'.

It is a projective R'-module of rank n.  The Dieudonné crystal does not depend on the choice of G', in the strong sense that the formation of M(G)(R'→ R) is functorial in both G and R'.

We will only deal with the case that R/p is a semiperfect ring, which means that the  Frobenius endomorphism ϕ R/p→ R/p is surjective.  In this case there is a universal p-adically complete PD thickening A_(R)→ R.  The construction goes this way:  let R^♭=_ϕ R/p be the “tilt” of R;  there is a natural surjective map W(R^♭)→ R, and then A_(R) is defined as the PD-hull of this surjection.  In this situation we simply write 

    M(G)=M(G)(A_(R)→ R),

and call this the Dieudonné module of G.  Considering the quotient EG'→ G', we find an R-linear surjective map

    β_G  M(G)⊗_A_(R) R→ G.

Evidently A_(R) only depends on R/p, and M(G) only depends on G⊗_R R/p.  We also write B_^+(R)=A_(R)[1/p], so that M(G)[1/p] is a projective B_^+(R)-module.

The Dieudonné module M(G) comes with two important structures:  Frobenius and Verschiebung.  To define these, we first assume that R is a semiperfect ring of characteristic p, with Frobenius map ϕ.  Let G^(p)=G⊗_R,ϕ R, and let F G→ G^(p) be the relative Frobenius morphism.  Let V G^(p)→ G be Cartier dual to the Frobenius morphism G^∨→ G^∨,(p);  we have the fundamental relations VF=p in G and FV=p in G^(p).

The Frobenius endomorphism induces by functoriality an endomorphism of A_(R) which we continue to call ϕ.  The morphisms F and V induce maps on Dieudonné modules as in the following definition. 

 Let R be p-adically complete ring such that R/p is semiperfect.  The category of Dieudonné modules _R is the category of projective A_(R)-modules M together with A_(R)-linear maps
F M⊗_A_(R),ϕ A_(R) → M and V M→ M⊗_A_(R),ϕ A_(R) satisfying VF=p and FV=p.  We let _R^0=_R⊗ be the corresponding isogeny category.

We write G↦ M(G) (resp., M^0(G)) for the Dieudonné module functor _R→_R (resp., _R^0→_R^0). 


  In fact F G→ G^(p) induces the map V on M(G) and vice versa.  This notational awkwardness is avoided when one uses the contravariant Dieudonné module.   For G=(_p/_p)_R the constant p-divisible group, the relative Frobenius on G is induced from the identity on G, and so M(G)=A_e for a basis element e satisfying F(e⊗ 1)=pe and V(e)=e⊗ 1.  
 


    

In the special case that R=k is a perfect field of characteristic p, we have A_(k)=W(k), since the kernel of W(k)→ k is (p) and p^n/n!∈_p for all n≥ 0.  In this case ϕ is an automorphism of A_(k).  A classical result of Fontaine <cit.> states that the Dieudonné module functor _k→_k is an equivalence.   In retrospect we may define the p-divisible group G_n∈__p by its Dieudonné module:  M(G_n)=_p^⊕ n⊗__p W(_p), and F=F_0⊗ϕ, where F_0∈_p^⊕ n is the matrix

    [         1; p        ;   p      ;     ⋱;   p   ].



For more general semiperfect rings R, the Dieudonné module functor can fail to be an equivalence, but under a weak assumption on R the functor is at least fully faithful up to isogeny.

  Assume R is the quotient of a perfect ring of characteristic p by a finitely generated ideal.  Then the Dieudonné module functor 

    M^0_R^0→_R^0
 
is fully faithful.





 §.§ The Fargues–Fontaine curve


Let R be a perfectoid algebra over _p.  This means that R is a Banach _p-algebra whose subring of power-bounded elements R^∘ is p-adically complete, and there exists an element p^1/p∈ R^∘ which is (up to a unit) a pth root of p, such that the pth power map R^∘/p^1/p→ R^∘/p is an isomorphism of rings.  It follows from this that R^∘/p is semiperfect.  For instance, if C is a complete algebraically closed field containing _p, then C is a perfectoid algebra.  

If a p-divisible group G over R^∘ is given, its Dieudonné module M(G) is a projective A_(R^∘)-module equipped with a ϕ-semilinear action of Frobenius.  Therefore it ought to descend to a vector bundle over the quotient of A_(R^∘) by ϕ.  The Fargues–Fontaine curve is morally something like this quotient, although one must invert p.  

Define a graded _p-algebra by
    
    P_R = ⊕_d≥ 0 B_^+(R^∘)^ϕ=p^d,

and a scheme X_R by 
    
    X_R =  P_R.

The quotient map B_^+(R^∘)→ R induces a closed immersion i R→ X_R.  


The foundational work <cit.> is concerned with the “absolute curve” X_C appearing in the case that C is an algebraically closed perfectoid field.  Despite not being finitely generated over a field, X_C was discovered to share properties with the Riemann sphere:  it is a Noetherian scheme of dimension 1, and furthermore the complement in X_C of any closed point is the spectrum of a PID.  

The construction of the Fargues–Fontaine curve globalizes, so that we have a scheme X_S for any perfectoid space S.  The reader is warned that there is no morphism X_S→ S.  Nonetheless, it is still something like a family of curves, in that one gets an absolute curve for each rank 1 geometric point of S.

What we have presented here is the “schematic” curve;  there is also an “adic” version which is more suitable to applications in p-adic geometry.  The two curves have the same category of quasi-coherent sheaves, by “GAGA for the curve” <cit.>.


There is a functor M↦(M) from rational Dieudonné modules over R to quasi-coherent sheaves on X_R.  For an object M of ^0_R, think of F as a ϕ-semilinear map ϕ M→ M, and then define ℰ(M) as the quasi-coherent sheaf on X_R associated to the graded P_R-module

    ⊕_d≥ 0 M^ϕ=p^d+1.

The definition has been normalized so that if M=M^0(_p/_p) is the rational Dieudonné module of the constant p-divisible group, then (M) is the trivial line bundle _X_R.  If M is the rational Dieudonné module of our fixed p-divisible group G_n over _p, and if a ring homomoprhism _p→ R^∘/p is given, we obtain a vector bundle (M) of rank n over X_R, and we denote this _X_R(1/n), or simply (1/n) if the base curve X_R is understood.

A major accomplishment of <cit.> is the complete classification of vector bundles on the absolute curve X_C, by means of a Harder–Narasimhan formalism.  Every vector bundle is isomorphic to a direct sum of vector bundles (λ) parameterized by rational numbers λ (“slopes”).  For each λ, the degree and rank of (λ) are the numerator and denominator of λ in lowest terms, and the automorphism group of (λ) is the group of units in a central division algebra over _p of invariant λ.  In particular, every automorphism of (1/n) arises from an automophism of G_n in the isogeny category:  (1/n) D_n^∗. 

The study of families of vector bundles on the Fargues–Fontaine curve (that is, vector bundles over X_S for a general perfectoid space S) is very rich, and has been shockingly fruitful in attacking the local Langlands correspondence <cit.>.  An important first result <cit.> is that vector bundles form a stack for the pro-étale topology (in fact this is true for the finer v-topology). Another result concerns families whose geometric fibers are all “isoclinic” (admitting only one slope); these are vastly simpler.

  Let S be a perfectoid space over W(_p).  Let λ be a rational number whose denominator d≥ 1 (in lowest terms) is a divisor of n.  The following groupoids are equivalent:
 

  * Vector bundles of rank n over X_S which are everywhere on S isoclinic of slope λ, which is to say isomorphic to _0=(λ)^⊕ (n/d).

  * Pro-étale J_λ-torsors over S, where J_λ is the group of units in the unique central simple algebra over _p of degree n and invariant λ.  

The equivalence sends a vector bundle  to the sheaf (_0,) whose R'-points (for a perfectoid R-algebra R') parameterize isomorphisms between the vector bundles after pullback to X_R'.


We spell out the consequences of Proposition <ref> in the two cases relevant to us:

In the case λ=0, we have J_λ=_n(_p).  Note that a pro-étale _n(_p)-torsor is the same thing as a _p-local system of rank n.  If T is a _p-local system of rank n, let us write T⊗__p_X_S for the vector bundle on X_S corresponding to the _p-local system  T[1/p].  

In the case λ=1/n, we have J_λ=D_n^∗.  As a corollary of the proposition, suppose S is a general perfectoid space, with no structure morphism to W(_p).  Then there is an equivalence of groupoids between (a) vector bundles of rank n on X_S which are everywhere of slope 1/n, and (b) pro-étale _n^0-torsors over S admitting a map to (K,W); or all the same, sections of the stack [(K,W)^♢/_n^0] over S.  




 §.§ p-divisible groups over a perfectoid ring


Keep the assumption that R is a perfectoid _p-algebra, and let S=(R,R^∘).  Let G be a p-divisible group of height n over R^∘.  We gather here some of the structures associated with G which live over the Fargues–Fontaine curve X_S.  

First we have the 
Tate module

    TG = _ν G[p^ν](R^∘) __R(_p/_p,G).

