Theory and Applications of Categories, Vol. 32, No. 32, 2017, pp. 889–969.

BIEXTENSIONS, BIMONOIDAL FUNCTORS, MULTILINEAR FUNCTOR CALCULUS, AND CATEGORICAL RINGS

ETTORE ALDROVANDI

Abstract. We associate to a bimonoidal functor, i.e. a bifunctor which is monoidal in each variable, a nonabelian version of a biextension. We show that such a biextension satisfies additional triviality conditions which make it a bilinear analog of the kind of spans known as butterflies and, conversely, these data determine a bimonoidal functor. We extend this result to n-variables, and prove that, in a manner analogous to that of butterflies, these multi-extensions can be composed. This is phrased in terms of a multilinear functor calculus in a bicategory. As an application, we study a bimonoidal category or stack, treating the multiplicative structure as a bimonoidal functor with respect to the additive one. In the context of the multilinear functor calculus, we view the bimonoidal structure as an instance of the general notion of pseudo-monoid. We show that when the structure is ring-like, i.e. the pseudo-monoid is a stack whose fibers are categorical rings, we can recover the classification by the third Mac Lane cohomology of a ring with values in a bimodule.

Contents

1 Introduction  890

I  897
2 Preliminaries  897
3 Biadditive morphisms  903
4 Biextensions by braided crossed modules  904
5 Biadditive morphisms and biextensions  908
6 Symmetry properties of biextensions  911
7 Butterflies  913
8 Biadditive morphisms and butterflies  916
9 Commutative structures  920

II  922
10 Multiextensions and compositions  922

Received by the editors 2015-08-05 and, in final form, 2017-07-28. Transmitted by Larry Breen. Published on 2017-08-02. 2010 Mathematics Subject Classification: 18D10, 13D03, 18G55, 55P43, 14A20. Key words and phrases: Categorical ring, biextension, bimonoidal, ring-like stack, butterfly, multi-

extension, multi-category, multi-functor, Mac Lane cohomology.

\begin{matrix}c \\ ◯ \end{matrix} Ettore Aldrovandi, 2017. Permission to copy for private use granted.

11 Compositions of n-butterflies  924
12 The bi-multicategories of group-like stacks and crossed modules  929

III  931
13 Bimonoidal structures and weakly categorical rings  931
14 The presentation of a categorical ring  934
15 Decomposition of categorical rings and the cohomology of rings  937

IV  950
A Multi-variable compositions  950
B (Unbiased) Monoids and monoidal structures  953
C Pentagons  956
D Some lemmas on symmetric braidings  960
E Hypercohomology computations  961
References  966

1. Introduction

Let ℋ and 𝒢 be monoidal stacks in a topos T and F a monoidal functor F: ℋ → 𝒢. It is well known that F can be represented by a special kind of span. More precisely, if the monoidal laws are group-like, or if we restrict to the invertible objects (restricting to the invertible objects does not affect the characteristic class), we can find presentations H_{•} for ℋ and G_{•} for 𝒢 by crossed modules of T so that F is represented by a diagram of group-objects of the form

\xymatrix{H_{1} \ar[dd] \ar[dr] &  & G_{1} \ar[dd] \ar[dl] \\  & E \ar[dl] \ar[dr] \\ H_{0} &  & G_{0}}

where the salient feature is that the sequence G_{1} → E → H_{0} of objects of T is an extension of H_{0} by G_{1} → G_{0}: an exact sequence in which the conjugation action of E on G_{1} is compatible with that of G_{0} [AN09]. Such an extension is conveniently described in geometric terms as a G_{1}-bitorsor E over H_{0}, with the property that one of the actions is obtained by way of the crossed module structure of G_{•} [Bre90]. A converse of this correspondence is also available, establishing an equivalence between the groupoid of monoidal functors from ℋ to 𝒢 and that of diagrams like the one above (a morphism between two diagrams with the same wings H_{•} and G_{•} is a homomorphism of E → E′ compatible with all the maps).

One of the main results of this paper is an analogous result for bimonoidal functors F: ℋ × 𝒦 → 𝒢, that is, bifunctors which are monoidal in each variable.

The concept of bimonoidal functor requires 𝒢 to be braided. In addition, F must satisfy a compatibility condition requiring that applying the monoidal condition on both of its variables in the two possible orders lead to the same result. (We can think of F as an obvious generalization of a bilinear map, in which case this condition is trivially satisfied.) This condition is formally equal to the compatibility one between the two partial multiplication laws of a biextension [Mum69, Gro72]. Recall that for abelian groups A, B, C of T, a biextension of B and C by A consists of an A-torsor E over B × C equipped with two partial commutative and associative product laws, making E, for each generalized point b ∈ B, an extension of C by A, and, similarly, for each c ∈ C, an extension of B by A. The coincidence between these conditions is not purely formal: the bilinear point of view is to regard a biextension as providing a bifunctor φ_{E}: B × C → Tors(A), which is then monoidal (i.e. a homomorphism) in each variable. Notable examples are certain duality pairings where, in particular, C = B^{∨} and A = G_{m} (see, e.g. [Boy10, Dat10]).

Our result begins from a generalization of these ideas, starting with that of biextension. Thus, for two groups H and K of T, and a braided crossed module (G_{1}, G_{0}), we define a biextension of H, K by (G_{1}, G_{0}) as a G_{1}-bitorsor (or, more appropriately, using the terminology of [Bre90], a (G_{1}, G_{0})-torsor) E over H × K such that, for each point x ∈ H (resp. y ∈ K), E is an extension of K (resp. of H) by the crossed module G_{1} → G_{0}. The braiding is required by the compatibility between the two partial product laws of E. The relative diagram is formally the same as in the classical situation, with some differences due to the fact that none of the groups involved is assumed abelian—although the crossed module G_{1} → G_{0} and its associated stack are braided.

Now, let F be a bimonoidal functor F: ℋ ×𝒦 → 𝒢, where each of 𝒦 , ℋ , and 𝒢 has a presentation by a crossed module. We show that F determines a biextension of H_{0}, K_{0} by the crossed module G_{1} → G_{0}, equipped with a pair of compatible trivializations for the two pullbacks along the maps ∂ ×id: H_{1} ×K_{0} → H_{0} ×K_{0} and id ×∂: H_{0} ×K_{1} → H_{0} ×K_{0}. This is what we call, in this context, a butterfly —the bilinear version of the notion introduced in [AN09] (earlier, over a point, in [Noo05]). Viceversa, given a biextension E of (H_{0}, K_{0}) by G_{1} → G_{0}, we obtain a bimonoidal functor

φ_{E}: H_{0} × K_{0} ⟶ Tors(G_{1}, G_{0}),

where the right hand side denotes the stack of (G_{1}, G_{0})-torsors equivalent to 𝒢, and where H_{0} and K_{0}, are interpreted as discrete monoidal stacks. The additional data provided by the two compatible trivializations allow to conclude that φ_{E} is compatible with descent along the presentations \xymatrix{H_{0} ⋉ H_{1} \ar@{->>}[r] & H_{0}} → ℋ and \xymatrix{K_{0} ⋉ K_{1} \ar@{->>}[r] & K_{0}} → 𝒦 by the groupoids determined by their respective crossed modules, so it determines a bimonoidal functor F_{E}: ℋ × 𝒦 → 𝒢. With the obvious notion of morphism, we obtain:

1.1. Theorem. [Theorem 8.1 in the main text] There is an equivalence of (pointed) groupoids

u: Biext(H_{•}, K_{•}; G_{•}) ⟶^{∼} Hom(ℋ , 𝒦 ; 𝒢),

where the right hand side denotes the groupoid of bimonoidal functors, and left hand side that of biextensions equipped with the aforementioned trivializations.

This result is actually valid over a variable object S of T, hence we have a similar statement where the left and right side above are replaced by the corresponding stacks, which one obtains by letting S → Biext(H_{•}|_{S}, K_{•}|_{S}; G_{•}|_{S}), and similarly for the right hand side.

Bimonoidal categories or stacks provide examples of bimonoidal morphisms. If ℛ is bimonoidal, it has two monoidal structures, say ⊞ and ⊠, satisfying an appropriate set of axioms [Lap72a, Lap72b]; the distributivity one, in particular, says that ⊠ is a bimonoidal functor ⊠: ℛ × ℛ → ℛ with respect to the other structure, ⊞. Now, assuming ⊞ to be group-like (see [BDRR13] for passing from a merely additive monoidal to a full group-like one, i.e. from “rig” to “ring”) and ℛ to have a presentation of the form \xymatrix{R_{0} ⋉ R_{1} \ar@{->>}[r] & R_{0}} → ℛ as above, the bifunctor ⊠ can be described by a biextension E_{⊠} of (R_{0}, R_{0}) by R_{1} → R_{0}. We are interested in extracting information about ℛ, especially of a cohomological nature such as the characteristic class, from the biextension E_{⊠}. Information of this kind, which reduces to (usually complicated) cocycle calculations, ordinarily come from the coherence diagrams of ⊠, among others. These diagrams involve morphisms like ⊠ ∘ (⊠ × id) and ⊠ ∘ (id ×⊠), which are monoidal in each of the three variables, and higher iterations of compositions involving ⊠ and id. Ideally, each of those multi-functors corresponds to some kind of iteration of E_{⊠}. So we need to generalize the representation of bimonoidal functors by biextensions to an arbitrary number of variables.

Extending the concept of bimonoidal functor to n variables is immediate, the only difference being the same compatibility condition we have in the n = 2 case must hold for each pair of variables. Therefore, for the multivariable analog of the right hand side in the above statement, the only substantial difference is in the bookkeeping aspect, and we immediately see that monoidal functors in n variables can be composed—provided we restrict ourselves to considering braided objects. While this is easy to verify, it has the far-reaching consequence that braided monoidal categories, or more generally stacks, comprise the 2-categorical analog of a multicategory [CS10, Shu10] we denote M𝔅𝔊𝔯𝔖𝔱, where 𝔅𝔊𝔯𝔖𝔱 is the 2-category they form relative to ordinary unary monoidal functors.

As for the left hand side of the equivalence in the theorem, we must extend the concept of biextension to n variables and investigate whether such objects admit a composition law. We define a multi-extension, or an n-extension, by G_{1} → G_{0} as a (G_{1}, G_{0})-torsor (same as for biextensions) over an n-fold product, this time equipped with n partial product laws which are required to be pairwise compatible in the same manner as those in a biextension. (The generalization of a biextension to n variables, in the fully abelian context, is outlined in a remark in [Gro72].) Given n crossed modules H_{i,•} = (∂: H_{i,1} → H_{i,0}), i = 1, …, n, we are interested in the n-extensions of H_{1,0} × ⋯ × H_{n,0} by G_{1} → G_{0} which are also equipped with n compatible trivializations of their pullbacks along each of the n morphisms ∂_{i} = id × ⋯ × ∂ × ⋯ × id. We call them n-butterflies. Now, a direct extension of the above theorem (cf. Theorem 10.1 below) provides an equivalence

MExt(H_{1,•}, …, H_{n,•}; G_{•}) ⟶^{∼} Hom(ℋ_{1}, …, ℋ_{n}; 𝒢)

of pointed groupoids, where each H_{i,•} is a presentation of the corresponding ℋ_{i}, and G_{•} of 𝒢, the left hand side is the groupoid of n-extensions, and the right hand side denotes the groupoid of n-monoidal functors ℋ_{1} × ⋯ × ℋ_{n} → 𝒢.

These multi-butterflies can be composed in a way reminiscent of the unary case discussed in [AN09, Noo05], that is by “wing juxtaposition,” which turns out to be associative up to coherent isomorphism. Wing juxtaposition (we informally use this term thanks to the shape of the composition diagrams in the unary case) is an operation which allows to associate to butterflies F_{1}, …, F_{n}, E, where E is an n-butterfly, a new one, denoted E(F_{1}, …, F_{n}), whose underlying bitorsor is a quotient of the pullback of E to the product F_{1} ×⋯×F_{n}—each F_{i} is equipped with an equivariant section with values in the i^{th} factor in the base of E; the pullback is along the product of these maps. The upshot is that braided crossed modules of T now form a bi-multicategory (i.e. bicategorical analog of a multi-category) of their own if we use these multi-butterflies as morphisms. In addition, the composition is compatible with the previous equivalence. More precisely, we have:

1.2. Theorem. [Theorem 11.1, Propositions 11.3 and 11.4, and Theorem 12.3] Braided crossed modules of T, equipped with the groupoids MExt(H_{1,•}, …, H_{n,•}; G_{•}) as Homcategories, form a bi-multicategory M𝔅𝔛𝔐𝔬𝔡. Further, there is an equivalence of bimulticategories

M𝔅𝔛𝔐𝔬𝔡 ⟶^{∼} M𝔅𝔊𝔯𝔖𝔱,

induced by the associated stack functor, carrying each MExt(H_{1,•}, …, H_{n,•}; G_{•}) to Hom(ℋ_{1}, …, ℋ_{n}; 𝒢).

This is the multi-categorical analog of the unary case treated in [AN09]^{1}, which states the bicategory 𝔅𝔛𝔐𝔬𝔡 of braided crossed modules, equipped with groupoids of spans as categories of morphisms, is equivalent (as a bicategory) to 𝔅𝔊𝔯𝔖𝔱. The proof, in particular that of Theorem 11.1, is geometrical, unlike that of corresponding result in [AN09, Noo05]. It requires showing first that the pullback of the bitorsor E to the product F_{1} × ⋯ × F_{n} is equivariant under the left and right actions of H_{1,1} × ⋯ × H_{n,1}, and that it descends to the base of F_{1} × ⋯ × F_{n}; second, that this descended bitorsor has the structure of a multi-butterfly.

We can now discuss bimonoidal categories from the broader and more convenient perspective afforded by the idea of a bi-multicategory. If Mℭ is a bi-multicategory, we informally look into monoids with respect to the multi-composition structure of Mℭ. Formally, we say that an object X of Mℭ is a (pseudo-)monoid of ℭ, the underlying bicategory of Mℭ, if it is a pseudo-algebra over the club N, where N, after [Kel72], denotes the natural numbers equipped with the category structure described in section A; a club is a multicategory with one object.^{2} The monoidal structure carried by X is of the “unbiased” kind [Lei04], namely we have an n-ary operation m_{n}: X × ⋯ × X → X for each object n ∈ N, as opposed to a privileged binary one, with the various coherence conditions being taken care of by the pseudo-algebra structure.

1In fact the unary version proved therein holds without any commutativity assumption.

2For the sake of simplicity we are ignoring the “extraordinary” structure given by the action of permutations.

We apply the previous observation to ℭ = 𝔅𝔊𝔯𝔖𝔱 or ℭ = 𝔅𝔛𝔐𝔬𝔡. We say that ℛ is weakly ring-like (for want of a better name) if it is a pseudo-monoid of 𝔅𝔊𝔯𝔖𝔱. The “weakly” adverb refers to the fact that the underlying categorical group structure of ℛ is only braided commutative, whereas we usually define categorical rings as having underlying braided symmetric categorical groups (see, e.g. [JP07]). Thus “weak” does not mean “not strict,” but it signals the commutative law of the additive structure of ℛ is weaker than usually required.

We make a similar definition for a pseudo monoid in 𝔅𝔛𝔐𝔬𝔡: this is a novel object comprised of a braided crossed module R_{•} and multi-extensions E_{n} of (R_{0}, …, R_{0}) (n factors) by R_{1} → R_{0} for each n ∈ N, subject to the conditions dictated by the pseudoalgebra structure. We show that even in the realm of a genuine bicategory such as 𝔅𝔛𝔐𝔬𝔡, we obtain coherence conditions resembling a weak version of Mac Lane’s pentagon (see sections B and C, in particular diagram (C.2)).

Since pseudo-monoids are transported across equivalences, pseudo-monoids in 𝔅𝔊𝔯𝔖𝔱 are presented by those in 𝔅𝔛𝔐𝔬𝔡. We have:

1.3. Proposition. [Proposition 13.5, Proposition 14.1, and Corollary 14.2] Let ℛ be (weakly) ring-like. Let \xymatrix{R_{0} ⋉ R_{1} \ar@{->>}[r] & R_{0}} → ℛ be a presentation by a braided crossed module. Then R_{•} is equipped with biextensions E_{n}, n ∈ N as above so that (R_{•}, E_{•}) is a pseudomonoid in 𝔅𝔛𝔐𝔬𝔡. Furthermore, if A = π_{0}(ℛ) ≃ coker ∂ and M = π_{1}(ℛ) ≃ ker ∂, then A is a (possibly non-unital) ring of T and M is an A-bimodule. The converse also holds.

We may ask whether the weakly ring-like structure is strong enough to force the underlying braiding to become symmetric. The answer is affirmative, at least in the more interesting (to us) case where the multiplicative structure has a unit object. As we shall see, this is tied to the seemingly unrelated problem of computing a characteristic class for weakly categorical rings.

Our procedure is a bit nonstandard in two aspects. The first is that to compute the class corresponding to ℛ we only choose local data for the underlying braided group-like structure of ℛ, or which is the same thing, for the presentation. We compute the rest of the cocycle from the pseudo-monoidal structure carried by the presentation, by examining the biextension’s behavior with respect to the chosen local data.

The second aspect is that we choose to initially dispense with the symmetry condition, therefore we start with a choice of local data for the underlying braided group-like structure of ℛ. This yields an invariant in the Eilenberg Mac Lane group H^{4}(K(A, 2), M), lifting to H^{5}(K(A, 3), M) if braiding is symmetric [JS93, Bre99].^{3} In fact the initial choice for the local data steers the computation into being based on the Eilenberg Mac Lane iterated bar construction [EML53, EML54], in place of the more customary Mac Lane’s Q-construction. (In the stack context, the iterated bar construction emerges as part of the decomposition of ℛ as a gerbe over A, together with its additional braiding structure [Bre94a, Bre99].)

3In a general topos we should interpret them as hypercohomology groups. See the discussion in [Bre69, §2], [Bre94a, §7], [Bre99, §6].

Over a point, a standard categorical ring gives rise to an element in the third Mac Lane cohomology group HML^{3}(A, M) [JP07] (a gap was filled in [Qua13]; see also [Lod98, chap. 13] for a general definition), so the question is whether our procedure yields anything standard, and in particular the expected invariant in HML^{3}(A, M).

That it does, at least in the unital case, is due to the fact that, by [ML58, §11], the third Mac Lane cohomology can also be computed by employing the infinite bar construction B^{∞}(A) := B(A, 1) ⊆ B(A, 2) ⊆ ⋯ instead of the cubical complex Q(A).^{4} One needs a product structure on B^{∞}(A), explicitly up to degrees corresponding to the subcomplex B(A, 2), to use in the multiplicative bar construction. It follows that we can plug one of B(A, 2), B(A, 3), or B^{∞}(A) to calculate a cohomology of A with values in the bimodule M. In the unital case, all these choices give rise to isomorphic cohomologies. The product structure on B(A, 2), which is unusual, is explicitly given in Mac Lane’s work. We recover that product, and hence the rest of the structure, directly from the pseudo-monoidal structure attached to the presentation of ℛ, namely the biextension plus the various compatibility conditions it satisfies.

We do not obtain the representing cocycle for the class of ℛ in the form corresponding to class in HML^{3}(A, M) right away. Instead, we find that the class of ℛ is represented by a twisted cocycle in the multiplicative bar resolution for B(A, 2), but the twisting disappears in the unital case, which yields the desired result. In particular this implies that the braiding is necessarily symmetric. More precisely, we have:

1.4. Theorem. [Theorem 15.1] There is a bijective correspondence between equivalence classes of weak ring-like stacks ℛ, with A = π_{0}(ℛ) and M = π_{1}(ℛ), and twisted classes in H̃_{2}^{3}(A, M) defined below (cf. Definition 15.9). In the unital case, i.e. when the external monoidal structure of ℛ has a unit element, A is likewise unital and the underlying braiding of ℛ is necessarily symmetric. Hence the weak ring-like structure of ℛ is fully ring-like, and [ℛ] ∈ H̃_{2}^{3}(A, M) ≃ HML^{3}(A, M).

Hence, unital categorical rings in the sense of the present paper are categorical rings tout-court and, by [ML58], we re-obtain their classification in terms of HML^{3}(A, M). Had we chosen to work with braided symmetric objects from the beginning there is little doubt the same procedure—i.e. a decomposition of the underlying group-like stack followed by an analysis of the attached biextension—would have yielded a Mac Lane cocycle in the right group right from the start. However, part of the interest was to test to what extent our framework successfully captures the standard theory.

Organization of the material. We have collected various preliminary items in section 2, including a quick review of standard biextensions. The recapitulation of the relation between group-like stacks and their presentations by crossed modules, though standard, involved a contravariance issue which becomes relevant, therefore we provided a brief outline.

4Recall that classically MacL̃ane cohomology is defined as the Hochschild cohomology of the complex Q(A). The latter computes the stable homology of the Eilenberg-Mac Lane spaces, that is, H_{q}(Q(A)) ≃ H_{n+q}(K(A, n)), n > q.

Aside from the preliminary section, we can divide the rest in roughly three parts, plus another containing some appendices.

We develop the generalization of biextensions and their symmetry properties in sections 3 to 6, and the butterfly special kind, including the representation of bimonoidal functors, from section 7 to 9.

The extension to an arbitrary number of variables takes sections 10 to 12. In particular, the composition of butterflies is discussed in section 11. There, Theorem 11.1 establishes the existence of the wing juxtaposition operation. Its proof is technically involved bookkeeping, but it ultimately is straightforward; it is reproduced in full because, while the juxtaposition product has features similar to the unary case, the proof itself is not an immediate generalization of the corresponding one in [AN09, Noo05]. Note also that the constructions in the proof are used in the cocycle computations of sect. 15.13.

The bi-multicategories comprised by braided group-like stacks on one side, and braided crossed modules of T on the other, are discussed in section 12.

The idea of categorical ring, or ring-like stack, and its presentation as a pseudo-monoids in their respective bicategories is expounded in sections 12 and 14, where we also discuss some specific facts about the presentations. In section 15 we discuss the cohomology of rings and the computation of the characteristic class. We do this in slightly simplified form by omitting the additional computations which arise because the choices for the various trivializations required for the cocycle computations are necessarily local. The complete computations are deferred to Appendix E.

To avoid unnecessary and possibly long detours, parts of the material have been placed in a number of appendices. Some of this material is necessary for self-consistency but known to the experts. Thus, bi-multicategories are discussed in appendices A and B, the analog of the pentagon in a bicategory in C, and appendix D contains just some technical lemmas pertaining to section 6. Finally, the resolution of a simplicial object by hypercovers and the complete hyper-cohomology computations necessary for a full calculation of the characteristic class are contained in Appendix E.

What to read. One possibility is to only read Part I followed by Part III, if the reader is willing to only skim through Part II. For the latter, it is possible to just read the statements, in particular for the n-fold composition of multi-extensions theorem proved in sect. 11, only referring to the proof of 11.1 when needed in sect. 15.13. Alternatively, one can read Part III, in particular section 15, only skimming through the previous two, and omitting most of section 15. Parts II and III depend on the multivariable functor calculus in a bicategory, an account of which is contained in the first two appendices, which may only be referred to when needed. Most their content should be known to the experts. An account of what the pentagonal axiom would look like in a bicategory is contained in Appendix C, and in general a reader will need only equation (C.1) and diagram (C.2). While in section 15.13, the reader can refer to Appendix E for the full hyper-cohomology arguments.

Notations and conventions. The convention we use is to equate “additive” with “monoidal,” therefore the term “biadditive” means “bimonoidal” in the sense used above in the introduction. By extension, “n-additive” means “n-monoidal,” i.e. monoidal in each of the n-variables.

A notation of the form G_{•} or G denotes a crossed module (G_{1}, G_{0}, ∂), where ∂: G_{1} → G_{0} is the homomorphism, and the (right) action is denoted element-wise by (x, g) ↦ g^{x}, but is not otherwise labeled. Very often we will simply write (G_{1}, G_{0}).

For ease of notation we will often use the convention: (G_{1}, G_{0}) ≅ (G, Π) and, later in the paper, (R_{1}, R_{0}) ≅ (R, Λ) when we discuss ring-like structures.

Up to section 13, all monoidal structures are notated multiplicatively, with I denoting the identity object. We switch to an additive notation for one of the two monoidal structures in a bimonoidal situation. In that case, the identity object is 0. If this is the structure for which we construct a presentation by a crossed module, then we use an additive notation for the groups in the crossed module, even though they are by no means assumed commutative.

The notation G_{S} denotes the base-change of G to S, that is, S×G → S. If G is a group object, then G_{S} is group over S, with group law (s, g)(s, g′) = (s, gg′), in set-theoretic notation.

Acknowledgements. I would like to thank Sandra Mantovani, Giuseppe Metere, Cristiana Bertolin, and Enrico Vitale for interesting discussions and the kind hospitality extended to me by their institutions in Milan, Turin, and Louvain-la-Neuve, where parts of this work were presented.

Special thanks are also due to the anonymous referee, whose pointed criticisms and inquiries led to a much improved version of this work.

Part I

2. Preliminaries

In the following, let T be a topos which is assumed to be Sh(C) for a site C. For the reader’s convenience we recall some well know facts on monoidal stacks and their relations with crossed modules of T.

2.1. Group-like stacks, crossed modules, and bitorsors. We follow [Bre90, AN09], to which we refer for further details. A crossed module of T consists of a group homomorphism ∂: G → Π of T and a right action of Π on G such that: (1) ∂ is a morphism of right Π-objects, with Π acting on itself by conjugation; (2) the action of G on itself induced by the Π-action via ∂ coincides with conjugation. If g, h are (generalized) points of G and x of Π we have the familiar conditions:

∂(g^{x}) = x^{−1} ∂(g) x
g^{∂(h)} = h^{−1} g h,

where the exponent notation denotes the action.

The crossed module G gives rise to a groupoid Γ: \xymatrix{Π ⋉ G \ar@{->>}[r] & Π}, and hence, via the nerve construction, to a simplicial group N G whose object in degree p is (⋯ (Π ⋉ G) ⋉ ⋯ ⋉ G) (p factors). The groups π_{1} = ker ∂ and π_{0} = coker ∂ are the homotopy (sheaves of) groups of G_{•} are in fact the only two nonzero homotopy sheaves of N G. It is easily verified that π_{1} is abelian and central in G and that π_{0} is the sheaf of connected components of the groupoid corresponding to G (and hence of N G).

The group laws of G and Π ⋉ G equip the groupoid Γ with a structure of strict categorical group, that is, a morphism

m: Γ × Γ ⟶ Γ

of groupoids satisfying the usual axioms of a group-like, strict monoidal category (see also [JS93]).

The associated stack construction performed on Γ yields a stack 𝒢 of T, which inherits the—now lax—group-like structure. (“The” associate stack is only unique up to equivalence. The specific model we consider is the one whose fibered categories consist of descent data for Γ and their morphisms—see [LMB00, Lemme 3.2]. It is easy to see that in the case at hand these descent data are cocycles with values in Γ—see[AN09, §3.3-4].) Thus, 𝒢 is a monoidal stack, in the sense that there exists a stack morphism

m: 𝒢 × 𝒢 ⟶ 𝒢 ,

and an associator μ: m ∘ (m × 1) ⇒ m ∘ (1 × m) satisfying the axioms of a monoidal category. We specify the rest of the group-like structure by requiring the existence of a unit object I, which we can identify with a morphism I: e → 𝒢 (where e is the terminal object of T), and that left and right-multiplications by a fixed object x induce functorial equivalences

m(x, −): 𝒢 ⟶ 𝒢 m(−, x): 𝒢 ⟶ 𝒢 .

The group-like stack obtained in this way from a crossed module will be denoted [∂: G → Π]^{∼}.

There is a compelling geometric picture for the associated stack obtained in this way from a crossed module. Recall the stack Tors(G, Π) is the stack of right G-torsors P whose structure group extension P ∧^{G} Π is isomorphic to the trivial torsor [Del79]. Equivalently, an object of this stack is a pair (P, s) where P is a G-torsor and s: P → Π a G-right-equivariant morphism, where G acts on Π on the right via multiplication. Objects of Tors(G, Π) are called (G, Π)-torsors. The notion makes sense for each group homomorphism, but if ∂: G → Π is a crossed module, then Tors(G, Π) becomes grouplike. Indeed, each torsor is also equipped with a left G-action defined by

gu = ug^{s(u)},

where u ∈ P and g ∈ G, which makes it into a G-bi torsor. The monoidal structure is given by the contracted product

m((P, s), (Q, t)) = (P ∧^{G} Q, st),

and it is easily proved to be group-like. Finally, the stack Tors(G, Π) is equivalent, as a group-like stack, to [∂: G → Π]^{∼}, obtained via the associated stack construction described above.