Here we write _p/_p for the constant p-divisible group over R^∘.  
Since R^∘ is integrally closed in R and each G[p^ν] is finite over R^∘, the natural map G[p^ν](R^∘)→ G[p^ν](R) is an isomorphism.  On the other hand G[p^ν]_R is an étale group scheme of rank p^nν over R.  This shows that TG can be given the structure of a _p-local system of rank n over S_.  Let TG⊗__p_X_S be the corresponding vector bundle on X_S which is everywhere on S isomorphic to ^⊕ n.
 
On the other hand we have the rational Dieudonné module M^0(G), which gives rise to a quasi-coherent sheaf on X_S, which we notate (G). 
If G arises from an R^∘-point of _^n, it means there exists pro-étale locally on R^∘/p a quasi-isogeny G_n→ G, and thus an isomorphism M^0(G_n)→ M^0(G).  Translating to vector bundles, we find that (G) is a vector bundle on X_S which is everywhere on S isomorphic to (1/n).  

A morphism _p/_p→ G representing a section of TG induces a map of vector bundles (_p/_p)→(G), which is just a global section of (G) since (_p/_p) is the trivial line bundle.  We have defined a map TG→ H^0((_p/_p)) of sheaves of groups on the pro-étale site of S;  this is adjoint to a map of vector bundles

    α_G    TG⊗__p_X_S→(G).


  Let R be a perfectoid algebra over _p, and let S=(R,R^∘).  Let G be a p-divisible group over R^∘.  The morphism α_G fits into an exact sequence of _X_S-modules:
    
    0→ TG⊗__p_X_Sα_G→ℰ(G) → i_* G[1/p]→ 0,

where i R→ X_S is the closed immersion defined above.



The injectivity of the map α_G comes from the faithfulness part of <ref>.  

The surjective map (G)→ i_* G[1/p] arises via adjunction from the surjective map i^*(G)=M^0(G)⊗_B_^+(R^∘) R → G[1/p] from (<ref>).  

Next we claim that the sequence in (<ref>) is actually a complex.  A section s of TG⊗__p_X_R is represented by a morphism _p/_p→ G.  The image of s in (G) is the image of the canonical generator of (_p/_p)= under (_p/_p)→(G).  We now use the commutativity of

    (_p/_p) [r] [d]     i_*_p/_p [d] 
    (G) [r]     i_* G

together with _p/_p=0 to conclude that the image of s in i_* G is zero.  

The exactness of (<ref>) can be checked on every geometric point of S, so let us assume R=C is an algebraically closed field.  The result follows from a numerical argument:  since α_G is an injective map of vector bundles, its cokernel is torsion of degree equal to G;  for this to surject onto i_* G must mean that the cokernel is exactly i_* G.





 §.§ The diamond stack _n


We return now to the study of the stack _^n over _p.  
Our next order of business is to pass from _^n to its generic fiber to obtain an analytic object living over _p.  

 Suppose X is a presheaf of sets or groupoids on the category of admissible _p-algebras.  The diamond generic fiber X^♢__p of X is the sheafification of the presheaf

    (R,R^+)↦ X(R^+)

on perfectoid (_p,_p)-algebras with respect to the pro-étale topology.

Thus for a perfectoid space S over (_p,_p), an object of X__p^♢(S) is given by a pro-étale cover of S by affinoids (R_i,R^+_i), and for each i an object of X(R^+_i), together with a descent datum to S.

 
Let 
    
    _n = (_FG^n)^♢__p,

and call this the diamond stack of 1-dimensional formal groups of height n.  


Suppose S is a perfectoid space over (_p,_p), and let H be an S-point of _n.  To specify H is to give the data of a covering of S_ by affinoid perfectoid spaces (R,R^+) and a formal group over each R^+ representing an R^+-point of _^n.  We shall refer to H as a formal group of height n over S_.  

We remark here on the following “partial properness” property of _n.  Suppose (R,R^+) is an affinoid perfectoid space, so that R^+ is an open integrally closed subring of the p-adically complete ring R^∘.  We claim that the natural map _^n(R^+)→_^n(R^∘) is an equivalence.  By <ref>, it is enough to check that _n(R^+)→_n(R^∘) is a bijection.  This is true because _n(R^∘) is the (n-1)-fold cartesian product of the set of topologically nilpotent elements of R^∘, and these all lie in R^+ because R^+ is integrally closed in R^∘.  We conclude that _n is partially proper, meaning that its groupoid of (R,R^+)-points depends only on R and not on the subring R^+⊂ R.

Suppose H is a formal group of height n over S_.  Since quasi-coherent -modules satisfy pro-étale descent, H[1/p] defines an invertible -module on the analytic site S_.  

  The following stacks are equivalent:
    
        
  * _n;
        
  * the stack _n whose points over S=(R,R^∘) classify modifications
            
    0→ T⊗_X_S→ℰ→ i_*L → 0,

        where:
            
                
  * T is a _p-local system on S_,
                
  * ℰ is a vector bundle on X_S which is isomorphic to (1/n) everywhere on S,
                
  * L is a locally free _S-module of rank 1;
            
        
  * the quotient stack [_n,K^♢/_n];
        
  * the quotient stack [^n-1,♢/_n(_p)], where ^n-1 is Drinfeld's symmetric space.
    
The equivalence between _n and _n sends a formal group H to the exact sequence associated in <ref> to G=H[p^∞].  Furthermore, let H^ be the universal formal group over (_^n)^♢__p, so that H^[1/p] is an invertible -module on this stack.  With respect to the isomorphism to [^n-1,♢/_n(_p)], H^[1/p] corresponds to the restriction of the _n(_p)-equivariant tautological line bundle (1) from ^n-1__p to ^n-1.  



  The equivalence between [_n,K^♢/_n] and [^n-1,♢/_n(_p)] is the “isomorphism between the Lubin–Tate and Drinfeld towers”.  We phrase it here through the medium of modifications of vector bundles on the Fargues–Fontaine curve.  Detailed proofs can be found in <cit.>. 

The equivalence between (1) and (3) is a consequence of <ref>.  
  
Let W=W(_p) and K=W[1/p].  Let _n be the sheaf whose value on an affinoid perfectoid space S=(R,R^+) over (K,W) is the set of modifications
    
    0→ T⊗__p_X_S→_X_S(1/n)→ i_*L→ 0,

    where T is a pro-étale _p-local system on S, and L is a locally free _S-module of rank 1.  
Then _n is an example of a local shtuka space as introduced in <cit.>.  There is a morphism _n^♢→_n lying over (K,W), described as follows:  if R is a perfectoid K-algebra, and if (G,ρ) represents a (R,R^∘)-point of _n^♢, then ρ induces an isomorphism _X_S(1/n)→ℰ(G), and then <ref> produces a (R,R^∘)-point of _n.  The map _n^♢→_n is an equivalence:  this is a special case of <cit.>.

There is an obvious morphism _n→_n, which is a torsor for the automorphism group of the pair (_p,(1/n)), which is exactly _n^0.  Therefore

    _n [_n/_n^0] [_n^♢/_n^0] [_n^♢/_n],
 
where the last isomorphism derives from <ref>.  This is the equivalence between (2) and (3).  


For the equivalence between (2) _n and (4) [^n-1/_n(_p)]:  We interpret S-points of projective space ℙ^n-1 to mean invertible _S-modules ℓ which are direct summands of _S^⊕ n.  Observe that such an ℓ determines
a vector bundle  containing _X_S^⊕ n, whose sections are the same except they are allowed to have a simple pole along the hyperplane determined by ℓ.  Thus ℓ determines a modification 
    
    0→_X_S^⊕ n→ℰ→ i_*L → 0,

where L is an invertible _S-module.  (In fact L is a twist of ℓ by the tangent line to the curve X_C along the point i; this tangent line is canonically a Tate twist of C.)  The kernel of _S^⊕ n=i^*_X_S^⊕ n→ i^*ℰ is exactly ℓ.  

Let (C,C^∘) be a geometric point of S, and let ℓ_C⊂ C^n and ℰ_C be the pullbacks of those objects to C.  The following statements are all equivalent:


    
  * The line ℓ_C is contained in a _p-rational hyperplane.
    
  * There exists a decomposition of _p^n as the direct sum of two nonzero _p-linear subspaces, such that ℓ is contained in one of them.
    
  * There exists a decomposition of _X_C^⊕ n as the direct sum of two nonzero sub-bundles of slope 0 (i.e., they are both direct sums of _X_C), such that ℓ is contained in the fiber of one of them.  (The equivalence of this statement with the last one boils down to H^0(X_C,_X_C)=_p.)
    
  * The modified vector bundle  admits a trivial summand.
    
  * The modified vector bundle  is not isomorphic to (1/n).     
For the equivalence of the last two statements:  a priori  has degree 1 and rank n, and all its slopes are nonnegative because it contains ^⊕ n as a maximal rank sub-bundle.  Therefore  must be isomorphic to (1/i)⊕^⊕ (n-i) for some i=1,…,n.  This contains a nontrivial -summand if and only if i<n.       

A morphism S→ℙ^n-1 determines a line ℓ, which determines in turn a modification;  the above discussion shows that the modified vector bundle  is everywhere isomorphic to (1/n) if and only if S→ℙ^n-1 factors through Drinfeld symmetric space ^n-1, defined as the complement in ℙ^n-1 of all _p-rational hyperplanes.  