Conversely, for any group-like stack 𝒢 there exists a presentation by a crossed module, namely there exists a crossed module G = (G, Π, ∂) and an equivalence 𝒢 ≃ [∂: G → Π]^{∼}. For any group-like stack 𝒢 we can define π_{1} = Aut_{𝒢} (I), and π_{0} as the sheaf associated to the presheaf of connected components. Upon choosing a presentation, its homotopy sheaves are isomorphic to the ones of 𝒢.

In view of the previous geometric interpretation, 𝒢 is also equivalent to Tors(G, Π). In addition, the sequence

(2.1)  \xymatrix{G \ar[r]^{∂} & Π \ar[r]^{π} & 𝒢},

where each object is regarded as a stack, is exact in a homotopical sense. If 𝒢 is identified with Tors(G, Π), the projection Π → 𝒢 sends x ∈ Π to the bitorsor (G, x), that is the trivial right G-torsor whose left action on itself is given by g · u = g^{x}u, with u, g ∈ G. (This follows from the fact that x is identified with the equivariant section that assigns x to the unit section e_{G}.) In particular, (G, e_{Π}) can be identified with the unit object of 𝒢.

2.2. Remark. Since a morphism φ: (P, s) → (Q, t) in Tors(G, Π) has the form

\xymatrix{P \ar[dr]_{s} \ar[rr]^{φ} &  & Q \ar[dl]^{t} \\  & Π}

it is immediately seen that the equivalence

[∂: G → Π]^{∼} ⟶ Tors(G, Π)

is contravariant. In particular, if g: x → x′ is a morphism in the strict categorical group Γ, that is, x′ = x ∂g, the corresponding morphism of (G, Π)-torsors is (G, x′) → (G, x).^{5}

A morphism of group-like stacks is a stack morphism F: ℋ → 𝒢 preserving the monoidal structure. A morphism of crossed modules determines one between the associated group-like stacks in the obvious way. In the converse direction, a morphism F: ℋ → 𝒢 only determines a morphism in the homotopy category between corresponding crossed modules. This morphism can be represented by a butterfly, namely a diagram of group objects of T of the form:

\xymatrix{H \ar[dd]_{∂} \ar[dr]^{σ} &  & G \ar[dd]^{∂} \ar[dl]_{ı} \\  & E \ar[dl]_{π} \ar[dr]^{ȷ} \\ Σ &  & Π}

5There appears to be no good way to get around the issue. A “fix” is to replace the crossed module with a left one. This restores the expected direction of the arrows, at the cost of turning one of the monoidal laws into the opposite one (see e.g. [Bre90, after Théorème 4.5]).

(where the vertical arrows are crossed modules) from which a fraction representing the morphism F: ℋ → 𝒢 can be obtained (see loc. cit. and below for more details).

2.3. Braidings. A braided crossed module, (see [JS93, AN09]), is a crossed module G = (G, Π, ∂) equipped with a bracket

{−, −}: Π × Π ⟶ G

such that ∂{x, y} = y^{−1}x^{−1}yx, for all points x, y of Π. The bracket satisfies:

(2.2)  \begin{aligned}\{x, yz\} & = \{x, y\}^{z}\{x, z\}, & \{x, ∂h\} & = h^{−1}h^{x}, \\ \{xy, z\} & = \{y, z\}\{x, z\}^{y}, & \{∂g, y\} & = g^{−y}g, \end{aligned}

for all x, y, z ∈ Π and g, h ∈ G. These properties all arise from the observation that the bracket {−, −} corresponds to a braiding in the usual sense for the categorical group determined by G_{•}. Thus c_{x,y} = {x, y} gives a family of functorial isomorphisms

c_{x,y}: xy ⟶ yx

for each pair (x, y) of objects in Γ, so that c: m ∘ τ ⇒ m: Γ × Γ → Γ, where τ is the interchange functor. The properties above can be derived from functoriality and Mac Lane’s hexagon diagrams. Conversely, if Γ is a braided strict categorical group, setting {x, y} = y^{−1}x^{−1}c_{x,y}: e → y^{−1}x^{−1}yx defines a braiding on the corresponding crossed module.

The braiding is symmetric if it has the property that {y, x} = {x, y}^{−1} for all x, y ∈ Π. A symmetric braiding is Picard if in addition it satisfies the condition {x, x} = e. These conditions match the corresponding ones for the categorical group Γ. As observed in [Bre94a, §1], for a Picard crossed module the bracket is a full lift of the commutator map.

The braided, symmetric, and Picard structures translate in the expected manner to the associated group-like stack 𝒢 ≃ [∂: G → Π]^{∼}, and conversely, if 𝒢 is a braided (resp. symmetric, Picard) stack with presentation given by [∂: G → Π], then the latter acquires a braided (resp. symmetric, Picard) structure as above. Let us observe here that a braiding on 𝒢 = Tors(G, Π) induces one on the crossed module by way of the butterfly representing the morphism m: 𝒢 × 𝒢 → 𝒢. In the Picard case this leads to two possible presentations: by a braided crossed module satisfying the Picard condition, or, according to Deligne [Del73], by a length-one complex of abelian sheaves. We refer the reader to [AN09, §7] for full details on this correspondence. Here we limit ourselves to observe that if (P, s) and (Q, t) are two (G, Π)-torsors, the choice of two sections u ∈ P and v ∈ Q allows us to write the braiding morphism c_{P,Q}: P ∧^{G} Q → Q ∧^{G} P as c_{P,Q}(u ∧ v) = (v ∧ u) χ_{u,v}, where χ: P ∧^{G} Q → G, is a coordinate representation of c_{P,Q} subject to certain equivariance conditions dictated by the requirements:

c_{P,Q}(gu ∧ v) = g c_{P,Q}(u ∧ v),
c_{P,Q}(ug ∧ v) = c_{P,Q}(u ∧ g v),
c_{P,Q}(u ∧ vg) = c_{P,Q}(u ∧ v) g.

2.4. Lemma. The coordinate χ: P ∧^{G} Q → G has the expression:

(2.3)  χ_{u,v} = {s(u), t(v)}^{−1}.

Proof Proof (Sketch). The relation (2.3) is immediate for (P, s) = (G, x) and (Q, t) = (G, y), where x, y ∈ Π, using Remark 2.2. The general case follows by descent by exploiting the equivariance conditions for χ and for {−, −} recalled above.

2.5. Biextensions. Biextensions were introduced in Mumford [Mum69], and later reexamined by Grothendieck [Gro72]. We refer to the latter and the text by Breen [Bre83] for details on the standard concept of biextensions. We briefly recall the basic definitions, and later extend them to introduce crossed modules as coefficients.

Let H, K, G be groups of T, with G assumed to be abelian. A biextension of H, K by G is a G_{H×K}-torsor E on H × K equipped with two partial composition laws ×_{1} (resp. ×_{2}) making E a central extension of H_{K} (resp. K_{H}) by G_{K} (resp. by G_{H}). This means that ×_{1} (resp. ×_{2}) is only defined on E ×_{K} E (resp. E ×_{H} E), where E ×_{K} E (resp. E ×_{H} E) means fiber product over K (resp. H) with respect to the obvious projection. These composition laws are required to be compatible with one another. Usually one also requires ×_{1} and ×_{2} to be commutative, which make sense whenever H and K are abelian. Our definition, in which ×_{1} and ×_{2} are not required to be commutative is then referred to as a weak biextension. Even when H and K are commutative, this notion of biextension is weaker than that of loc. cit. in which the partial multiplication laws are commutative.

Analogously to extensions, the composition laws can be represented by morphisms of G-torsors. Let B H and B K be the standard classifying simplicial objects of T.^{6} The bisimplicial object B H × B K has face maps d^{h} and d^{v} (see e.g. [GJ99]). In particular, for i = 0, 1, 2 we consider d_{i}^{h}: H × H × K → H × K and d_{i}^{v}: H × K × K → H × K. Then ×_{1} and ×_{2} correspond to morphisms of G-torsors

(2.4a)  γ^{1} : d_{2}^{h}^{∗}E ∧^{G} d_{0}^{h}^{∗}E ⟶ d_{1}^{h}^{∗}E

and

(2.4b)  γ^{2} : d_{2}^{v}^{∗}E ∧^{G} d_{0}^{v}^{∗}E ⟶ d_{1}^{v}^{∗}E.

The associativity and commutativity diagrams for γ^{1} and γ^{2} are the obvious ones. More interesting is the compatibility condition of the two structures, which can be expressed as follows.

Let E_{h,k} be the fiber of E over a generalized point (h, k) of H × K. Then the compatibility condition reads

6That is, B H = N H[1], where H[1] is the groupoid with a single object e and Aut(e) = H.

(2.5)  \xymatrix{ & E_{h,k}E_{h′,k}E_{h,k′}E_{h′,k′} \ar[rr] \ar[dl]_{γ_{h,h′;k}^{1}γ_{h,h′;k′}^{1}} &  & E_{h,k}E_{h,k′}E_{h′,k}E_{h′,k′} \ar[dr]^{γ_{h;k,k′}^{2}γ_{h′;k,k′}^{2}} \\ E_{hh′,k}E_{hh′,k′} \ar[drr]_{γ_{hh′;k,k′}^{2}} &  &  &  & E_{h,kk′}E_{h′,kk′} \ar[dll]^{γ_{h,h′;kk′}^{1}} \\  &  & E_{hh′,kk′}}

where we have suppressed the torsor contraction symbol. The horizontal arrow is the canonical symmetry map swapping the factors in Tors(G), which exists whenever G is abelian. A more intrinsic way to express the same thing is to observe that the above compatibility condition amounts to the equality

(2.6)  γ_{d_{1}^{h}^{∗}E}^{2} ∘ (d_{2}^{v}^{∗}γ^{1}d_{0}^{v}^{∗}γ^{1}) = (1 × c × 1) ∘ γ_{d_{1}^{v}^{∗}E}^{1} ∘ (d_{2}^{h}^{∗}γ^{2}d_{0}^{h}^{∗}γ^{2})

as morphisms from (d_{0}^{h}d_{2}^{v})^{∗}E ∧ (d_{2}^{h}d_{0}^{v})^{∗}E ∧ (d_{0}^{h}d_{0}^{v})^{∗}E ∧ (d_{1}^{h}d_{1}^{v})^{∗}E to (d_{1}^{h}d_{1}^{v})^{∗}E over H × H × K × K. c is the symmetry morphism which swaps the two terms in the middle.

With the straightforward notion of morphism, when H, K are also abelian, biextensions of T form a Picard category Biext(H, K; G). The relative version of it, where we consider the various Biext(H_{S}, K_{S}; G_{S}) over a variable base S, is a Picard stack, denoted Biext(H, K; G). As observed in Grothendieck [Gro72] and Breen [Bre83], Biext(H, K; G) is biadditive in all three variables.

2.6. Schreier-Grothendieck-Breen theory of extensions. It is well known that an extension E of K with a possibly nonabelian kernel G determines a morphism ȷ: E → Aut(G) from the action of E on G by conjugation (see, e.g. [ML95]). A refinement of this situation is when G is part of a crossed module ∂: G → Π. Since part of the crossed module data is precisely a homomorphism Π → Aut(G), an extension of K by that crossed module, or by (G, Π) for short, is a commutative diagram of group objects of the form [Bre90, §8]

\xymatrix{1 \ar[r] & G \ar[r]^{ı} \ar[d]_{∂} & E \ar[r]^{p} \ar[dl]^{ȷ} & K \ar[r] & 1 \\  & Π}

where the row is exact and ı(g)e = eı(g^{ȷ(e)}), for points g of G and e of E.

An equivalent characterization is that E is a (G_{K}, Π_{K})-torsor over K, equipped with (G, Π)-torsor isomorphisms

γ_{k,k′}: E_{k} ∧^{G} E_{k′} ⟶^{∼} E_{kk′}

satisfying the standard associativity condition [Gro72, Bre90, AN10], such that the equivariant structural morphism to Π is a homomorphism. By loc. cit. and [AN10], such an extension corresponds to a group-like stack morphism F: K → [G → Π]^{∼}, where K is identified with the (discrete) group-like stack. In this sense the above diagram is a butterfly representing this morphism.

In all these characterizations the morphism ȷ arises as the structural equivariant section of the (G, Π)-torsor, whereas ı is the identification G ≃ E_{1}, where the (G, Π)-torsor E_{1}, isomorphic to the unit one, is the fiber over the unit section of K. This identification is explicitly given as ı(g) = g e_{0} = e_{0} g, where e_{0} is the central section of E_{1} corresponding to the unit section of G. Since ȷ is a homomorphism, we obtain the commutativity of the “wing” in the diagram above. The conjugation property satisfied by ı and ȷ reflects the structure of (G, Π)-torsor of E. In particular, we have that ı(g)e (resp. eı(g)), product in E, agrees with the left (resp. right) G_{1}-action on E.

3. Biadditive morphisms

If H, K, G are groups of T, a biadditive (or bimultiplicative) morphism is a map f: H × K → G which is a homomorphism in each variable. For abelian groups this is none other than a Z-bilinear morphism.

More generally, if H, K, and G are categories, which we assume to be group-like, we say that a bifunctor is biadditive (or bimultiplicative) if it is monoidal in both variables, in a compatible way.^{7} For this, we must also assume G be endowed with a braiding c. More precisely, we have:

3.1. Definition. A bifunctor F: H × K → G is biadditive if:

1. it has the structure of additive functor with respect to each variable, namely there exist functorial (iso)morphisms

λ_{h,h′;k}^{1} : F (h, k)F (h′, k) ⟶ F (hh′, k) and λ_{h;k,k′}^{2} : F (h, k)F (h, k′) ⟶ F (h, kk′)

satisfying the standard associativity conditions and compatibility with the braiding of G;

2. for all objects h, h′ of H and k, k′ of K we have a functorial commutative diagram (3.1)

\xymatrix{(F (h, k)F (h′, k)) (F (h, k′)F (h′, k′)) \ar[rr]|{ĉ} \ar[d]_{λ_{h,h′;k}^{1}λ_{h,h′;k′}^{1}} &  & (F (h, k)F (h, k′)) (F (h′, k)F (h′, k′)) \ar[d]^{λ_{h;k,k′}^{2}λ_{h′;k,k′}^{2}} \\ F (hh′, k)F (hh′, k′) \ar[r]_{λ_{hh′;k,k′}^{2}} & F (hh′, kk′) & F (h, kk′)F (h′, kk′) \ar[l]^{λ_{h,h′;kk′}^{1}}}

where the upper horizontal arrow is the “commuto-associativity” morphism

(x y) (z w) ⟶ (x z) (y w)

7A better choice would be to use “bi-monoidal” in place of biadditive. The latter is motivated by continuity with the naming convention in [AN09], where morphisms between gr-stacks are called additive—a convention derived from the particular but significant Picard case.

arising from the braiding of G (an explicit definition can be found in [JP07]);

3. the two morphisms F (I_{H}, I_{K}) → I_{G} that can be deduced from the first condition coincide.

The second condition ensures that the two possible ways to compute F (hh′, kk′) agree. The obvious similarity with Diagram (2.5) will be exploited below.

The definition can be extended to the case where all three categories are braided. In this case, we add to the biadditivity the condition that F be braided in each variable, namely that the following diagrams commute:

\xymatrix{F (h, k)F (h′, k) \ar[r]^{λ^{1}} \ar[dd]_{c_{F,F}} & F (hh′, k) \ar[dd]^{F (c,k)} &  & F (h, k)F (h, k′) \ar[r]^{λ^{2}} \ar[dd]_{c_{F,F}} & F (h, kk′) \ar[dd]^{F (h,c)} \\  &  & and \\ F (h′, k)F (h, k) \ar[r]_{λ^{1}} & F (h′h, k) &  & F (h, k′)F (h, k) \ar[r]_{λ^{2}} & F (h, k′k)}

A natural transformation φ: F ⇒ F′ between two biadditive morphisms is a natural transformation of bifunctors such that in each variable the conditions to be a natural transformation of additive functors are satisfied. We denote by Hom(H, K; G) the resulting groupoid of biadditive morphisms.

It is clear that the same definitions can be stated in the case of group-like stacks as opposed to categories. Similarly to the pointwise case, we obtain a groupoid Hom(ℋ , 𝒦 ; 𝒢) of biadditive morphisms. For the latter, by way of a standard process of restricting to variable base S, we obtain a stack Hom(ℋ , 𝒦 ; 𝒢) arising from the various fiber categories S → Hom(ℋ |_{S}, 𝒦 |_{S}; 𝒢 |_{S}).

4. Biextensions by braided crossed modules

We extend the notion of biextension by adapting it to include braided crossed modules as coefficients.

Let G = (∂: G → Π, {·, ·}) be a braided crossed module of T, and let H, K be groups of T.

4.1. Definition. A biextension of H, K by (G, Π) is a (G, Π)_{H×K}-torsor E over H ×K equipped with two partial composition laws ×_{1} (resp. ×_{2}) making E into an extension of H_{K} (resp. K_{H}) by (G, Π)_{K} (resp. by (G, Π)_{H})—cf. sect. 2.6. We require the composition laws to be compatible in the manner described by a diagram formally identical to (2.5) in sect. 2.5.

To be precise, in the last condition of the above definition we must specify that the horizontal arrow in diagram (2.5), adapted to the present case,

(4.1) ĉ: (E_{h,k} ∧^{G} E_{h′,k}) ∧^{G} (E_{h,k′} ∧^{G} E_{h′,k′}) ⟶ (E_{h,k} ∧^{G} E_{h,k′}) ∧^{G} (E_{h′,k} ∧^{G} E_{h′,k′})

is given by the braiding of the stack 𝒢 = [∂: G → Π]^{∼} in a manner analogous to the horizontal arrow of diagram (3.1) in Definition 3.1. A global version of the compatibility condition is given by equation (2.6).

The two partial composition laws give rise to morphisms of (G, Π)-torsors

γ_{h,h′;k}^{1} : E_{h,k} ∧^{G} E_{h′,k} ⟶^{∼} E_{hh′,k}

and

γ_{h;k,k′}^{2} : E_{h,k} ∧^{G} E_{h,k′} ⟶^{∼} E_{h,kk′}

each satisfying the obvious associativity condition, relative to the relevant variables, which can be obtained, mutatis mutandis, from refs. [Gro72, Bre83]. Again, a global version is given by (2.4).

The morphism (4.1) can be explicitly computed through the braiding of G as follows.

4.2. Proposition. Let u, u′, v, v′ be points of E_{h,k}, E_{h′,k}, E_{h,k′}, and E_{h′,k′}, respectively. Then we have

(4.2)  (u ×_{1} u′) ×_{2} (v ×_{1} v′) = (u ×_{2} v) ×_{1} (u′ ×_{2} v′) {f(u′), f(v)}^{−f(v′)},

where f is the equivariant section into the trivial Π-torsor.

Proof. Use Lemma 2.4 to express the morphism E_{h′,k} ∧^{G} E_{h,k′} → E_{h,k} ∧^{G} E_{h′,k}. The right action by f(v′) arises from the relation between the left and right G-torsor structures of E.

4.3. Example. A standard biextension E of H, K by G (all groups assumed commutative) is a biextension in the sense specified above with coefficients in the complex G[1] : [G → 0]. The interchange law, when expressed in terms of points, takes the standard form

(u ×_{1} u′) ×_{2} (v ×_{1} v′) = (u ×_{2} v) ×_{1} (u′ ×_{2} v′).

Dropping the commutativity of the partial multiplication laws, E becomes a biextension of the sort mentioned in the “variants” in [Gro72]. For such a biextension each extension determined by the partial multiplication laws is a central extension.

4.4. Remark. We do not require the partial multiplication laws to be commutative, even if the groups H and K are. If we do require the multiplication laws to be commutative, then G must be abelian and the composite of ∂: G → Π with Π → Aut(G) arising from the crossed module structure, factors through zero. This follows from the fact that the strict commutativity forces the braiding to be identically equal to the identity morphisms, and hence, by Lemma 2.4, we have χ = 1. By evaluating on torsors of the form (G, x) we obtain {·, ·} = 1 identically on Π × Π. Then from the properties (2.2) it follows at once that G and Π must be abelian. As a consequence, the action of ∂g, for any g ∈ G, is trivial, therefore the biextension reduces to one of the type mentioned in Example 4.3 above. See however below for a more appropriate notion of commutativity.

4.5. Definition. A morphism of biextensions of H, K by (G, Π) is a morphism of the underlying (G, Π)-torsors compatible with the partial composition laws.

The compatibility with the composition laws means that there are commutative diagrams of (G, Π)-torsors

\xymatrix{E_{h,k} ∧^{G} E_{h′,k} \ar[dd]_{φ∧φ} \ar[r]^{γ_{1}} & E_{hh′,k} \ar[dd]^{φ} &  & E_{h,k} ∧^{G} E_{h,k′} \ar[dd]_{φ∧φ} \ar[r]^{γ_{2}} & E_{h,kk′} \ar[dd]^{φ} \\  &  & and \\ E′_{h,k} ∧^{G} E′_{h′,k} \ar[r]_{γ′_{1}} & E′_{hh′,k} &  & E′_{h,k} ∧^{G} E′_{h,k′} \ar[r]_{γ′_{2}} & E′_{h,kk′}.}

The trivial biextension is I = (G_{H×K}, e_{Π}). More generally, a biextension will be considered trivial if it is isomorphic, in the sense specified above, to the trivial one. Such an isomorphism amounts to the existence of a “central” section of the underlying (G, Π)- torsor, namely a global s: H × K → E commuting with the left and right actions of G on E, obtained as the image by φ of the unit section e_{G} of I. Then the two above diagrams show that s is a homomorphism in each variable.

With the above notion of morphism and trivial object, biextensions of H, K by G form a pointed groupoid Biext(H, K; G). A standard argument based on descent shows that over a variable base S, the collection of all Biext(H_{S}, K_{S}; G_{S}) forms a stack, equally pointed, denoted Biext(H, K; G). The question of whether either of Biext(H, K; G) and Biext(H, K; G) possess a monoidal structure is more delicate, and it will require additional commutativity properties from G, as discussed below.

The groupoid Biext(H, K; G) is evidently additive with respect to the last variable, namely

Biext(H, K; G_{1} × G_{2}) ≃ Biext(H, K; G_{1}) × Biext(H, K; G_{2}),

and the same holds for Biext(H, K; G). Additivity on the other two variables relies on the commutativity of the partial multiplications for the same reason as [Bre83, §1.2], hence it will not hold in our case.

Biextensions by (G, Π) have certain structural properties analogous to those of extensions reviewed in sect. 2.6, more precisely, we have:

4.6. Lemma. If E → H × K is a biextension of H, K by (G, Π), then E is equipped with a map ȷ: E → Π which is a homomorphism for each partial composition law. For all points e ∈ E_{h,1}, or e ∈ E_{1,k}, and g ∈ G we have the relation

(4.3)  ı(g)e = eı(g^{ȷ(e)}).

and the diagram

(4.4)  \xymatrix{G \ar[d]_{∂} \ar[r]^{ı} & E \ar[r]^{p} \ar[dl]^{ȷ} & H × K \\ Π}

commutes, where ı is the identification (G, e_{Π}) →^{∼} E_{1,1} as (G, Π)-torsors.

The identification of the fiber E_{1,1} above the unit (e_{H}, e_{K}) of H × K with the unit torsor I = (G, e_{Π}) amounts to the definition of exactness of the row in (4.4).

Proof. The arguments are virtually the same as those of [Gro72], except for the part concerning the morphism ȷ. Following loc. cit., we can use the facts about extensions recalled in sect. 2.6 by working over the terminal object K (resp. H) when we view E as an extension of H_{K} (resp. K_{H}) by (G, Π)_{H} (resp. (G, Π)_{K}).

With this in mind, let f = (ȷ, p): E → Π_{H×K}, with ȷ: E → Π, be the equivariant morphism which is part of the (G, Π)_{H×K}-torsor structure. As an extension of H_{K} by (G, Π)_{K}, E has a structure of (G, Π)_{K} ×_{K} H_{K}-torsor over H_{K}, hence f: E → Π_{K} ×_{K} H_{K} ≃ Π × H × K is written as f = (ȷ_{K}, p), where ȷ_{K}: E → Π_{K}. We have ȷ_{K} = (ȷ, p_{K}), where p_{K} (resp. p_{H}) is the composite of p with the projection to K. A similar picture holds by exchanging H with K. Since ȷ_{K} and ȷ_{H} are homomorphisms for the corresponding product structure, ȷ is a homomorphism for both.

Let e_{H/K} = (e_{H}, id_{K}) be the unit section of H_{K} and e_{K/H} = (id_{H}, e_{K}) that of K_{H}. Since the fibers E_{1,k} = e_{H/K}^{∗}E and E_{h,1} = e_{K/H}^{∗}E are respectively identified with G_{K} and G_{H} as (G, Π)-torsors, we have the unit section e_{E/K} (resp. e_{E/H}) in E_{1,k} (resp. E_{h,1}). Now let E_{1,1} = (e_{H}, e_{K})^{∗}E be the fiber over the unit section (e_{H}, e_{K}) of H × K. This is the common pullback of both E_{1,k} and E_{h,1}. The same argument as in [Gro72, §2.2] shows that the restriction e_{E/K} and e_{E/H} to (e_{H}, e_{K}) agree, so let e_{E} ∈ E_{1,1} be their common value. From [Gro72, §2.1] e_{E} is central, since both e_{E/K} and e_{E/H} are, and we define ı by ı(g) = g e_{E} = e_{E} g. Note, ı is the common restriction of ı_{K} and ı_{H} respectively defined by e_{E/K} and e_{E/H}.

Analogously to the case of extensions, the commutativity of the “wing” in (4.4) follows from the equivariance of f, and hence of ȷ, having observed that ȷ(e_{E}) must be equal to the unit of Π. Finally, the relation (4.3) follows from the definition of the bitorsor structure underlying that of a (G, Π)-bitorsor (cf. the end of sect. 2.1).

4.7. Remark. In the converse direction to Lemma 4.6 a biextension of H, K by (G, Π) can be recovered from the datum (E, ×_{1}, ×_{2}, ı, ȷ), where p: E → H × K is an object of T/_{H×K} equipped with two partial composition laws satisfying the interchange law, and ı and ȷ satisfy the conditions of the lemma.

If E′ is a second biextension, with maps ı′: G → E and ȷ′: E → Π satisfying the conditions of Lemma 4.6, a biextension morphism φ: E → E′ makes the following diagram

\xymatrix{G \ar[rr]^{ı′} \ar[d]_{∂} \ar[r]^{ı} & E \ar[r]^{φ} \ar[dl]^{ȷ} & E′ \ar[dll]^{ȷ′} \\ Π}

commutative.

For a biextension which is globally of the form (G_{H×K}, x), where x: H × K → Π characterizes the equivariant structural morphism, we obtain a decomposition in terms of a pair of nonabelian cocycles, which we briefly describe.

The partial multiplication laws are described by

(4.5a)  (h, k, a)(h′, k, a′) = (hh′, k, g_{1}(h, h′; k)a^{x(h′,k)}a′),
(4.5b)  (h, k, b)(h, k′, b′) = (h, kk′, g_{2}(h; k, k′)b^{x(h,k′)}b′).

The actions stem from the change from right to left action,

(h, k, a)(h′, k, a′) = ((h, k, e_{G})a)(h′, k, a′) = (h, k, e_{G})(a(h′, k, a′)) = (h, k, e_{G})(h′, k, a^{x(h′,k)}a′),

and similarly for the other one. From the associativity property and the fact that the equivariant section G_{H×K} → Π must be a homomorphism for both laws, we find that (g_{1}, g_{2}, x) must satisfy a pair of nonabelian cocycle conditions:

(4.6a)  g_{1}(hh′, h′′; k)g_{1}(h, h′; k)^{x(h′′,k)} = g(h, h′h′′; k)g_{1}(h′, h′′; k)
(4.6b)  x(h, k)x(h′, k) = x(hh′, k) ∂g_{1}(h, h′; k)

and

(4.6c)  g_{2}(h; kk′, k′′)g_{2}(h; k, k′)^{x(h,k′′)} = g_{2}(h; k, k′k′′)g_{2}(h; k′, k′′)
(4.6d)  x(h, k)x(h, k′) = x(h, kk′) ∂g_{2}(h; k, k′)

The cocycles (g_{1}, x) and (g_{2}, x) are not independent: from the compatibility between the partial multiplication laws, using Proposition 4.2, we find:

(4.7) g_{2}(hh′; k, k′)g_{1}(h, h′; k)^{x(hh′,k′)}g_{1}(h, h′; k′) =

g_{1}(h, h′; kk′)g_{2}(h, k, k′)^{x(h′,kk′)}g_{2}(h′; k, k′){x(h′, k), x(h, k′)}^{−x(h′,k′)}.