Thus we find a morphism ℋ^n-1→_n carrying a line ℓ to its corresponding modification and forgetting the trivialization of the _p-local system T of rank n.  This morphism a torsor for the group _n(_p).





 §.§ The determinant map _n→_1.


To access the determinant sphere 𝕊_K(n)[] as a Picard class on _n, we will need to understand the role of the determinant in the study of 1-dimensional formal groups.  

  
There exists a determinant morphism

    _n→_1

making the following diagram commute:
    
    [_n,K^♢/_n] [rr]^-[d]__      _n [rr]^-[d]_       [^n-1,♢/_n(_p) ] [d]^_
    
            [_1,K^♢/_1] [rr]_-      _1 [rr]_-       [^0,♢/_p^∗].

In the diagram:

    
  * the map labeled _ is induced from:
    
        
  * the structure map _n,K→_1,K(K,W);
        
  * the determinant map _n→_1, explained in the remarks below,
    
    
  * the map labeled _ is induced from:
    
        
  * the structure map ^n-1→^0(_p,_p);
        
  * the determinant map _n(_p)→_p^∗.
    
    


Before beginning the proof, we offer some remarks.  For a perfectoid space S over (_p,_p), the S-points of the morphism _n→_1 should input a formal group H of dimension 1 and height n over S_, and output a formal group ⋀^nH over S_ of dimension 1 and height 1. The crucial computation that makes this work comes from the case of Dieudonné modules over _p:  if M(G_n) is the Dieudonné module of G_n as described in <ref>, then the top exterior power ⋀_W(_p)^n M(G_n) is isomorphic to M(G_1).  Consequently we have an isomorphism of line bundles on X_S for any perfectoid space S:  ⋀^n _X_S(1/n)_X_S(1).  

Considering the action of _n=(G_n,_p) on ⋀_W(_p)^n M(G_n) M(G_1), we find a determinant map
    
    _n→_1.

This map is isomorphic to the profinite completion of the reduced norm map D_n^∗→_p^∗, where D_n= G_n ⊗__p_p.  

 One might wonder whether general exterior powers exist beyond the determinant.  
    For a general p-divisible group G of height n over a perfect field k of characteristic p, the exterior powers ⋀^i_W(k) M(G) are Dieudonné modules if and only if G≤ 1, so one can only expect exterior powers of G itself to exist in this case.  A  construction of exterior powers of p-divisible groups of dimension ≤ 1 over the ring of integers in a nonarchimedean local field appeared previously in <cit.>.


 
Let S be a perfectoid space over (_p,_p), and let H be a formal group of dimension 1 and height n over S_, representing an S-point of _n.  Let 

    0 → T⊗__p_X_Sα_H→(H)→ i_* L→ 0

represent the corresponding S-point of _n.  

The determinant of α_H is an injective morphism of line bundles:

    α_H⋀^n__pT ⊗__p_X_S→⋀^n (H),

and ⋀^n(H) is locally isomorphic to _X_S(1).  The cokernel of α_H has degree 1 at every geometric point of S and is supported on the image of i, so it can be none other than the line bundle i_*i^*⋀^n.  Therefore α_H determines an S-point of _1, corresponding to a formal group ⋀^n H of height 1 over S_.  

For any n≥ 1, the stack [(_p,_p)^♢/_n(_p)] classifies _p-local systems of rank n. The commutativity of the right square of the diagram is exactly the statement that there is a functorial isomorphism of _1(_p)=_p^∗-torsors

    T(⋀^n H)⋀^n__p TH,

and this is in turn an artifact of our construction of ⋀^n H.  

Similarly, the commutativity of the left square of the diagram traces to the statement that there is a functorial isomorphism of vector bundles

    (⋀^n H)⋀^n (H);

once again this is an artifact of our construction.






§ THE FUNDAMENTAL EXACT SEQUENCE


The goal of this section is to construct what we dub the fundamental exact sequence, describing the group H^1_(_n,A_n^∗∗) we are interested in in terms of H^1_(_1,A_1^∗∗) and H^1_(_C^n-1,^∗∗)^Π_n. The key inputs are the isomorphism of towers <ref>, the determinant map, as well as a logarithm exact sequences for the sheaf of principal units (<ref>) to compute the terms. Throughout, special care is needed for the “anomalous” case n=p=2. 

The objects of study in this section are closely related to those considered by Ertl, Gilles, and Nizioł in <cit.>, whose methods inspired the approach in this section. 




 §.§ The sheaf of principal units ^∗∗


As a general reference for the material we recall in this subsection, we refer the interested reader to <cit.> and <cit.>.

For a topological ring A, let A^∘ be the subset of power-bounded elements, and let A^∘∘⊂ A^∘ be the subset of topologically nilpotent elements. Let K be a nonarchimedean field of characteristic (0,p).  This means that K is complete with respect to a real-valued ultrametric  which satisfies 0<p<1.  
In particular K contains _p as a subfield.  The subset K^∘ (also called K^+ or _K) is a valuation subring of K, equal to the set of all f∈ K with f≤ 1.  

Recall the notion of a Huber pair (A,A^+) over (K,K^+).  In this situation A is a topological K-algebra containing an open subring A_0 (a ring of definition) which carries the p-adic topology, and A^+⊂ A is an open and integrally closed K^+-subalgebra contained in A^∘.  The pair (A,A^+) is complete if A contains a p-adically complete ring of definition.

 Let (A,A^+) be a complete Huber pair over (K,K^+).  

    
  * For f∈ A, there exists n≥ 0 such that p^nf∈ A^+.
    
  * pA^∘⊂ A^+.
    
  * A^∘∘ = (pA^+), the radical of pA^+.
    
  * 1+A^∘∘ is an open subgroup of A^∗.  



 


  * Let f∈ A.  Since p^n→ 0 in A_0⊂ A, and multiplication in A is continuous, we have p^nf→ 0.   Since A^+ is open we have p^nf∈ A^+ for n large enough. 

  * If f∈ A^∘, the sequence f^n is bounded, so that (pf)^n→ 0.  For n large enough (pf)^n∈ A^+, and then pf∈ A^+ since A^+ is integrally closed.

  * If f∈ A^∘∘, then f^n→ 0, so for n large enough f^n∈ pA^+.  (Note that pA^+⊂ A is open since multiplication by p^-1 is continuous on A.) This implies f∈ A^+ (once again since A^+ is integrally closed) and so f∈(pA^+).  For the reverse inclusion, it is enough to show pA^+⊂ A^∘∘, and indeed pA^+⊂ pA^∘⊂ A^∘∘ since p is topologically nilpotent.

  * The subset 1+A^∘∘ is clearly closed under multiplication.  To see that it contains inverses, let x∈ A^∘∘, and note that (1+x)^-1=1-x+x^2-⋯ converges to an element of 1+A^∘∘.  It is open because it contains the neighborhood 1+pA_0 of 1.  


In the context of the lemma we write 

    A^∗∗=1+A^∘∘,

and we call this the group of principal units of A.  

Let X be an adic space over (K,K^+).  Thus X is covered by affinoid adic spaces (A,A^+), where (A,A^+) is a complete Huber pair over (K,K^+).  For x∈ X, let K_x denote the completed residue field of x, and let K_x^+ denote its valuation ring.  Then (K_x,K_x^+) is an affinoid field.  If f is an analytic function defined in a neighborhood of x, let f(x) denote the image of f in K_x.

Recall the topologies X_,X_,X_ on X, listed here in increasing order of fineness.  Each topology has a basis consisting of affinoid objects U→ X which are open immersions in the case of X_, étale morphisms in the case of X_, and pro-étale morphisms from perfectoid affinoids in the case of X_.  

Each of the three topologies carries a structure sheaf, written _X or simply .  In the case of X_, what we mean is the completed structure sheaf (written  in <cit.>).  Let ^∗⊂ be the sheaf of invertible elements.  For each topology t∈,,, the first cohomology group _t X:=H^1_t(X,^∗) classifies invertible -modules on X_t.  As one has descent of -modules in the étale topology, the natural map _(X)→_(X) is an isomorphism.  However, _(X) is generally strictly larger than _(X).  A systematic treatment of Picard groups in finer topologies is given in <cit.>; see also <cit.> for the case that X is a Stein space.

Let ^+⊂ be the subsheaf of integral elements;  that is, for every U as above we have

    ^+(U)=f∈(U)|f(x)≤ 1, all x∈ U.

Finally, define the sheaf
of principal units ^∗∗⊂^+∗ by 

    ^∗∗(U)=f∈(U) |f(x)-1<1, all x∈ U.


 If U=(A,A^+) is an affinoid object of X_, X_, or X_, then ^∗∗(U) = A^∗∗.


 The lemma can be restated as follows:  Given f∈ A, we have that f(x)<1 for all x∈ U if and only if f∈ A^∘∘.  Consider the rational subset 
    
    U(1/f)=x∈ U |f(x)≥ 1.

We have U(1/f)=(R,R^+), where R=R_0[1/p], and R_0 is the completion of A_0[1/f].  We have that U(1/f) is empty if and only this completion is the zero ring, which is true if and only if f is 
topologically nilpotent.  