These cocycles are a coboundary if there exists u: H × K → G such that:

(4.8a)  u(hh′, k) g_{1}(h, h′; k) = u(h, k)u(h′, k)
(4.8b)  u(h, kk′) g_{2}(h; k, k′) = u(h, k)u(h, k′)
(4.8c)  x(h, k) = ∂ u(h, k).

5. Biadditive morphisms and biextensions

Let E be a biextension of H, K by (G, Π). Then E defines a biadditive morphism

F_{E}: H × K ⟶ 𝒢 ,

in the sense of Definition 3.1, where 𝒢 is the group-like associated stack, by assigning to a point (h, k) ∈ H×K the (G, Π)-torsor E_{h,k}. Indeed, it is easily seen that the isomorphisms γ^{1} and γ^{2} plus the compatibility of the two composition laws expressed by (2.5) satisfy the required conditions. It is also easily verified that a morphism of biextensions φ: E → E′ induces a natural transformation (denoted with the same symbol)

φ: F_{E} ⇒ F_{E′}: H × K ⟶ 𝒢 .

This establishes a functor

u: Biext(H, K; G) ⟶ Hom(H, K; 𝒢),

and in fact one between stacks

u: Biext(H, K; G) ⟶ Hom(H, K; 𝒢),

These functors are evidently fully faithful. In the converse direction, we have:

5.1. Proposition. Let F: H × K → 𝒢 be a biadditive morphism, and let 𝒢 have a presentation by way of the braided crossed module G = (G, Π, ∂, {·, ·}). Then the pullback of the sequence (2.1) along F determines a biextension E = E_{F} of H, K by G.

Proof. Let E = (H × K) ×_{𝒢} Π be the pullback. Set p: E → H × K (resp. ȷ: E → Π) equal to the first (resp. second) projection. We claim that E is a biextension of H, K by G.

The sequence (2.1) can be seen as the universal extension by (G, Π). In particular, Π can be identified with the universal G, Π)-torsor (the identity id_{Π} is tautologically the structural equivariant section). This makes apparent that the pullback to H × K is a (G, Π)-torsor with structural map ȷ. As for the partial multiplication laws, observe that, with the same notation as sect. 2.5, the biadditivity of F amounts to a pair of natural transformations

(F ∘ d_{2}^{h}) (F ∘ d_{0}^{h}) ⟹ (F ∘ d_{1}^{h}): H × H × K ⟶ 𝒢 ,
(F ∘ d_{2}^{h}) (F ∘ d_{0}^{h}) ⟹ (F ∘ d_{1}^{h}): H × K × K ⟶ 𝒢 ,

and the equality of two transformations (see eqns. (2.6) and (3.1))

(F d_{0}^{h}d_{2}^{v}) (F d_{2}^{h}d_{0}^{v}) (F d_{0}^{h}d_{0}^{v}) (F d_{1}^{h}d_{1}^{v}) ⟹ (F d_{1}^{h}d_{1}^{v}): H × H × K × K ⟶ 𝒢 .

This translates into the required properties for E once we observe that the universality of Π (as a (G, Π)-torsor over 𝒢) implies that for any pair f, g: S → 𝒢 we have an isomorphism f^{∗}Π ∧^{G} g^{∗}Π ≃ (fg)^{∗}Π of (G, Π)-torsors. Last, the morphism ı is the image, under a further pullback along the unit section (e_{H}, e_{K}) ∈ H × K, of ∂: G → Π. It is clear ı and ȷ have the properties stated in Lemma 4.6.

In more down-to-earth terms, everything can be explicitly checked by writing out explicit expressions for the points of the pullback. Thus, a point e ∈ E is a triple e = ((h, k), a, x), where (h, k) ∈ H × K, x ∈ Π, and a: F (h, k) → (G, x). The Gaction becomes evident by looking at two points e, e′ in the same fiber. We must have a commutative diagram of (G, Π)-torsors

\xymatrix{F (h, k) \ar[r]^{a′} \ar[dr]^{a} & (G, x′) \ar[d]^{g} \\  & (G, x)}

where the vertical arrow, as a map of (G, Π)-torsors whose underlying G-torsor is trivial, is completely determined by a point g ∈ G (the image of the unit section e_{G}) such that

x′ = x ∂g.

The situation for the left action is completely analogous, with x′ = ∂g x.

When (h, k) = (e_{H}, e_{K}), the point e_{E} = ((e_{H}, e_{K}), l, e_{Π}), where l is the unique morphism coming from the third condition in Definition 3.1, is the central unit section of E. This gives the morphism ı: G → E. The partial multiplications are obtained as follows. Let e = ((h, k), a, x) and e = ((h′, k), a′, x′). Then ee′ = ((hh′, k), a′′, xx′), where the morphism a′′ is obtained from the diagram

\xymatrix{F (h, k)F (h′, k) \ar[d]_{λ_{hh′,k}^{1}} \ar[r]^{aa′} & (G, xx′) \\ F (hh′, k) \ar[ur]_{a′′}}

The second multiplication is defined in the same way. The interchange law immediately follows from the above and (3.1).

Since it is clear that if φ: F ⇒ F′ is a natural transformations of biadditive morphisms by pullback we obtain a corresponding isomorphism E_{F} → E_{F′} of biextensions, the construction of Proposition 5.1 provides functors

v: Hom(H, K; 𝒢) ⟶ Biext(H, K; G)

and

v: Hom(H, K; 𝒢) ⟶ Biext(H, K; G).

5.2. Proposition. The functors u and v are quasi-inverses.

Proof. We need to show that the functor u is essentially surjective.

For each F biadditive, we construct a morphism F → F′ = F_{E}, where E = (H × K) ×_{𝒢} Π. The biadditive morphism determined by E assigns to each (h, k) ∈ H × K the (G, Π)-torsor E_{h,k} consisting of pairs (f, x) such that

f: F (h, k) ⟶^{∼} (G, x).

Having observed this, the proof proceeds in a manner identical to that of Proposition 4.4.2 in [AN09] (in particular, cf. 4.4.2.1 and 4.4.2.2).

To summarize the previous discussion, Hom(H, K; 𝒢) and Biext(H, K; G) are equivalent, pointed groupoids. The distinguished point is the trivial biextension, corresponding to the trivial biadditive morphism sending (h, k) ∈ H × K to the trivial (G, Π)-torsor. The same holds for Hom(H, K; 𝒢) and Biext(H, K; G).

6. Symmetry properties of biextensions

It is well known (and see the discussion at the end of [AN10]) that the groupoid of extensions of K by (G, Π) admits a monoidal structure if and only if the crossed module ∂: G → Π (or equivalently the associated stack 𝒢) is braided. This monoidal structure is itself equipped with a commutativity isomorphism if and only if the braiding carried by 𝒢 is symmetric.

For biextensions the situation is more rigid. This is already apparent from the fact that in the very definition of a biextension a braided structure is required in order to be able to formulate the compatibility condition between the two multiplication laws.

If E and F are biextensions, we can use the braiding of 𝒢 to construct the partial multiplication laws in the expected manner: (6.1)

(E_{h,k} ∧^{G} F_{h,k}) ∧^{G} (E_{h′,k} ∧^{G} F_{h′,k}) ⟶^{ĉ} (E_{h,k} ∧^{G} E_{h′,k}) ∧^{G} (F_{h,k} ∧^{G} F_{h′,k}) ⟶^{γ^{1}∧μ^{1}} E_{hh′,k} ∧ F_{hh′,k} ,

and similarly for the second partial multiplication.

6.1. Proposition. For any pair of biextensions E and F of (H, K) by (G, Π), the partial multiplication laws (6.1) comprise a biextension structure on the contracted product (as (G, Π)-torsors) E ∧^{G} F if the braiding on the crossed module G = (G, Π, ∂, {−, −}) (or equivalently 𝒢) is symmetric.

Proof. With reference to (6.1), let us simply denote the first multiplication morphism so obtained by γ^{1}μ^{1}, and let γ^{2}μ^{2} denote the second multiplication structure obtained in exactly the same way.

For the compatibility condition between the two multiplications we must write diagram (2.5) for γ^{1}μ^{1} and γ^{2}μ^{2}. To simplify notations and readability, let us suppress the symbol ∧^{G} and the associativity morphisms. Set: a = E_{h,k}, x = F_{h,k}, y = E_{h′,k}, z = F_{h′,k}, u = E_{h,k′}, v = F_{h,k′}, w = E_{h′,k′}, b = F_{h′,k′}. Also, let us use parentheses to denote the various codomains of γ, μ, namely (ay) = E_{hh′,k}, (xz) = F_{hh′,k}, etc. Consider the following diagram:

\xymatrix{ & axyzuvwb \ar[rrrrrr] \ar[dr] \ar[dddl] &  &  &  &  &  & axuvyzwb \ar[dl] \ar[dddr] \\  &  & ayxzuwvb \ar[dr] \ar[ddll] &  &  &  & auxvywzb \ar[dl] \ar[ddrr] \\  &  &  & ayuwxzvb \ar[rr] \ar[d] &  & auywxvzb \ar[d] \\ (ay)(xz)(uw)(vb) \ar[rrr] \ar[drrrr] &  &  & (ay)(uw)(xz)(vb) \ar[dr] &  & (au)(yw)(xv)(zb) \ar[dl] &  &  & (au)(xv)(yw)(zb) \ar[lll] \ar[dllll] \\  &  &  &  & E_{hh′,kk′}F_{hh′,kk′}}

The outer pentagon corresponds to the sought-after compatibility for the partial multiplications of E∧^{G}F, whereas the little inner one is obtained by juxtaposing the compatibility conditions for E and F. Thus the inner pentagon commutes. The triangles commute by definition, since they just hold the definitions of γ^{1}μ^{1} and γ^{2}μ^{2}. The quadrangles are commutative by inspection. This leaves the big hexagon in the middle. The position of the external variables is unchanged throughout, hence the analysis of the diagram reduces to the one in Lemma D.2. This proves the proposition.

Once the coefficient crossed module carries a symmetric braiding, we are provided with a product of biextensions. An easy argument, based on the definition of the partial multiplication laws in the proof of Proposition 6.1 and Lemma D.1, or alternatively [AN10, §8.2], shows that the braiding provides a morphism

c: E ∧^{G} F ⟶ F ∧^{G} E

of biextensions, which is evidently symmetric. We summarize the previous discussion as

6.2. Proposition. For any H, K and G = (G, Π, ∂, {−, −}), Biext(H, K; G) (respectively, Biext(H, K; G)) is a braided symmetric group-like groupoid (respectively, stack) if and only if G carries a symmetric braiding.

By equivalence, the same statement holds for Hom(H, K; 𝒢) and Hom(H, K; 𝒢), respectively; the resulting symmetric monoidal structure is given by the pointwise product of functors (with respect to the monoidal structure of 𝒢).

Let 𝒢 be symmetric. If F_{1}, F_{2} : H × K → 𝒢 correspond, via Proposition 5.2, to the biextensions E_{1} and E_{2}, then it is not difficult to see that E_{1} ∧^{G} E_{2} is the biextension corresponding to F_{1} F_{2} : H × K → 𝒢, where F_{1} F_{2} is computed pointwise, i.e. for all h ∈ H and k ∈ K we set (F_{1} F_{2})(h, k) := F_{1}(h, k) F_{2}(h, k), where the juxtaposition on the right indicates the monoidal structure in 𝒢. Thus, Proposition 5.2 is upgraded to an equivalence of symmetric group-like groupoids or stacks. More precisely, we get:

6.3. Proposition. The assignment defined by u in Proposition 5.2 gives an equivalence

u: Biext(H, K; G) ⟶^{∼} Hom(H, K; 𝒢)

(resp.

u: Biext(H, K; G) ⟶^{∼} Hom(H, K; 𝒢))

of symmetric group-like categories (resp. stacks).

7. Butterflies

Let H_{•} : H_{1} →^{∂} H_{0} and K_{•} : K_{1} →^{∂} K_{0} be a pair of crossed modules, and let ∂: G_{1} → G_{0} be a crossed module equipped with a braiding {−, −}. We denote by ℋ , 𝒦 , and 𝒢 the corresponding associated stacks.

7.1. Definition. A butterfly from H_{•}×K_{•} to G_{•} is a biextension E ∈ Biext(H_{0}, K_{0}; G_{•}) equipped with trivializations of its pullbacks (∂, id)^{∗}E and (id, ∂)^{∗}E, i.e. maps s_{1} : H_{1} × K_{0} → E and s_{2} : H_{0} × K_{1} → E subject to the following conditions:

1. (Restriction) s_{1} and s_{2} agree on H_{1} × K_{1}: (id, ∂)^{∗}s_{1} = (∂, id)^{∗}s_{2}.

2. (Compatibility) For all (h, z) ∈ H_{1} × K_{0} and (y, k) ∈ H_{0} × K_{1}, and e ∈ E_{y,z}, s_{1} and s_{2} satisfy:

(7.1)  \begin{aligned}s_{1}(h, z) ×_{1} e & = e ×_{1} s_{1}(h^{y}, z) , \\ s_{2}(y, k) ×_{2} e & = e ×_{2} s_{2}(y, k^{z}) . \end{aligned}

A morphism φ: (E, s_{1}, s_{2}) → (E′, s′_{1}, s′_{2}) is a morphism of the underlying biextensions preserving the trivializations. Butterflies form a pointed groupoid, denoted Biext(H_{•}, K_{•}; G_{•}). The distinguished object is the trivial biextension.

7.2. Remark. As noted, a trivialization of a biextension is given by a (central) global section. In the previous definition the pullbacks are (∂, id)^{∗}E = (H_{1} × K_{0}) ×_{H_{0}×K_{0}} E and (id, ∂)^{∗}E = (H_{0} ×K_{1})×_{H_{0}×K_{0}} E, so, for example, a global section ŝ_{1} : H_{1} ×K_{0} → (∂, id)^{∗}E is equivalently given by a global map s_{1} : H_{1} × K_{0} → E. Same for s_{2}.