This lemma allows us to use the notation ^∗∗(U) and A^∗∗ interchangeably, and we will do so below. In particular, we note:

  Let A=_KT_1,…,T_d, and let D= A.  Let D_K be the adic generic fiber of D, so that D_K is the nonvanishing locus of p in (A,A).  Then (in whatever topology) H^0(D_K, ^∗∗)  1+M, where M⊂ A is the maximal ideal.  Applied to Lubin–Tate space, we find H^0(_n,K,) A_n^ = 1 + 𝔪.  





 §.§ The logarithm exact sequence


The following logarithm exact sequence will allow us to access the cohomology of ^∗∗ on the pro-étale site.  

Since ^∗∗(U) is a topological group for all U in a basis for the topology, we can think of ^∗∗ as a sheaf of condensed abelian groups.

 Let X be a rigid-analytic variety.  There is an exact sequence of sheaves of condensed abelian groups on X_ (and hence on X_):
    
    0 →μ_p^∞→^∗∗log→→ 0.

Here μ_p^∞ is the sheaf of pth power roots of 1.  


 Suppose U→ X is an affinoid object of X_, with U=(A,A^+).  The description of ^∗∗ in the previous lemma shows that the usual logarithm series converges on ^∗∗(U) with values in (U).  

We claim first that the kernel of log^∗∗(U)→(U) is μ_p^∞(U).  Given f∈^∗∗(U), there exists n≥ 0 large enough so that f^p^n-1∈ p^2A_0.  Then 

    p^n log f =log f^p^n =∑_i≥ 1 (-1)^i-1(f^p^n-1)^i/i∈ (f^p^n-1)(1+pA_0).

(Here we have used 2(i-1)≥ v_p(i)+1 for all i≥ 2, where v_p is the p-adic valuation on integers.)  
Thus if log f = 0, we must have f^p^n=1.

Now we claim that log^∗∗→ is surjective.  Let U=(A,A^+) as above.  For f∈(U)=A, let n≥ 0 be large enough so that p^nf ∈ p^2 A^+.  Then exp(p^nf) converges to an element e∈ A^∗∗ such that log e = p^n f.  Let B=A[e^1/p^n];  since e∈ A^∗ and p is invertible in A, we have that B is a finite étale A-algebra.  We find that if V=(B,B^+) (with B^+= integral closure of A^+ in B), then V→ U is an étale cover for which ^∗∗(V) contains a preimage of f under log, namely e^1/p^n.  





 §.§ The fundamental exact sequence


In this section we leverage the isomorphisms between stacks in <ref> to control the H^1_(_n,A_n^∗∗).  By <ref>, A_n^∗∗^∗∗(_n,K).  

We repeatedly apply the following general formalism:   Suppose a stack 𝔛 on the pro-étale site of __p admits a presentation 𝔛 [Y/G] for a diamond Y and a profinite group G which acts on Y.  Let ℱ be a sheaf of condensed abelian groups on 𝔛. Then we have a low-degree exact sequence 
    
    0→ H^1_(G,ℱ(Y)) → H^1_(𝔛,ℱ) → H^1_(Y,ℱ)^G

arising from the descent spectral sequence of E_2-signature
    
    H^i_(G,H^j_(Y,ℱ))  H^i+j_(𝔛,ℱ).

(For background on sheaves of condensed abelian groups on the pro-étale site, see <cit.>, and especially Proposition 3.6.3 for an example of how continuous G-cohomology appears naturally in the context of a pro-étale G-torsor.)

The formation of the exact sequence in (<ref>) is functorial in the following sense:  If a morphism of pairs (Y_1,G_1)→ (Y_2,G_2) is interpreted to mean a homomorphism ϕ G_1→ G_2 and ψ Y_1→ Y_2 such that ψ(gy)=ϕ(g)ψ(y) for g∈ G_1, y∈ Y_1, then such a morphism induces a morphism of descent spectral sequences and hence a morphism between the corresponding low-degree exact sequences.

Now suppose there are two such presentations for 𝔛:
    
    𝔛 [Y_1/G_1] [Y_2/G_2].

The corresponding low-degree exact sequences assemble to give a diagram

    0 [d]    
        P [r] [d]     H^1_(G_1,ℱ(Y_1)) [d]    
    
                0 [r]     H^1_(G_2,ℱ(Y_2)) [r]     H^1_(𝔛,ℱ) [r] [d]     H^1_(Y_2,ℱ)^G_2
           H^1_(Y_1,ℱ)^G_1,

where P is the indicated pullback in the diagram. In particular, there is an exact sequence

    0 → P → H^1_(G_1,ℱ(Y_1)) → H^1_(Y_2,ℱ)^G_2

as well as a similar one which swaps the roles of the two presentations.  The formation of the diagrams in (<ref>) and (<ref>) are functorial on the category of data consisting of (Y_1,G_1),(Y_2,G_2), and an isomorphism [Y_1/G_1] [Y_2/G_2] and with morphisms consisting of a pair of maps of pairs and a 2-commuting square of stacks.
  
Let C be the completion of an algebraic closure of _p, and let 
    
    Γ__p=(_p/_p),

so that (_p,_p)(C,C^∘)/Γ__p.  <Ref> shows that for each n≥ 1 there is a pair of presentations of the stack _n:

    _n  [_n,K^♢/_n]  [_C^n-1,♢/(Γ__p×_n(_p))].

Here, we have replaced ^n-1,♢ with its equivalent presentation _C^n-1,♢/Γ__p. It will be convenient to use the notation 
    
    Π_n = Γ__p×_n(_p).


Recall from <ref> that the determinant morphism _n→_1 is compatible with the two presentations of _n given in (<ref>).  Applying (<ref>) functorially to the determinant morphism, we obtain a commutative diagram:

    scale=0.55,center          0 
       P_1      H^1_(Π_1,(_C^0))    0 
          P_n      H^1_(Π_n,(_C^n-1))
    
    	0    H^1_(_1,(_1,K))      H^1_(_1,)      H^1_(_1,K,)^_1
        0    H^1_(_n,(_n,K))      H^1_(_n,)      H^1_(_n,K,)^_n
             H^1_(^0_C,)^Π_1
                H^1_(_C^n-1,)^Π_n.["q"', from=2-2, to=3-3]
    	[from=3-5, to=5-5]
    	[from=2-5, to=3-5]
    	[from=5-5, to=7-5]
    	[from=3-3, to=3-5]
    	[from=2-4, to=4-4]
    	["_^∗", from=2-4, to=3-5]
    	[from=2-2, to=2-4]
    	[from=1-4, to=2-4]
    	[from=4-4, to=5-5]
    	[from=4-4, to=6-4]
    	[from=6-4, to=7-5]
    	[from=3-3, to=5-3]
    	[from=5-3, to=5-5]
    	[from=5-5, to=5-7]
    	["_^∗",from=4-2, to=5-3]
    	["r", from=2-2, to=4-2]
    	[from=4-2, to=4-4]
    	[from=4-4, to=4-6]
    	["_^∗", from=4-6, to=5-7]
    	
    	[from=4-1, to=4-2]
    	[from=5-2, to=5-3]

We proceed with six lemmas concerning objects in the diagram (<ref>).


    Considering _p^=1+p_p with its trivial _n(_p)-action, we have:
        
    H^1_(_n(_p),_p^) 
                        0,     (n,p) ≠ (2,2); 
    /2     (n,p) = (2,2).

In the case (n,p) = (2,2), the group H^1_(_n(_p),_p^) is generated by the homomorphism given by the composition 
    
    _2(_2)_2(/2) S_3{± 1}⊂_2^.


 Since _p^ is abelian, H^1_(_n(_p),_p^) is isomorphic to the group of continuous homomorphisms _n(_p)^→_p^, where 
    
    _n(_p)^ab = _n(_p) / [_n(_p),_n(_p)].

Since _n() is dense in _n(_p), the image of _n()^ab→_n(_p)^ab is also dense.  We apply the well-known result <cit.>:
    
    _n()^ab = 
    0,     n ≠ 2; 
    /12     n =2.

This settles the claim for n≠ 2, and shows that in all cases _n(_p)^ab is a (possibly trivial) torsion cyclic group. For p ≠ 2, _p^≅_p is torsion free, so this settles the claim for p ≠ 2.

We are left with the case (n,p) = (2,2). As _2^≅_2⊕/2 and _2(_2)^ab is torsion cyclic we see that H^1_(_2(_2),_2^) is of order at most 2. To conclude we observe that the map _2(_2)→_2^ described above determines a non-trivial element in H^1_(_2(_2),_2^).


 
For (n,p)≠ (2,2), the map

    _^∗ H^1_(Π_1,(_C^0))
    → 
    H^1_(Π_n,(_C^n-1))

is an isomorphism.  For (n,p) = (2,2), _^∗ is an inclusion of a direct summand with complement /2 generated by the class α∈ H^1_(Π_2,C^) defined as the composite 
    
    Π_2 →_2(_2) _2(/2) ≃ S_3 {± 1}⊂ C^,

where the first map is the canonical projection.



We first recall from <cit.> that every bounded-by-1 function on ^n-1__p is constant:  H^0(^n-1__p,^+) ≅_p.  This follows from the existence of an integral model __p^n-1 for ^n-1, whose special fiber __p^n-1 is a connected scheme each of whose irreducible components is projective (see <cit.>). Indeed, this shows that H^0(__p^n-1,)≅_p, which suffices because H^0(^n-1__p,^+) H^0(^n-1__p,) is p-adically complete. Consequently H^0(_C^n-1,)≅ C^.