If we identify the sections corresponding to the trivializations with s_{1} : H_{1} × K_{0} → E and s_{2} : H_{0} × K_{1} → E, condition 1 in the Definition implies that

s_{1}(h, ∂k) = s_{2}(∂h, k).

Let s: H_{1} × K_{1} → E be the resulting morphism. The next lemmas characterize these objects in terms of explicit diagrams (see (7.2) and (7.5) below). Their shape justifies the name.

7.3. Lemma. The trivializations s_{1} and s_{2}, as maps s_{1} : H_{1}×K_{0} → E and s_{2} : H_{0}×K_{1} → E, render the following diagrams commutative

(7.2)  \xymatrix{H_{1} × K_{0} \ar[dd]_{(∂,id)} \ar[dr]^{s_{1}} &  & G_{1} \ar[dd]^{∂} \ar[dl]_{ı} & H_{0} × K_{1} \ar[dd]_{(id,∂)} \ar[dr]^{s_{2}} &  & G_{1} \ar[dd]^{∂} \ar[dl]_{ı} \\  & E \ar[dl]^{p} \ar[dr]_{ȷ} &  &  & E \ar[dl]^{p} \ar[dr]_{ȷ} \\ H_{0} × K_{0} &  & G_{0} & H_{0} × K_{0} &  & G_{0}}

with ȷ ∘ s_{1} and ȷ ∘ s_{2} equal to the trivial map (identically equal to the unit e_{0} of G_{0}). In addition, s_{1} and s_{2} have the following properties:^{8}

1. (Multiplicative) s_{1} and s_{2} are multiplicative in each variable, namely

(7.3)  \begin{aligned}s_{1}(h, z) ×_{1} s_{1}(h′, z) & = s_{1}(hh′, z), & s_{1}(h, z) ×_{2} s_{1}(h, z′) & = s_{1}(h, zz′), \\ s_{2}(y, k) ×_{1} s_{2}(y′, k) & = s_{2}(yy′, k), & s_{2}(y, k) ×_{2} s_{2}(y, k′) & = s_{2}(y, kk′); \end{aligned}

2. (Central) For all g ∈ G_{1}, and whenever ∂h (resp. ∂k) is equal to the unit section of H_{0} (resp. K_{0}), we have:

(7.4)  \begin{aligned}s_{1}(h, z)ı(g) & = ı(g)s_{1}(h, z), \\ s_{2}(y, k)ı(g) & = ı(g)s_{2}(y, k). \end{aligned}

Proof. The relations (7.3), as well as the fact that the compositions ȷ∘s_{1} and ȷ∘s_{2} must be equal to the unit element, follow from the condition that the pullbacks (∂, id)^{∗}E and (id, ∂)^{∗}E split as biextensions. Indeed, as it was previously described, the trivialization of, say, (∂, id)^{∗}E amounts to an isomorphism of biextensions G_{H_{1}×K_{0}} ≃ (∂, id)^{∗}E. The image of the unit central section of G_{H_{1}×K_{0}} defines the trivializing one for (∂, id)^{∗}E, and, hence, the map s_{1}. In practice, the isomorphism is written (h, z, g) ↦ s_{1}(h, z)g, and the central condition on the trivializing section means that

s_{1}(h, z) g = g s_{1}(h, z) ,

where the juxtaposition indicates the G_{1}-action. Of course, similar considerations hold for s_{2}, as well.

Then ȷ ∘ s_{1} and ȷ ∘ s_{2} are trivial because, under the isomorphisms determined by s_{1} and s_{2}, they must indeed be equal to the equivariant section for the trivial bitorsor which simply sends the unit e_{1} ∈ G_{1} to e_{0} ∈ G_{1}.

The relations (7.3) follow from the fact that the isomorphisms determined by s_{1} and s_{2} are compatible with the partial multiplication laws. For example:

(s_{1}(h, z) g) ×_{1} (s_{1}(h′, z) g′) = s_{1}(h, z) ×_{1} (g s_{1}(h′, z) g′) = s_{1}(h, z) ×_{1} s_{1}(h′, z) gg′ , on the other hand in the trivial bitorsor (h, z, g) ×_{1} (h′, z, g′) = (hh′, z, gg′) is mapped to s_{1}(hh′, z) gg′, which proves the first of (7.3). The others are similar.

8In some of the following formulas we explicitly write the product symbols ×_{1} and ×_{2} to avoid ambiguities.

The relations (7.4) follow from the centrality and (4.3) of Lemma 4.6.

7.4. Remark. An alternative way to characterize a butterfly is to say that it consists of a biextension E of (H_{0}, K_{0}) by G_{•} such that s_{1} and s_{2} provide it with the structure of butterflies in the ordinary sense of [AN09] from (H_{•})_{K_{0}} (resp. (K_{•})_{H_{0}}) to G_{•} in a compatible way.

It is clear that properties similar to those pertaining to s_{1} and s_{2} stated in Lemma 7.3 hold for their common restriction s: H_{1} × K_{1} → E. In particular, the following diagram

(7.5)  \xymatrix{H_{1} × K_{1} \ar[dd]_{(∂,∂)} \ar[dr]^{s} &  & G_{1} \ar[dd]^{∂} \ar[dl]_{ı} \\  & E \ar[dl]^{p} \ar[dr]_{ȷ} \\ H_{0} × K_{0} &  & G_{0}}

is commutative, with ȷ ∘ s equal to the trivial map (identically equal to the unit e_{0} of G_{0}). In addition, s is multiplicative in each variable:

(7.6)  \begin{aligned}s(h, k) ×_{1} s(h′, k) & = s(hh′, k) \\ s(h, k) ×_{2} s(h, k′) & = s(h, kk′). \end{aligned}

Relations (7.4) (collapsed into one) and (7.1) also hold. As an easy consequence we have the following

7.5. Lemma. Let (E, s_{1}, s_{2}) ∈ Biext(H_{•}, K_{•}; G_{•}). The pullback (∂, ∂)^{∗}E is isomorphic (via s) to the trivial biextension in Biext(H_{1}, K_{1}; G_{•}).

7.6. Remark. The correspondence (E, s_{1}, s_{2}) ↦ (E, s) determines a morphism

Biext(H_{•}, K_{•}; G_{•}) ⟶ H Ker(\xymatrix{Biext(H_{0}, K_{0}; G_{•}) \ar[r]^{(∂,∂)^{∗}} & Biext(H_{1}, K_{1}; G_{•})})

where H Ker denotes the homotopy kernel (recall both groupoids are pointed) which is not an equivalence, in general. Requesting that the biextension E become trivial when pulled back to H_{1} × K_{1} is a weaker condition for it does not provide for the two other pullbacks to be trivializable.

7.7. Remark. A butterfly has a description in terms of cocycles if the underlying (G_{1}, G_{0})- torsor E is globally trivial as a right G_{1}-torsor. Therefore, as seen at the end of sect. 4, if E has the form (H_{0}×K_{0}×G_{1}, x), with x: H_{0}×K_{0} → G_{0}, then a cocyclic description (4.5), (4.6), (4.7) (with h, h′ ∈ H_{0} and k, k′ ∈ K_{0}) is available. A butterfly will described by a cocycle consisting of that set plus additional relations for the trivializations of the two pullbacks to H_{1} × K_{0} and H_{0} × K_{1}. (The required triviality conditions are the same as those for a nonabelian 1-cocycles of the sort that appears in the theory of extensions.)

Since E = H_{0} × K_{0} × G_{1}, there exist u_{1} : H_{1} × K_{0} → G_{1} and u_{2} : H_{0} × K_{1} → G_{1}, such that s_{1} and s_{2} are written as s_{1}(h, z) = (∂h, z, u_{1}(h, z)^{−1}) and s_{2}(y, k) = (y, ∂k, u_{2}(y, k)^{−1}). The conditions in Lemma 7.3 then express the triviality of the cocycle pair (g_{1}, g_{2}) when pulled back to H_{1} × K_{0} and H_{0} × K_{1}, namely we get:

(7.7a)  u_{1}(hh′, z) g_{1}(∂h, ∂h′; z) = u_{1}(h, z) u_{1}(h′, z)
(7.7b)  u_{1}(h, zz′) g_{2}(∂h; z, z′) = u_{1}(h, z) u_{1}(h, z′)
(7.7c)  x(∂h, z) = ∂ u_{1}(h, z)

and

(7.7d)  u_{2}(yy′, k) g_{1}(y, y′; ∂k) = u_{2}(y, k) u_{2}(y′, k)
(7.7e)  u_{2}(y, kk′) g_{2}(y; ∂k, ∂k′) = u_{2}(y, k) u_{2}(y, k′)
(7.7f)  x(y, ∂k) = ∂ u_{2}(h, z)

for all pairs (h, z), (h′, z) ∈ H_{1} × K_{0} and (y, k), (y, k′) ∈ H_{0} × K_{1}. Moreover, since the two trivializations must agree when pulled back to H_{1} × K_{1}, it follows that u_{1}(h, ∂k) = u_{2}(∂h, k). Denoting this common restriction by u: H_{1} × K_{1} → G_{1}, the two previous sets coalesce into

(7.8a)  u(hh′, k) g_{1}(∂h, ∂h′; ∂k) = u(h, k)u(h′, k)
(7.8b)  u(h, kk′) g_{2}(∂h; ∂k, ∂k′) = u(h, k)u(h, k′)
(7.8c)  x(∂h, ∂k) = ∂ u(h, k).

for all h, h′ ∈ H_{1} and k, k′ ∈ K_{1}.

8. Biadditive morphisms and butterflies

We now consider biadditive morphisms

F: ℋ × 𝒦 ⟶ 𝒢

with 𝒢 braided (cf. sect. 3). As before, we assume we have presentations by crossed modules ℋ ≃ [H_{1} → H_{0}]^{∼}, 𝒦 ≃ [K_{1} → K_{0}]^{∼}, and 𝒢 ≃ [G_{1} → G_{0}]^{∼}, the latter equipped with a braiding structure {−, −}. Our purpose is to prove the following

8.1. Theorem. There is an equivalence of (pointed) groupoids

u: Biext(H_{•}, K_{•}; G_{•}) ⟶^{∼} Hom(ℋ , 𝒦 ; 𝒢).

In essence, the theorem states that any biadditive functor F: ℋ × 𝒦 → 𝒢 can be represented by a butterfly involving the presentations. This equivalence is compatible with restriction and base-change so that, via the usual mechanism of considering the above equivalence relative to a variable object S of T, we obtain a corresponding equivalence

u: Biext(H_{•}, K_{•}; G_{•}) ⟶^{∼} Hom(ℋ , 𝒦 ; 𝒢).

Following [AN09] and sect. 5, we exhibit a pair of quasi-inverse functors. We will essentially confine ourselves to just exhibit the relevant definitions, for the methods are quite similar to those in [AN09, §4], with the exception of biadditivity. The latter is discussed explicitly, where the need arises.

Let F̄ be the composition of the projection π: H_{0} × K_{0} → ℋ × 𝒦 with F. It is evidently biadditive. Consider the pullback E = (H_{0} × K_{0}) ×_{F̄,𝒢 ,π} G_{0}. By Proposition 5.1 it is a biextension.

8.2. Lemma. E defined above satisfies the conditions in the statement of Lemma 7.3.

Proof. We must verify the pullbacks of E to H_{1} ×K_{0} and H_{0} ×K_{1} are trivializable. This is a direct consequence of the fact that \xymatrix{H_{1} \ar[r] & H_{0} \ar[r] & ℋ} and \xymatrix{K_{1} \ar[r] & K_{0} \ar[r] & 𝒦} are (homotopically) exact. Furthermore, the decomposition (∂, ∂) = (id, ∂) ∘ (∂, id) = (∂, id) ∘ (id, ∂) and condition 3 of definition 3.1 ensure that the restriction condition in definition 7.1 is satisfied.

In explicit terms, a point of the pullback (∂, id)^{∗}E is given by a triple ((h, z), f, x) where (h, z) ∈ H_{1} × K_{0}, x ∈ G_{0} and f is an isomorphism

\xymatrix{F (π(∂h), π(z)) \ar[r]^{f} & (G_{1}, x).}

We have π(∂h) = (H_{1}, ∂h), so there must be an isomorphism I_{ℋ} = (H_{1}, e_{h_{0}}) ⟶ (H_{1}, ∂h). Thus, there is a chain of isomorphisms

\xymatrix{F (I_{ℋ} , π(z)) \ar[d] \ar[r] & F (π(∂h), π(z)) \ar[r]^{f} & (G_{1}, x) \\ I_{𝒢}}

so that we must have x = ∂g, g ∈ G_{1}. This provides an explicit trivializing section s_{1} for (∂, id)^{∗}E. Similarly for the other one, s_{2}, and their common restriction s.

A computation based on the biadditivity of F and the same technique at the end of the proof of 5.1 shows that both s_{1} and s_{2} are multiplicative, i.e. each satisfies condition 1 of lemma 7.3, with respect to both variables. Condition 2 also follows from a direct calculation, as in [AN09, §4.3.6].

8.3. Proposition. There exists a functor v: Hom(ℋ , 𝒦 ; 𝒢) → Biext(H_{•}, K_{•}; G_{•}) defined by assigning to an object F the butterfly whose underlying bitorsor is E = (H_{0} × K_{0}) ×_{F̄,𝒢 ,π} G_{0}.

Proof. The pullback biextension construction is functorial (cf. the end of section 5). In particular, if φ: F ⇒ F′ is a morphism of biadditive functors, the resulting morphism of biextensions is compatible with the trivializations. This observation proves Proposition 8.3.

To get a biadditive functor from a butterfly in functorial way, we proceed as in ref. [AN09, §§4.3.2–4.3.4]. Specifically, an object of ℋ × 𝒦 will be represented by a pair of torsors, as ((Y, y), (Z, z)) where (Y, y) is an (H_{1}, H_{0})-torsor and (Z, z) an (K_{1}, K_{0})-torsor. Let (E, s) be an object of Biext(H_{•}, K_{•}; G_{•}). We define a F_{E,s_{1},s_{2}} : ℋ × 𝒦 → 𝒢 by assigning to the above pair the (G_{1}, G_{0})-torsor (X, x) where:

X := Hom_{H_{1},K_{1}}(Y, Z; E)_{(y,z)}, \begin{aligned}x: & X ⟶ G_{0} \\ & e ⟼ ȷ ∘ e. \end{aligned}

X consists of separately H_{1} or K_{1}-equivariant local lifts of (y, z): Y × Z → H_{0} × K_{0} to E. By this we mean that the lift of (y ∂h, z) is related to that of (y, z) by e(y ∂h, z) = e(y, z) s_{1}(h, z). Similarly, e(y, z ∂k) = e(y, z) s_{2}(y, k). The notion is consistent thanks to Conditions 1 and the multiplicativity of s_{1} and s_{2}. This construction is obviously functorial with respect to each of its arguments. It gives the defining part of the

8.4. Proposition. There exists a functor

u: Biext(H_{•}, K_{•}; G_{•}) ⟶ Hom(ℋ , 𝒦 ; 𝒢),

whose value at (E, s_{1}, s_{2}) is the bifunctor F_{E,s_{1},s_{2}} defined above.

Proof. We must verify the biadditivity property, namely that F (Y_{1}, Z)∧^{G_{1}} F (Y_{2}, Z) ⟶^{∼} F (Y_{1} ∧^{H_{1}} Y_{2}, Z), which follows from the diagram:

\xymatrix{ &  & E ×_{K_{0}} E \ar[r]^{γ^{1}} \ar[d]^{(p,p)} & E \ar[d]^{p} \\ Y_{1} × Y_{2} × Z \ar[r]^{Δ_{Z}} & Y_{1} × Z × Y_{2} × Z \ar[r] \ar[ur]^{(e_{1},e_{2})} & (H_{0} × K_{0}) ×_{K_{0}} (H_{0} × K_{0}) \ar[r] & H_{0} × K_{0}}

It is easy (and left to the reader) to check that the diagram is invariant under replacing (e_{1}g, e_{2}) by (e_{1}, ge_{2}) and (y_{1}h, y_{2}) by (y_{1}, hy_{2}). (The first arrow to the left is the diagonal of Z followed by the swap of the two inner factors.)

8.5. Remark. A coordinate version of the construction of the functor u above is as follows. According to the beginning of section 5, if E is the underlying biextension of an object in Biext(H_{•}, K_{•}; G_{•}), we obtain a biadditive morphism H_{0} × K_{0} → 𝒢 by sending the pair (y, z) to the (G_{1}, G_{0})-torsor E_{y,z}. Since E is part of a butterfly, this construction is compatible with morphisms in ℋ and 𝒦 (in fact, in the prestacks defined by the presentations) because, if say y′ = y ∂h, we have

E_{y′,z} ⟵ E_{y,z} ∧^{G_{1}} E_{∂h,z} ⟵^{∼} E_{y,z},

since from the definition the existence of s_{1} implies E_{∂h,z} is a trivial (G_{1}, G_{0})-torsor. (Similarly for E_{y,∂k}.) We have a similar calculation whenever z′ = z ∂k, and the properties of the butterfly (plus the compatibility of ×_{1} and ×_{2}) ensure we obtain a unique morphism

E_{y,z} ⟶ E_{y′,z′} ,

which allows us to define u(E) on more general objects by descent. The connection with the global version above is of course that Hom_{H_{1},K_{1}}(Y, Z; E)_{(y,z)} reduces to E_{y,z} when (Y, y) = (H_{1}, y) and (Z, z) = (K_{1}, z).

Proof of Theorem 8.1. We prove that u and v are quasi-inverses. To this end, recall that for a G_{1}-torsor P we have the isomorphism

(8.1)  Hom_{G_{1}}(G_{1}, P) →^{∼} P,  m ↦ m(e_{G_{1}}),

[Gir71, Proposition III 1.2.7]. This extends to (G_{1}, G_{0})-torsors by assigning to m: G_{1} → P the element s(m(e_{G_{1}})) ∈ G_{0} (see [AN09, n. 4.4.2]). We apply this observation to both v(u(E)) and u(v(F)), where (E, s_{1}, s_{2}) is a butterfly with underlying biextension E and F: ℋ × 𝒦 → 𝒢 is biadditive. In the first case, a point of the biextension v(u(E)) is given by a tuple ((y, z), f, x) with (y, z) ∈ H_{0} × K_{0} and x ∈ G_{0}, such that

f: E_{y,z} ⟶^{∼} (G_{1}, x).

Considering the fiber over (y, z) we have the following chain of morphisms of (G_{1}, G_{0})- torsors:

(8.2)  v(u(E))_{y,z} ⟶^{∼} Hom_{G_{1}}(G_{1}, E_{y,z}) ⟶^{∼} E_{y,z},

where the first arrow sends (f, x) to f^{−1}, and the projection (f, x) → x, namely the equivariant section, to ȷ∘f^{−1}. Therefore we have obtained an isomorphism of biextensions v(u(E)) →^{∼} E, by virtue of the result quoted at the beginning. This is (tautologically) an isomorphism of butterflies. For this, consider the composite isomorphism I_{𝒢} = (G_{1}, e) →^{∼} E_{∂h,z} →^{∼} (G_{1}, x) (resp. I_{𝒢} = (G_{1}, e) →^{∼} E_{y,∂k} →^{∼} (G_{1}, x)) which provides the trivialization of the pullback v(u(E)) to H_{1} × K_{0} (resp. H_{0} × K_{1}), as in the proof of Lemma 8.2. The resulting identifications of the pair (f, x) with an element g ∈ G_{1} is clearly compatible via the chain (8.2), with the trivialization (G_{1}, e) →^{∼} E_{y,z}.

In the second case, v(F) = (H_{0} × K_{0}) ×_{𝒢} G_{0} and u(v(F)) is the biadditive morphism that assigns to (y, z) ∈ H_{0} × K_{0} the (G_{1}, G_{0})-torsor

v(F)_{y,z} = {(f, x)|F (π(y), π(z)) ⟶^{f} (G_{1}, x)} ⟶^{∼} F (π(y), π(z)),

where the isomorphism on the right is by way of (8.1) and loc. cit. The pullback v(F) has the required trivializations by Lemma 8.2, which are evidently compatible with F (π(∂h), π(z)) →^{∼} (G_{1}, e) and F (π(y), π(∂k)) →^{∼} (G_{1}, e), showing that the above isomorphism holds for general objects and is functorial, proving there is an isomorphism u(v(F)) ≃ F.

8.6. Theorem. [Theorem 8.1, symmetric version] Let the monoidal structure of 𝒢 be symmetric. Then the equivalence in Theorem 8.1 extends to one of group-like groupoids or stacks.

Proof Proof (Sketch). We need only check the trivializations. Suppose F, F′ are two biadditive morphisms and E, E′ are the corresponding butterflies. Let us also use the letters E, E′ to denote the underlying torsors, and let s_{1}, s_{2} and s′_{1}, s′_{2} be the trivialization morphisms of E and E′, respectively. Let us denote by F ∧ F′ the biadditive morphism ℋ ×𝒦 → 𝒢, (y, z) ↦ F (y, z)F′(y, z). (The unnamed juxtaposition denotes the product structure in 𝒢.) Consider the bitorsor E ∧^{G} E′, equipped with the biextension structure of Proposition 6.1.

We claim that the pairs (s_{1}, s′_{1}) and (s_{2}, s′_{2}) give the two trivialization morphisms turning the biextension E ∧^{G} E′ into a butterfly.

Indeed, the restriction condition 1 in Definition 7.1 is immediate. The compatibility condition 2 can be verified by a simple computation, using the definition of the partial multiplications from the proof of Proposition 6.1 and Lemma 2.4.

9. Commutative structures

With the equivalence between u: Biext(H_{•}, K_{•}; G_{•}) → Hom(ℋ , 𝒦 ; 𝒢) at our disposal, we can discuss commutativity conditions for biextensions (cf. Remark 4.4 above).

9.1. Definition. Let ℋ , 𝒦 , and 𝒢 all be equipped with a braiding structure. Then we say that (E, s_{1}, s_{2}) ∈ Biext(H_{•}, K_{•}; G_{•}) (or simply E by abuse of language) is braided if the biadditive morphism u(E) is braided in the sense of sect. 3.

In order to turn the definition into actual diagrams, we use the explicit variance of the biextension E, Remark 8.5. Here the point of Remark 2.2 becomes relevant. Whereas the morphism u(E): ℋ × 𝒦 → 𝒢 is covariant in each variable, the biextension E itself is a (G_{0}, G_{1})-torsor over H_{0} × K_{0}, and Tors(G_{1}, G_{0}) is anti-equivalent to 𝒢.

Using the trivialization s_{1} : H_{1} × K_{0} → E and s_{2} : H_{0} × K_{1} → E, and the braidings {−, −}_{H} and {−, −}_{K} for the presentations, we have morphisms η^{1} and η^{2}:

η_{y,y′;z}^{1} : E_{y′y,z} ⟶ E_{yy′,z}  η_{y;z,z′}^{2} : E_{y,z′z} ⟶ E_{y,zz′}
e ⟼ e ×_{1} s_{1}({y′, y}_{H}, z)  e′ ⟼ e′ ×_{2} s_{2}(y, {z′, z}_{K}).

Thus, for y, y′ ∈ H_{0} and z, z′ ∈ K_{0} the following diagrams must commute:

(9.1)  [figure: E_{y,z} ∧^{G_{1}} E_{y′,z}; γ_{y,y′;z}^{1}; c_{y,y′;z}; E_{yy′,z}; η_{y,y′;z}^{1}; E_{y′,z} ∧^{G_{1}} E_{y,z}; γ_{y′,y;z}^{1}; E_{y′y,z}; and; E_{y,z} ∧^{G_{1}} E_{y,z′}; γ_{y;z,z′}^{2}; c_{y;z,z′}; E_{y,zz′}; η_{y;z,z′}^{2}; E_{y,z′} ∧^{G_{1}} E_{y,z}; γ_{y;z′,z}^{2}; E_{y,z′z}]

In both diagrams the vertical arrow to the left comes from the braiding in 𝒢. The two vertical arrows on the right express the functoriality with respect to the braiding structures of H_{•} and K_{•}.

Alternatively, the directions in the right vertical arrows of both diagrams in (9.1) can be restored if we interpret them as morphism in Tors(G_{1}, G_{0}) arising from the morphisms

(H_{0}, yy′) ⟶ (H_{0}, y′y) and (K_{0}, zz′) ⟶ (K_{0}, z′z)

in Tors(H_{1}, H_{0}) and Tors(K_{1}, K_{0}), respectively. This requires writing η^{1} and η^{2} (going in the opposite direction) in terms of

s_{1}({y′, y}_{H}^{−1}, z) and s_{2}(y, {z′, z}_{K}^{−1}).

Using Lemma 2.4 to express the braiding morphisms c_{y,y′;z} and c_{y;z,z′} we arrive at the expressions, valid for e ∈ E_{y,z}, e′ ∈ E_{y′,z}:

(9.2a)  e′ ×_{1} e = (e ×_{1} e′ ×_{1} s_{1}({y′, y}_{H}^{−1}, z)) {ȷ(e), ȷ(e′)}_{G};

and for e ∈ E_{y,z}, e′ ∈ E_{y,z′}:

(9.2b)  e′ ×_{2} e = (e ×_{2} e′ ×_{2} s_{2}(y, {z′, z}_{K}^{−1})) {ȷ(e), ȷ(e′)}_{G}.

It follows that the (ordinary) butterfly corresponding to each variable must be braided in the sense of [AN09, §7.4.1].

Theorems 8.1 and 8.6 specialize to this situation. We will use superscript ( )^{b} to denote the groupoids (or stacks) of braided biadditive morphisms and butterflies.

9.2. Theorem. [Theorems 8.1, 8.6 fully commutative case] Let 𝒦 , ℋ , and 𝒢 be all braided and have presentations by braided crossed modules. The equivalence u of Theorem 8.1 restricts to an equivalence

u: Biext^{b}(H_{•}, K_{•}; G_{•}) ⟶^{∼} Hom^{b}(ℋ , 𝒦 ; 𝒢)

of groupoids. Furthermore, if 𝒢 is braided symmetric, u (from Theorem 8.6) becomes an equivalence of group-like groupoids. Similar statements hold for Biext^{b}(−, −; −) and Hom^{b}(−, −; −).

As an application of these additional assumptions, we discuss whether Biext(H_{•}, K_{•}; G_{•}), Hom(ℋ , 𝒦 ; 𝒢), etc. are biadditive in their variables. This means asking whether, for example, we have an equivalence:

Hom^{b}(ℋ_{1} × ℋ_{2}, 𝒦 ; 𝒢) ≃ Hom^{b}(ℋ_{1}, 𝒦 ; 𝒢) × Hom^{b}(ℋ_{2}, 𝒦 ; 𝒢) ,

and a similar one relative to the variable 𝒦 , as well as similar statements for Biext(ℋ , 𝒦 ; 𝒢), Biext(H_{•}, K_{•}; G_{•}), and Hom(ℋ , 𝒦 ; 𝒢).

We have observed that regardless the commutativity assumptions (but keeping 𝒢 braided symmetric) they always are biadditive in the third variable. Biadditivity in the first and second variables only holds under the additional commutativity hypotheses. More precisely, we have 9.3. Proposition. The groupoid Hom^{b}(−, −; −) is biadditive in all variables for symmetric group-like stacks. (The same conclusion holds for Biext^{b}(−, −; −), Biext^{b}(−, −; −) and Hom^{b}(−, −; −).)

Proof. We can use the same argument as [Bre83, §1.2]. Specifically, consider the three (additive) functors d_{i}: ℋ × ℋ → ℋ (d_{1} is additive since ℋ is braided). Let F: ℋ × 𝒦 → 𝒢 be a biadditive morphism, and let E = v(F) be the corresponding butterfly. Then

λ^{1} : d_{2}^{∗}F d_{0}^{∗}F ⟶ d_{1}^{∗}F

is a morphism in Hom^{b}(ℋ × ℋ , 𝒦 ; 𝒢). As in loc. cit. this follows from the compatibility between λ^{1} and λ^{2} and the fact that λ^{1} is an additive morphism thanks to the (symmetric) braiding. In view of the equivalence Hom^{b}(−, −; −) ≃ Biext^{b}(−, −; −), the above morphism corresponds to a morphism of biextensions

γ^{1} : d_{2}^{∗}E ∧^{G_{1}} d_{0}^{∗}E ⟶ d_{1}^{∗}E

which is in fact a morphism of butterflies in Biext^{b}(H_{•} ×H_{•}, K_{•}; G_{•}), after one checks the trivializations. From Proposition 6.1 we see that the braiding must be symmetric. Now, for additive morphisms R, S: ℋ ′ → ℋ , we obtain a morphism

(R × I)^{∗}F (S × I)^{∗}F ⟶ ((RS) × I)^{∗}F

in Hom^{b}(ℋ ′, 𝒦 ; 𝒢), showing biadditivity in the first variable. The second variable is treated analogously.