The map _^∗ agrees with the one appearing in the inflation-restriction sequence for C^ and the map Π_n →Π_1 induced by the determinant map _n(_p)→_p^∗ with kernel _n(_p):
    
    0→ H^1_(Π_1, (C^)^_n(_p))  H^1_(Π_n,C^) → H^1_(_n(_p),C^)^Π_1.

Using that _n(_p) acts trivially on C^, Tate's theorem <cit.> which shows that C^Γ__p=_p, and postcomposing with H^1_(_n(_p),C^)^_p^∗⊆ H^1_(_n(_p),C^), we obtain an exact sequence 
    
    0→ H^1_(Π_1, C^)  H^1_(Π_n,C^) → H^1_(_n(_p),_p^).

For (n,p) ≠ (2,2), the claim follows from <ref>. In the case (n,p) = (2,2), it follows from <ref> and the fact that the class α∈ H^1_(Π_2,C^) is a lift of the nontrivial class in H^1_(_2(_2),_2^) with 2α=0, that the above sequence is split short exact.




The map
    
    r P_1→ H^1_(_1,(_1,K))

is an isomorphism.


    Consider the following exact sequence which is an instance of (<ref>) applied to (<ref>):
        
    0 → P_1  H^1_(_1,(_1,K)) → H^1_(^0_C,)^Π_1.

    Since _C^0 = (C,_C), we have H^1_(^0_C,)^Π_1 = 0.

















 
    The map 
        
    _^∗ H^1_(_1,K,)^_1→ H^1_(_n,K,)^_n

is injective.

  
It is enough to show that the map 
    
    _^∗ H^1_(_1,K,)→ H^1_(_n,K,)

is injective. This follows from the observation that any point (K,W)→_n,K induces a retract for ^∗.



 
Assume that (n,p)=(2,2). The map
    
    H^1_(Π_2,(_C^1)) → H^1_(_2,K,)

of the low degree exact sequence (<ref>) sends the class α from <ref> to 0.  Therefore, the class α lifts to P_2.


Consider the short exact sequence of sheaves:
    
    0→μ_2 →→ 0.

This short exact sequence gives rise to the following diagram with exact right vertical column:
    

    (_2,K)
          (_2,K)
    H^1_(Π_2,μ_2(_C^1))   H^1_(_2,μ_2)   H^1_(_2,K,μ_2)
    H^1_(Π_2,(_C^1))   H^1_(_2,)   H^1_(_2,K,).["( )^2"', from=1-3, to=2-3]
    	["∂"',from=2-3, to=3-3]
    	[from=3-1, to=3-2]
    	[from=3-1, to=4-1]
    	[from=3-2, to=3-3]
    	[from=3-2, to=4-2]
    	[from=3-3, to=4-3]
    	[from=4-1, to=4-2]
    	[from=4-2, to=4-3]

Let α̃∈ H^1_(Π_2,μ_2(_C^1)) be the class defined as the composite 
    
    Π_2 →_2(_2) _2(/2) ≃ S_3 {± 1}≃μ_2(_C^1).

The image of α̃ under the map H^1(Π_2,μ_2(_C^1)) → H^1(Π_2,(_C^1)) is α.
Let α∈ H^1(_2,K,μ_2) be the image of α̃  under the composition of the top horizontal maps. It now suffices to show that α lies in the image of ∂(_2,K) → H^1(_2,K,μ_2).   

The group H^1(_2,K,μ_2) classifies étale μ_2-torsors (i.e., double covers) of the rigid-analytic space _2,K, and the map ∂ sends a principal unit in A_2^ to the double cover of _2,K obtained by adjoining its square root.  The μ_2-torsor corresponding to α can be described this way:  Let H^_2 be the universal deformation of H_2 over A_2.  The 2-torsion subgroup H^_2[2] is a group scheme of order 4 over _2= A_2, whose generic fiber H^_2[2]_K is étale over _2,K.  Trivializations of H^_2[2] over _2,K (or, what is the same, markings of its three nonzero sections) form a torsor under the group _2(/2) S_3.  Finally, α is the quotient of this torsor corresponding to the sign homomorphism S_3→μ_2, which is to say it is the torsor classifying markings of the three nonzero sections of H^_2[2] up to cyclic permutation.  

Let E be a supersingular elliptic curve over _2, with universal deformation E^.  By Serre–Tate theory, there is an isomorphism between _2 and the formal scheme parameterizing deformations of E, which identifies H_2^ with the completion of E^ along its origin.  In particular H_2^[2] is identified with the 2-torsion group E^[2], and α is the double cover of _2,K which classifies markings of the three nonzero sections of E^[2] up to cyclic permutation.

A choice of nonvanishing differential ω on E^ determines a Weierstrass model for E^ over the local ring A_2.  Over A_2[1/2], the model can be brought into reduced form, with affine equation y^2=f(x), for a monic cubic polynomial f(x)∈ A_2[1/2][x].  The nonzero 2-torsion sections of E^ correspond to roots e_1,e_2,e_3 of f(x).  Therefore α is the double cover of _2,K obtained by adjoining the square root of Δ(E^,ω), where Δ(E^,ω)=∏_i<j (e_i-e_j)^2∈ A_2^∗ is the discriminant of the Weierstrass model determined by ω.  For a scalar λ∈ A_2^∗ we have Δ(E^,λω)=λ^12Δ(E^,ω);  since every element is a 12th power in _2^∗, we can find an ω with Δ(E^,ω)∈ A_2^∗∗.    We conclude that α lies in the image of ∂;  indeed it is the image of Δ(E^,ω). 




For (n,p) ≠ (2,2), the map q P_1→ P_n is an isomorphism. When (n,p) = (2,2), the map q is an inclusion of a direct summand with complement /2 generated by α.
    

Formation of the exact sequence (<ref>) is functorial, so we find a diagram

    0  [r]     P_1 [r] [d]_q     H^1_(Π_1,(_C^0)) [d]__^∗[r]     H^1_(_1,K,)^_1[d]__^∗
    
    0  [r]     P_n [r]      H^1_(Π_n,(_C^n-1)) [r]     H^1_(_n,K,)^_n.

When (n,p) ≠ (2,2), the map _^∗ is an isomorphism by <ref> and _^∗ is injective by <ref>, so q is an isomorphism. In the case (n,p) = (2,2), the map q is injective since _^∗ is injective by <ref>. The induced map from the cokernel of q to the cokernel of _^∗ is injective so has size at most /2, again by <ref>. By <ref>, α is a 2-torsion element of P_n that is not in the image of q. Since _^∗ is injective by <ref>, this class thus generates the /2 cokernel.


When p=2, we denote the image of α under the map P_2 → H^1(_2,A_2^∗∗) by α̂.
Combining the lemmas above, we deduce the following theorem.


    For (n,p) ≠ (2,2), there is an exact sequence
        
    0 → H^1_(_1,A_1^∗∗)  H^1_(_n,A_n^∗∗) b→ H^1_(_C^n-1,^∗∗)^Π_n,

    where b is the composite H^1_(_n,A_n^∗∗) → H^1_(_n,^∗∗) → H^1_(_C^n-1,^∗∗)^Π_n in (<ref>). 
    
    When (n,p) = (2,2) there is an exact sequence
        
    0 → H^1_(_1,A_1^∗∗) ⊕/2  H^1_(_2,A_2^∗∗) b→ H^1_(_C^1,^∗∗)^Π_2,

    where b is the composite H^1_(_2,A_2^∗∗) → H^1_(_2,^∗∗) → H^1_(_C^1,^∗∗)^Π_2 in (<ref>).     
         

    Consider the following commutative diagram, which is part of (<ref>):
        
    P_n [r] [d]     H^1_(Π_n,(_C^n-1)) [d] 
    
                H^1_(_n,A_n^∗∗) [r] [rd]_b     H^1_(_n,) [d] 
        H^1_(_C^n-1,^∗∗)^Π_n.

    As an instance of (<ref>), we obtain an exact sequence
        
    0 → P_n → H^1_(_n,A_n^∗∗)  H^1_(_C^n-1,^∗∗)^Π_n.

    <Ref> and <ref> provide isomorphisms
        
    P_n ≅
                        H^1_(_1,A_1^∗∗)    if  (n,p) ≠ (2,2); 
    
                        H^1_(_1,A_1^∗∗) ⊕/2    if  (n,p) = (2,2).

    Together with the commutativity of the left square in (<ref>) and the class α, this gives the two exact sequences of the theorem. 


When n=1, the theorem is uninteresting:  _^∗ is an isomorphism, and the target of b is 0 for dimension reasons.  The next objective is to identify H^1_(_C^n-1,^)^Π_n for n≥ 2.  
In the next section we will prove that this group is a free _p-module of rank 1 and the map b is surjective, generated by the image of ε_p(Σ^2𝕊_K(n)).  This claim implies <ref>.   





§ THE COHOMOLOGY OF DRINFELD SYMMETRIC SPACE _C^N-1


Via the fundamental exact sequence in <ref>, we have reduced the crucial <ref> to a question about H^1_(^n-1_C,), the cohomology of  on the pro-étale site of Drinfeld's symmetric space _C^n-1.  Recall that _C^n-1 is the complement in rigid-analytic ℙ^n-1_C of all _p-rational hyperplanes;  as such it admits an action of _n(_p).   In particular _C^1=ℙ^1_C\ℙ^1(_p) is analogous to the (upper and lower) half-plane ℙ^1(ℂ)\ℙ^1(ℝ).  

The cohomology of _C^n-1 was studied first by Drinfeld <cit.> in the case n=2, and then by Schneider–Stuhler <cit.> for general n.  The main theorem of <cit.> is that if H is a cohomology theory on rigid-analytic varieties satisfying certain reasonable axioms (for instance, homotopy-invariance relative to the unit disc D), then H^i(_C^n-1) is a generalized Steinberg representation of _n(_p) for 0≤ i≤ n, and is 0 otherwise.  (The construction of the generalized Steinberg representations are reviewed below.)  The result applies to ℓ-adic étale cohomology and to de Rham cohomology.

In contrast, étale cohomology with _p-coefficients does not satisfy homotopy-invariance; indeed, H^1_(D,_p) is very large.  One therefore might expect that there cannot be a meaningful description of H^i_(_C^n-1,_p), let alone H^i_(_C^n-1,_p).  Nonetheless, both groups were identified by Colmez–Dospinescu–Niziol <cit.>, <cit.>.  In the latter setting, H^i_(_C^n-1,_p) is an extension of a space of differential forms by a general Steinberg representation.  

Combining these results with the logarithm exact sequence (<ref>), we gain access to H^i_(_C^n-1,), see <ref> below.  Specializing to the case i=1, we find that H^1_(^n-1_C,)^Π_n is (for n≥ 2) isomorphic to the space of _n(_p)-invariants in a generalized Steinberg representation with coefficients in _p/_p.  This space of invariants is shown to be a free rank 1 _p-module in <ref>.




 §.§ Generalized Steinberg representations: definitions
 The following is a rephrasing of the constructions appearing in <cit.>.  For 0≤ r≤ n, let X_r be the set of flags
    
    V_1⊂ V_2⊂⋯⊂ V_r⊆_p^⊕ n,

where V_i is a _p-vector space of dimension i. Note that we in particular allow the empty flag. For  1≤ j≤ r, let X_r,j be set of flags as above with the jth vector space omitted.  The set X_r (resp., X_r,j) is a quotient of _n(_p) by a parabolic subgroup.  As such, it is a profinite set admitting a continuous action of _n(_p); note also the quotient map X_r→ X_r,j.  

For a profinite set S= S_i and a ring A, let (S,A) denote the ring of locally constant functions on S valued in A.  If A carries a topology, we give (S,A)=(S_i,A) the colimit topology, where each (S_i,A) (a finite free A-module) has the natural topology.  Define the generalized Steinberg (or special) representation by

    _r(A)=(X_r,A)/∑_j=1^r (X_r,j,A).

Each _r is a topological group admitting a continuous action of _n(_p). 
As a special case,

    _1(A)=(ℙ^n-1(_p),A)/A.

For any topological A-module M, let

    M^∗ = _(M,A),

meaning continuous A-module homomorphisms M→ A.  One can think of _r(A)^∗ as the module of distributions on X_r whose value is 0 on functions pulled back from any X_r,j.  

In the case r=1, identify ℙ^n-1(_p) with the set of linear forms in n variables up to scaling by _p^∗.  Let δ_ℓ denote the Dirac distribution at ℓ∈ℙ^n-1(_p).  Then any difference δ_ℓ-δ_ℓ' determines an element of _1(A)^∗.  

We give an ad hoc definition of the _n(_p)-module _r(_p/_p)^∗, defining this to be the quotient:

    0 →_r(_p)^∗→_r(_p)^∗→_r(_p/_p)^∗→ 0

Thus for instance elements of _1(_p/_p)^∗ are _p/_p-valued distributions on ^n-1(_p) with total measure 0.


  Assume n≥ 2.  The subset of _1(_p/_p)^∗ fixed by _n(_p) is a free _p-module of rank 1.  


 Let μ be a _p/_p-valued distribution on ^n-1(_p) = ^n-1(_p). For each m≥ 1, we have a _n(_p)-equivariant surjective map 
    
    π_m^n-1(_p)→^n-1(/p^m).

As m varies, the fibers of the π_m form a basis of open subsets of ^n-1(_p), so that μ is determined by its values on the fibers.  Now suppose μ is fixed by _n(_p).  Since _n(_p) acts transitively on ^n-1(/p^m), the value of μ is the same on all fibers of π_m;  let a_m denote the common value.  Then by the additivity of μ:
    
    μ(^n-1(_p)) = p^n-1/p-1a_1=p^n-1/p-1p^n-1a_2 = p^n-1/p-1p^2(n-1)a_3 = ⋯

Via μ↦ (a_1,a_2,…), the subgroup of _1(_p/_p)^∗ fixed by _n(_p) may be identified with the subgroup of ∏_m≥ 1_p/_p satisfying 
    
    0 = a_1 = p^n-1a_2 = p^2(n-1)a_3 = ⋯.

This group is _m /p^(n-1)m_p.


Taking invariants under _n(_p) in the exact sequence (<ref>), we find an injective  connecting map

    ∂ H^0(_n(_p),_1(_p/_p)^∗) → H^1(_n(_p),_1(_p)^∗).

Let μ∈ H^0(_n(_p),_1(_p/_p)^∗) be the generator of this group with a_m=1/p^(n-1)(m-1) for all m≥ 2.  


  Let ℓ∈^n-1(_p) be arbitrary.  Then ∂(μ) is represented by the cocycle
    
    g↦δ_ℓ-δ_g(ℓ).



The measure μ can be lifted to a (non-_n(_p)-invariant) measure μ̃∈_1(_p)^∗ defined by the formula (for m≥ 2)

    μ̃(π_m^-1(x)) = 1/p^(n-1)(m-1)-1,   π_m(ℓ)=x; 
    1/p^(n-1)(m-1),   otherwise.

Then ∂(μ) is represented by the cocycle sending g∈_n(_p) to g(μ̃)-μ̃, and this is easily seen to agree with δ_ℓ-δ_g(ℓ).




 §.§ Results of Colmez–Dospinescu–Niziol


We turn now to some crucial results of Colmez–Dospinescu–Niziol on the étale and pro-étale cohomology of Drinfeld symmetric space _C^n-1 over a complete algebraically closed field C/_p.  

Recall the difference between étale and pro-étale cohomology:  whenever X is a scheme (resp. stack, rigid-analytic space, etc.), one has the constant sheaf /p^n on the étale site of X, and then one defines

    H^i_(X,_p)   =    H^i_(X,/p^n) 
    
    H^i_(X,_p)   =    H^i_(X,_p)⊗__p_p.

The latter is not the cohomology of any sheaf “_p” on X_.  On the other hand, one has the constant sheaf _p on the pro-étale topology X_, defined by sending an object U∈ X_ to the ring of continuous maps U→_p, and then H^i_(X,_p) really is the sheaf cohomology of _p on X_.  The projection X_→ X_ induces a map 

    H^i_(X,_p)→ H^i_(X,_p)

which can fail to be an isomorphism if X is not quasi-compact (for instance, if X is the discrete infinite union of points).  

The cohomology groups H^i_(_C^n-1,_p) ≃ H^i_(_C^n-1,_p) and H^i_(_C^n-1,_p) were computed in <cit.> and <cit.>, respectively.  We now review these results, starting with H^i_.  
We need one more ingredient before we can give the statements, namely the Kummer map

    κ^∗(_C^n-1)→ H^1_(_C^n-1,_p(1)).

This map arises as follows: Consider the sequence of short exact sequences
    
    0 [r]    μ_p^n+1[r] [d]    ^∗[r]^-(-)^p^n+1[d]_(-)^p   ^∗[r] [d]^-id    0 
    
            0 [r]    μ_p^n[r]    ^∗[r]^-(-)^p^n   ^∗[r]      0.

Taking limits results in another short exact sequence, 
    
    0 [r]    _p(1) [r]    _(-)^p^∗[r]    ^∗[r]     0,

and κ is the boundary map in cohomology of this short exact sequence.


   There is a Γ__p×_n(_p)-equivariant isomorphism
    
    r_i_i(_p)^∗ H^i_(_C^n-1,_p(i))
    
such that (in the i=1 case) the relation
    
    r_1(δ_ℓ_1-δ_ℓ_2) = κ(ℓ_1/ℓ_2)

holds for all ℓ_1,ℓ_2∈ℙ^n-1(_p).  On the right side of the relation, ℓ_1 and ℓ_2 are interpreted as _p-rational linear forms in n variables, so that ℓ_1/ℓ_2∈^∗(_C^n-1) is well-defined up to _p^∗, and κ(ℓ_1/ℓ_2) is a well-defined class in H^1_(_C^n-1,_p(1)).


 The map r_i is defined in <cit.>;  for i=1 it is characterized by the relation r_1(δ_ℓ_1-δ_ℓ_2)=κ(ℓ_1/ℓ_2).  (There is a characterization of r_i for general i in terms of a “regulator map.”) The isomorphy of r_i is <cit.>.


Let us turn now to pro-étale cohomology.  For a rigid space X over C, the short exact sequence of abelian sheaves on X_ from <Ref>,
    
    0 →μ_p^∞→^∗∗log→→ 0,

gives rise, after taking sequential limits along the multiplication by p map to a short exact sequence of sheaves of _p-vector spaces on X_
    
    0 →_p(1)→_× plog→→ 0.

We thus get a connecting map ∂_log[-1]→_p(1) in the derived category of sheaves on X_.  The pro-étale cohomology of  was studied in <cit.>:  If ν X_→ X_ is the projection, then R^iν_*Ω^i(-i), where Ω^i=Ω^i_X/C is the sheaf of differential i-forms.  As a result there is a spectral sequence

    H^i_(X,Ω^j(-j)) H^i+j_(X,).

In the special case that X is a Stein space, the coherent sheaves Ω^j_X/C are acyclic, and so 
    
    H^i_(X,)Ω^i(X)(-i).

Therefore the connecting map [-1]→_p(1) induces a map
    
    expΩ^i-1(X)(1-i) → H^i_(X,_p(1)).
  
The map exp is related to the Bloch–Kato exponential, see <cit.> for a discussion.   When X admits a semistable formal model over the ring of integers of a local field, <cit.> identifies the kernel of exp with the module of closed forms d, and identifies the cokernel of exp in terms of the Hyodo–Kato cohomology of the special fiber of the model.  
We are now ready to state the theorem of Colmez–Dospinescu–Niziol on the pro-étale cohomology of _C^n-1.

  There is a Γ__p×_n(_p)-equivariant exact sequence of _p-vector spaces
    
    0→Ω^i-1(_C^n-1)/ dexp→ H^i_(_C^n-1,_p(i)) →_i(_p)^∗→ 0,

compatible with <ref>, in the sense that the evident diagram involving H^i_(_C^n-1,_p(i))→ H^i_(_C^n-1,_p(i)) commutes.


We readily deduce the pro-étale cohomology of μ_p^∞ on _C^n-1:


    There is a Γ__p×_n(_p)-equivariant exact sequence
        
    0→Ω^i-1(_C^n-1)/ dexp→ H^i_(_C^n-1,μ_p^∞(i-1)) →_i(_p/_p)^∗→ 0.



    <Ref> and <ref> together supply a map of exact sequences
        
    0 [r] [d]      H^i_(_C^n-1,_p(i)) [r]^-≅[d]    _i(_p)^∗[r] [d]     0 
    
                0 [r]    Ω^i-1(_C^n-1)/ d[r]^-exp     H^i_(_C^n-1,_p(i)) [r]    _i(_p)^∗[r]     0.

    Note that the right vertical map is injective with quotient _i(_p/_p)^∗, see (<ref>), while the middle vertical map comes from (i-1)-fold twist of the short exact sequence
        
    0 →_p(1)→_p(1) →μ_p^∞→ 0.

    The snake lemma then furnishes the short exact sequence of the corollary.




 §.§ Consequences for H^i_(_C^n-1,)

We gather here some consequences of <ref> and <ref>. The next results concern the -cohomology of _C^n-1.  

 

For i≥ 0 there is a short exact sequence
    
    0 →_i(_p/_p)^*(1-i)→ H^i_(_C^n-1,)→Ω^i,(_C^n-1)(-i) → 0,

where Ω^i, denotes the sheaf of closed differential i-forms on _C^n-1.


    Let ∂_log→μ_p^∞[1] be the connecting homomorphism  of the logarithm exact sequence (<ref>), and write
        
    ∂_log^i H^i_(_C^n-1,) → H^i+1_(_C^n-1,μ_p^∞)

    for the induced map on cohomology. It follows from the long exact sequence associatd to the logarithm sequence that there is a short exact sequence
        
    0 →(∂_log^i-1) → H^i_(_C^n-1,) →(∂_log^i) → 0.

    Using <ref>, we can identify the kernel and cokernel of ∂_log^i. To this end, first note that there is a lift
        
    [d]^∂_log@–>[dl]    
    _p(1)[1] [r]    μ_p^∞[1] [r]    _p(1)[2],

    where the bottom exact sequence is a rotation of the triangle coming from (<ref>). Indeed, this lift exists because the composite →μ_p^∞[1] →_p(1)[2] is trival, since the target is p-complete while the source has p acting invertibly. On cohomology, this lift is precisely the map (<ref>).

    Combining this observation with <ref> and (<ref>), we thus obtain a map of exact sequences
        
    0 [r]    Ω^i(_C^n-1)(-i) [r]^-≅@->>[d]     H^i_(_C^n-1,) [r] [d]^-∂_log^i    0 [d] 
    
                0 [r]    Ω^i(_C^n-1)/ d(-i) [r]^-exp    H^i+1_(_C^n-1,μ_p^∞) [r]    _i+1(_p/_p)^∗(-i) [r]     0

    By our discussion above, the left vertical map is the canonical quotient map. The snake lemma then implies that
        
    (∂_log^i) ≅Ω^i,(_C^n-1)(-i)     and    (∂_log^i) ≅_i+1(_p/_p)^∗(-i),

    as desired.


 
    Consider the short exact sequence of pro-étale sheaves on _C^n-1:
        
    0→_p(1) →lim_p→ 0.

    The associated long exact sequence for cohomology has zero boundary maps. 
    Therefore for every m ≥ 0 we obtain a short exact sequence of Π_n-modules
        
    0→ H^m_(ℋ_C^n-1,_p(1)) →  H^m_(ℋ_C^n-1,lim_p ) →  H^m_(ℋ_C^n-1, )→ 0.

    

The claim is equivalent to showing that the map
    
    H^m_(ℋ_C^n-1,lim_p ) →  H^m_(ℋ_C^n-1, )
 
is surjective for all m. Note that this map factors as
    
    H^m_(ℋ_C^n-1,lim_p ) →lim_p H^m_(ℋ_C^n-1, ) → H^m_(ℋ_C^n-1, ).

By the Milnor sequence, the first map  
    
    H^m_(ℋ_C^n-1,lim_p ) →lim_p H^m_(ℋ_C^n-1, )

is a surjection. The second map is surjective as well, since 
H^m_(ℋ_C^n-1,) is p-divisible. This follows from <ref>, which exhibits H^m_(ℋ_C^n-1,) as an extension of p-divisible abelian groups.  


Finally, we identify the Π_n-invariants in H^1_(_C^n-1,).  


  
There is a canonical isomorphism
    
    H^1_(_C^n-1,)^Π_n(_1(_p/_p)^∗)^_n(_p).

Thus if n≥ 2, then by <ref>, H^1_(_C^n-1,)^Π_n is a free _p-module of rank 1 with distinguished generator. When n=1 we have H^1_(_C^0,)^Π_1=0.


 Take Π_n-invariants in the exact sequence in the i=1 case of <ref>. Since ^n-1 is defined over _p we have an isomorphism
    
    Ω^1,(_C^n-1)(-1)Ω^1,(^n-1)⊗__p C(-1),
 
so that
    
    H^0(Γ__p,Ω^1,(_C^n-1)(-1))Ω^1,(^n-1)⊗__p H^0(Γ__p,C(-1))=0.

Therefore we have an isomorphism
    
    H^0(Π_n,_1(_p/_p)^∗) H^0(Π_n,H^1_(_C^n-1,)).

The group Π_n acts on _1(_p/_p)^∗ through its quotient _n(_p), and we have seen in <ref> that the module of invariants is rank 1 over _p for n≥ 2. For n=1, _C^0 = (C,_C) and thus H^1_(_C^0,
ℱ) is zero for any sheaf ℱ.




 §.§ A primitivity result
 Assume throughout that n≥ 2. 
Let us recall the fundamental exact sequence from <ref>, applying what we have learned from <ref>:

    0 [r]      H^1_(_1,A_1^∗∗) [r]     H^1_(_n,A_n^∗∗) [r]^-b    H^1_(_C^n-1,)^Π_n[d]_
             _p

Our current goal is to prove that the map labeled b is surjective.  To do this, it suffices to show that the image under b of the class ε_p(Σ^2𝕊_K(n)) is primitive, i.e. that it is a generator of the _p.  More precisely, we will prove:


 The map b in <ref> sends ε_p(Σ^2𝕊_K(n)) to the distinguished generator of the _p-module H^1_(_C^n-1,)^Π_n. 




The players in the proof of <ref> fit into the following commutative diagram:

    H^1_(_n,A_n^) [d]_g_0@/^-4pc/[dd]_b    H^1_(_n,^∗) 
    
        H^1_(_n,) [d]_e_1[ru]^-g_1[rd]_f_1    H^1_( Π_n, ^∗(_C^n-1)) [u]_g_2[d]^f_2
    
        H^0(Π_n,H^1_(_C^n-1,)) [r]_-ϕ_2[d]_i_1^≅    H^1_(Π_n, H^1_(ℋ_C^n-1,_p(1))) [d]^i_2
    
        H^0(_n(_p),_1(_p/_p)^∗) [r]^-∂    H^1_(_n(_p),_1(_p)^∗).

We explain the maps appearing in (<ref>). The map labeled g_0 arises from the equivalence _n [_n,K^♢/_n], g_1 arises from the inclusion →^∗, and g_2 arises from the equivalence _n [_C^n-1/Π_n]. The map labeled ϕ_2 appears as a connecting map when applying the long exact sequence in continuous Π_n-cohomology to the short exact sequence of Π_n-modules in 
<ref>.  The bottom map labeled ∂ is the connecting map from (<ref>); it is injective as noted in the discussion around (<ref>).  The map i_1 is the isomorphism from <ref>, and i_2 arises from the identification of H^1_(_C^n-1,_p(1)) with _1(_p)^∗ in <ref>. Finally, e_1 is the map from (<ref>) arising in the edge sequence, and b = e_1 ∘ g_0 and f_1 = ϕ_2 ∘ e_1 are the composites. 

For the construction of f_2, start with the short exact sequence of Π_n-equivariant sheaves 
on _C^n-1:

    0 →_p(1) →lim_p ^* →^* → 0.

Consider the boundary map 

    ^*(ℋ_C^n-1) → H^1_(ℋ_C^n-1,_p(1))

which is a map of Π_n-modules. Applying H^1_(Π_n,-) gives

    f_2  H^1_( Π_n, ^∗(_C^n-1)) → H^1_(Π_n, H^1_(ℋ_C^n,_p(1))).


 Let a_1=g_0ε(Σ^2(p^n-1)𝕊_K(n)), and let a_2∈ H^1_(Π_n,^∗(_C^n-1)) be the class of the cocycle g↦ (g(ℓ)/ℓ)^p^n-1, where ℓ∈^∗(_C^n-1) is a _p-rational linear form.  Then g_1(a_1)=g_2(a_2).



 Recall that H is the universal formal group over _n, and H[1/p] is its corresponding line bundle.  Also recall that by <ref>, under the isomorphism _n [_C^n-1/Π_n], the line bundle H[1/p] corresponds to (1)|__C^n-1.  Since ℓ is a nowhere vanishing section for (1)|__C^n-1, the image of a_2 in H^1_(_n,^∗) is ( H[1/p])^⊗ (p^n-1).  This equals g_1(a_1), ultimately because ε(Σ^2𝕊_K(n))∈ H^1_(_n,A_n^∗) is the class of H as a _n-equivariant invertible A_n-module.
 


    Consider two classes x_1∈ H^1_(_n,) and x_2 ∈ H^1_( Π_n, ^∗(_C^n-1)).
If g_1(x_1) = g_2(x_2), then f_1(x_1)  = f_2(x_2).


The proof of <ref> is postponed to the next subsection.

  By <ref> and <ref>, we have

    f_1g_0ε(Σ^2(p^n-1)𝕊_K(n))=(p^n-1)f_2(c),

where c the class of the cocycle g↦ g(ℓ)/ℓ in H^1_(Π_n,^∗(_C^n-1)).  The prime-to-p part of H^1_(_n,A_n^∗) has order (p^n-1), and so (p^n-1)ε_p(Σ^2𝕊_K(n))=ε(Σ^2(p^n-1)𝕊_K(n)).  Since (p^n-1) acts invertibly on the _p-module H^1_(Π_n,H^1_(_C^n-1,_p(1))), we must have f_1g_0ε_p(Σ^2𝕊_K(n))=f_2(c).  Apply i_2 and use the commutativity of the diagram in (<ref>) to find that ∂ i_1bε_p(Σ^2𝕊_K(n)) equals the cocycle g↦δ_g(ℓ)-δ_ℓ from <ref>.  Since ∂ is injective and i_1 is an isomorphism, we conclude that bε_p(Σ^2𝕊_K(n)) is the distinguished generator of H^1_(_C^n-1,)^Π_n.  





 §.§ Proof of <ref> and <ref>

We begin with some preparatory lemmas.


Assume that C and D are complexes in D() such that C ≃τ^[0,1]C and D ≃τ^[0,1]D (i.e., C,D are concentrated in cohomological degrees 0 and 1). Assume that H^*(C) is p-divisible and H^*(D) is derived p-complete.   Then 
every map f  C → D factors as 

    C → H^1(C)[-1] → H^0(D) → D,

where H^1(C) and H^0(D) are viewed as complexes concentrated in degree 0.


Since the cohomology groups H^*(D) are derived p-complete, D is derived p-complete, so the map f factors as
    
    C →C→ D,

where C is the derived p-completion of C. Since H^*(C) is p-divisible, we have H^i(C) = H^i+1(C). In particular, we have C≃τ^[-1,0]C. We thus get a decomposition 
    
    C →C→ H^0(C) → H^0(D) → D

and  H^0(C) ≅H^1(C)[-1]. We get that the composition C →C→ H^0(C) factors
as 
    
    C → H^1(C)[-1] →H^1(C)[-1]≃ H^0(C).

Combining the last two displayed factorizations gives the claim.


When A is a p-divisible sheaf (for any topology) there is a short exact sequence of sheaves

    0 → T_pA →_p A → A → 0,

where T_pA=_m A_p^m. This induces a map A → T_p A[1]. Now, the map τ^[0,1]RΓ(A) →τ^[0,1]RΓ(T_pA[1]) satisfies the requirements of <ref>. This provides us with a factorization:

    τ^[0,1]RΓ(A) [r] [d]     H^1(A)[-1] [d] 
    τ^[0,1]RΓ(T_pA[1])     H^1(T_pA)[0]. [l]


  In the derived category of abelian groups with Π_n-action,
    the map 
    
    H^1_(_C^n-1, _p(1))[0] →τ^[0,1]RΓ(_C^n-1,  _p(1)[1])

    is sent to an isomorphism  via the functor
    
    A↦ H^1_(Π_n,A):= H^1(A^hΠ_n).



    Consider the cofiber sequence 
        
    H^1_(ℋ_C^n-1, _p(1))[0] →τ^[0,1]RΓ_(_C^n-1,  _p(1)[1]) → H^2_(_C^n-1, _p(1))[-1].

    It is enough to show that 
        
    H^0(Π_n,H^2_(_C^n-1, _p(1))) = 0.

From <ref> we have H^2_(_C^n-1,_p(1))(_p)^∗(-1), which has no nonzero Π_n-fixed vectors because _p(-1)^Γ__p=0.



Consider the following diagram, in which all cohomology is computed on the pro-étale site of _C^n-1:


    τ^[0,1]RΓ() [ddl] [ddd] [rr]        H^1()[-1] [ddd] 
             
    τ^[0,1]RΓ(^∗) [dr]        H^0(^∗)[0] @-'[l][ll] [dr]    
       τ^[0,1]RΓ(_p(1)[1])        H^0(_p(1)[1]). [ll]


The diagram is commutative: the front rectangle commutes by <ref>, the bottom parallelogram commutes by naturality of the edge maps, and the left triangle arises from the composition ^∗∗→^∗→_p(1)[1].  

Applying the functor A↦ H^1(A^hΠ_n) to the above diagram yields: 


    !    H^1(_n, ) [ddl]_g_1[ddd]_ϕ_1[rr]^e_1       H^1(_C^n-1,)^Π_n[ddd]_ϕ_2
             
    
    H^1(_n, ^∗) [dr]_δ       H^1(Π_n,^∗(_C^n-1)) @-'[l][ll]_      g_2[dr]_f_2   
        H^1(Π_n,RΓ(_C^n-1,_p(1)[1]))        H^1(Π_n,H^1(_C^n-1,_p(1)[1])). [ll]_e_2^∼

Here e_2 is an isomorphism by <ref>.
We can now prove <ref>, in which f_1=ϕ_2∘ e_1.  Given that g_1(a_1)=g_2(a_2), we have 
    
    f_1(a_1)=ϕ_2e_1(a_1)=e_2^-1δ g_1(a_1) =e_2^-1δ g_2(a_2)=f_2(a_2),

as desired.


<Ref> is now proved.  We explain how it implies <ref>, which in turn implies our main theorem (<ref>), see the discussion following <ref>.  


    We first consider the case (n,p)≠ (2,2). <Ref> together with <ref> state that there is an exact sequence
        
    0→ H^1_(_1,A_1^)  H^1_(_n,A_n^)→_p → 0

    in which ε_p(Σ^2𝕊_K(n)) is sent to the generator of the _p.  Since _p is free, the exact sequence splits:
        
    H^1_(_n,A_n^)_p⊕ H^1_(_1,A_1^).
 
    Similarly for (n,p)= (2,2), we have an isomorphism
        
    H^1_(_2,A_2^)_2⊕ H^1_(_1,A_1^)⊕/2,
 
    where in both cases the _p summand is generated by ε_p(Σ^2𝕊_K(n)). The term H^1_(_1,A_1^) has been computed in <ref>; substituting this into (<ref>) resp. (<ref>) yields isomorphisms
        
    H^1_(_n,A_n^∗∗) ≅_p^2     if  p>2; 
    _2^2 ⊕ (/2)^⊕ 2   if  p=2  and  n>2; 
    _2^2 ⊕ (/2)^⊕ 3   if  p=2  and  n=2.

    As explained in <Ref>, a generator of the torsion-free part of H^1_(_1,A_1^) is sent to the determinant in H^1_(_n,A_n^), thus finishing the proof.




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