9.4. Remark. In the proof, if E_{R} and E_{S} are the butterflies corresponding to R, S: ℋ ′ → ℋ , we obtain the morphism of butterflies in Biext(H′_{•}, K_{•}; G_{•}) is

((E_{R} × I) ×_{H_{0}}^{H_{1}} E) ∧^{G_{1}} ((E_{S} × I) ×_{H_{0}}^{H_{1}} E) ⟶ ((E_{R} ∧^{H_{1}} E_{S}) × I) ×_{H_{0}}^{H_{1}} E

expressing the biadditivity in the first variable at the level of biextensions. Here I is a shorthand for the diagram corresponding to the identity morphism. (This kind of compositions is systematically studied in the next Part II.)

Part II

10. Multiextensions and compositions

The generalizations of the previous notions of biextension and biadditive morphism to the case of n-variables is straightforward, see [Gro72, §2.10.2], [Bre99, §7]. Let (G, Π, ∂, {−, −}) be a braided crossed module. A multiextension, or n-extension, of (H_{1}, … H_{n}) by (G, Π) is a (G, Π)_{H_{1}×···×H_{n}}-torsor E over H_{1} × ⋯ × H_{n} equipped with n partial multiplication laws ×_{1}, …, ×_{n}, plus a compatibility relation of the type (2.5) for each pair (×_{i}, ×_{j}).

Lemma 4.6 remains valid, as well as the analogs of Propositions 5.1 and 5.2. In fact, the notion of butterfly can be extended to this case. Let us consider ℋ_{1}, …, ℋ_{n} and 𝒢, the latter equipped with a braiding as usual. There is an evident notion of n-additive functor F: ℋ_{1} × ⋯ × ℋ_{n} → 𝒢, which can be defined by an appropriate generalization of Definition 3.1. If we suppose that each ℋ_{i} has a presentation ℋ_{i} ≃ [H_{i,1} → H_{i,0}]^{∼}, we can define a butterfly (or, more precisely, an n-butterfly) from H_{1,•} × ⋯ × H_{n,•} to G_{•} as an n-extension E of (H_{1,0} × ⋯ × H_{n,0}) by G_{•} equipped with n trivializations s_{i}: H_{1,0} × ⋯ × H_{i,1} × ⋯ × H_{n,0} → E, each satisfying the conditions of Definition 7.1, with the obvious modifications. Theorem 8.1 and its symmetric variant 8.6 generalize to this case. Let us state this independently for future reference. Denoting by MExt the groupoid of n-butterflies, we have

10.1. Theorem. There is an equivalence of (pointed) groupoids:

u: MExt(H_{1,•}, …, H_{n,•}; G_{•}) ⟶^{∼} Hom(ℋ_{1}, …, ℋ_{n}; 𝒢),

sending a multi-extension E, to the n-additive functor u(E) that to the object (y_{1}, …, y_{n}) of ℋ_{1} ×⋯×ℋ_{n}, where each y_{i} is in the essential image of H_{i,0}, assigns the (G, Π)-torsor E_{y_{1},...,y_{n}}. It is an equivalence of group-like groupoids whenever 𝒢 is symmetric.

Once again, the axioms of Definition 7.1 ensure this is compatible with the descent, ensuring u(E) is a well defined morphism. Just like for biextensions, the equivalence is compatible with localization, with the stacks MExt and Hom in place of the global groupoids.

Multi-additive functors can be composed in the following way. Let 𝒢, ℋ_{1}, …, ℋ_{n}, 𝒦_{i,1}, …, 𝒦_{i,m_{i}}, i = 1, …, n, be group-like stacks, with 𝒢 and ℋ_{1}, …, ℋ_{n} braided. Let F ∈ Hom(ℋ_{1}, …, ℋ_{n}; 𝒢), and for i = 1, …, n let G_{i} ∈ Hom(𝒦_{i,1}, …, 𝒦_{i,m_{i}}; ℋ_{i}). Then if x_{1,1}, …, x_{n,m_{n}}, collectively denoted x_{1}, …, x_{m}, are objects of 𝒦_{1,1}, …, 𝒦_{n,m_{n}}, define, as usual, F (G_{1}, …, G_{n}) by

(10.1)  F (G_{1}, …, G_{n})(x_{1} …, x_{m}) := F (G_{1}(x_{1}, …), …, G_{n}(…, x_{m})).

10.2. Proposition. The composition defined in (10.1) assigns to the tuple (F, G_{1}, …, G_{n}) a well defined object F (G_{1}, …, G_{n}) of Hom(𝒦_{1,1}, …, 𝒦_{n,m_{n}}; 𝒢). This composition is associative.

An identical statement holds with Hom replaced by Hom.

Proof. The only thing to check is that the composition F (G_{1}, …, G_{n}) satisfy the conditions (3.1) for each pair (i, j) of indices within the list {1, …, m_{1} + m_{2} + ⋯ + m_{n}}, where m_{k} is the arity of G_{k}. There are two cases depending on whether the variables corresponding to the pair (i, j) belongs to the same “slot,” say relative to G_{k}, or when i and j fall into two different slots, relative to G_{k} and G_{l}, with k ≠ l. We now indicate the main points of the verification, leaving the easy task of writing the complete diagrams to the reader.

In the former case, the mechanics of the verification are completely captured by considering the values n = 1, m_{1} = 2. First we write the pentagonal diagram corresponding to (3.1) for F (G). In it, we use the functoriality of F to reduce the arrow

(F (G(x, y)) F (G(x′, y))) (F (G(x, y′)) F (G(x′, y′)))

(F (G(x, y)) F (G(x, y′))) (F (G(x′, y)) F (G(x′, y′)))

to the arrow

\xymatrix{F ((G(x, y) G(x′, y)) (G(x, y′) G(x′, y′))) \ar[r] & F ((G(x, y) G(x, y′)) (G(x′, y) G(x′, y′)))},

and then use the fact that G itself satisfies (3.1). For the latter case, the general situation is captured by considering n = 2, m_{1} = m_{2} = 1, so we need to write the diagram (3.1) for F (G_{1}, G_{2}). For this, the interchange law for F gives us a unique morphism from

(F (G_{1}(x), G_{2}(y)) F (G_{1}(x′), G_{2}(y))) (F (G_{1}(x), G_{2}(y′)) F (G_{1}(x′), G_{2}(y′)))

to

F (G_{1}(x) G_{1}(x′), G_{2}(y) G_{2}(y′)).

Now the functoriality of F gives the morphism

F (G_{1}(x) G_{1}(x′), G_{2}(y) G_{2}(y′)) ⟶ F (G_{1}(xx′), G_{2}(yy′))

by way of the square

\xymatrix{F (G_{1}(x) G_{1}(x′), G_{2}(y) G_{2}(y′)) \ar[r] \ar[d] & F (G_{1}(x) G_{1}(x′), G_{2}(yy′)) \ar[d] \\ F (G_{1}(xx′), G_{2}(y) G_{2}(y′)) \ar[r] & F (G_{1}(xx′), G_{2}(yy′))}

The statement about associativity is immediate.

11. Compositions of n-butterflies

From Theorem 10.1, the multilinear functors can be expressed in terms of multi-extensions, and from Proposition 10.2 multilinear functors can be composed. Hence we expect an analogous composition exists for multi-extensions. In this section we provide a construction of this composition, generalizing the composition for single butterflies found in [AN09, §5.1].

For i = 1, …, n, and integers j_{i} = 1, …, m_{i}, let G_{•}, H_{1,•}, … H_{n,•} and K_{i,1,•}, …, K_{i,m_{i},•} crossed modules, of which G_{•}, H_{1,•}, … H_{n,•} are assumed to be braided. Let E be a braided n-butterfly from H_{1,•} × ⋯ × H_{n,•} to G_{•}, and for i = 1, …, n let F_{i} be an m_{i}-butterfly from K_{i,1,•} × ⋯ × K_{i,m_{i},•} to H_{i,•}. In the following we let

P = (F_{1} × ⋯ × F_{n}) ×_{H_{1,0}×···×H_{n,0}} E,

be the (H_{1,1} × ⋯ × H_{n,1})-equivariant (G_{1}, G_{0})-torsor over F_{1} × ⋯ × F_{n}.

11.1. Theorem. The (G_{1}, G_{0})-torsor P has a well defined quotient by H_{1,1} × ⋯ × H_{n,1}, denoted

(F_{1} × ⋯ × F_{n}) ×_{H_{1,0}×···×H_{n,0}}^{H_{1,1}×···×H_{n,1}} E,

which carries a structure of m_{1}+⋯+m_{n}-extension (=butterfly) from K_{1,1,•}×⋯×K_{n,m_{n},•} to G_{•}.

11.2. Definition. The composition E(F_{1}, …, F_{n}) of E with F_{1}, …, F_{n} is

E(F_{1}, …, F_{n}) := (F_{1} × ⋯ × F_{n}) ×_{H_{1,0}×···×H_{n,0}}^{H_{1,1}×···×H_{n,1}} E.

We call this composition the juxtaposition product, since it entails placing the butterflies wing-by-wing next to one another.

Proof of Theorem 11.1–Construction of the juxtaposition product. We will use element notation throughout. The procedure consists of several steps. The first is to form the fiber product exactly as in loc. cit.

P = (F_{1} × ⋯ × F_{n}) ×_{H_{1,0}×···×H_{n,0}} E,

which as noted is an (H_{1,1} × ⋯ × H_{n,1})-equivariant (G_{1}, G_{0})-torsor over F_{1} × ⋯ × F_{n}, and to mod out the left and right actions of H_{1,1} × ⋯ × H_{n,1}. Let us use the notation

(y_{1}, …, y_{n}) := (ȷ_{1}(v_{1}), … ȷ_{n}(v_{n})) = p(u)

for the tuple in H_{1,0} × ⋯ × H_{n,0}, where ȷ_{i}: F_{i} → H_{i,0} for each (H_{i,1}, H_{i,0})-torsor F_{i}, i = 1, …, n.

Consider first the right action of elements of the form (1, …, h_{i}, …, 1), where h_{i} ∈ H_{i,1}. The action on P is given by sending a point (v_{1}, …, v_{n}, u) to

(v_{1}, …, v_{i} h_{i}, … v_{n}, u ×_{i} s_{i}(y_{1}, …, h_{i}, …, y_{n})),

For the left action, we have

(11.1) (v_{1}, …, h_{i} v_{i}, … v_{n}, s_{i}(y_{1}, …, h_{i}, …, y_{n}) ×_{i} u)

= (v_{1}, …, v_{i} h_{i}^{y_{i}}, … v_{n}, u ×_{i} s_{i}(y_{1}, …, h_{i}^{y_{i}}, …, y_{n})),

which follows from Definition 7.1, adapted to the multi-variable case. These are actions thanks to the multiplicativity properties of s_{j} established in Lemma 7.3. These actions are compatible for i ≠ j. Acting with h_{i} and then with h_{j}, assuming for example that i < j, we have

(u ×_{i} s_{i}(y_{1}, …, h_{i}, …, y_{n})) ×_{j} s_{j}(y_{1}, …, y_{i}∂h_{i}, …, h_{j}, …, y_{n}) = (u×_{i}s_{i}(y_{1}, …, h_{i}, …, y_{n}))×_{j}(s_{j}(y_{1}, …, y_{i} …, h_{j}, …, y_{n})×_{i}s_{j}(y_{1}, …, ∂h_{i}, …, h_{j}, …, y_{n})), where we used the multiplicativity property of s_{j}. Now, using the interchange property for ×_{i} and ×_{j}, and the compatibility between the trivializations s_{i} and s_{j}, the right hand side becomes

(u ×_{j} s_{j}(y_{1}, …, h_{j}, …, y_{n})) ×_{i} (s_{i}(y_{1}, …, h_{i}, …, y_{n}) ×_{j} s_{i}(y_{1}, …, h_{i}, …, ∂h_{j}, …, y_{n})),

which coincides with the action of h_{j} first, followed by that of h_{i}. This ensures that following formula for the right action of a generic point (h_{1}, …, h_{n}) is well defined:

(11.2) (v_{1} h_{1}, … v_{n} h_{n}, (⋯ (u ×_{1} s_{1}(h_{1}, y_{2}, …, y_{n})) ×_{2} s_{2}(y_{1}∂h_{1}, h_{2}, …, y_{n}) ⋯) ×_{n} s_{n}(y_{1}∂h_{1}, …, y_{n−1}∂h_{n−1}, h_{n})).

The reader will have no difficulty in writing the corresponding formulas for the left action. Since each F_{i} is an (H_{i,1}, H_{i,0})-torsor, the action is free.

Note also that the G_{1}-actions on P (both left and right) are compatible with the action of H_{1,1} × ⋯ × H_{n,1} thanks to the relations (7.4) in Lemma 7.3 (property 2). In particular, the G_{1}-actions happen by way of those on the last element of the tuple; denoting the class of a tuple by brackets we have:

[v_{1}, …, v_{n}, u] g = [v_{1}, …, v_{n}, u g],
g [v_{1}, …, v_{n}, u] = [v_{1}, …, v_{n}, g u] = [v_{1}, …, v_{n}, u g^{ȷu}].

Finally, the equivariant section from the quotient of P to G_{0} is defined to be

(v_{1}, …, v_{n}, u) ⟼ ȷ(u).

Next, we define m_{1} + ⋯ + m_{n} partial product structures ×_{i,j} for i = 1, …, n and j = 1, …, m_{i} as follows. Using our established index convention, let us use the notation

(z_{1,1}, …, z_{i,j}, …, z_{n,m_{n}}) ∈ K_{1,1,0} × ⋯ × K_{i,j,0} × ⋯ × K_{n,m_{n},0}

for a point of the base of P/(H_{1,1} × ⋯ × H_{n,1}). To begin with, consider the special case of two points (v_{1}, …, v_{j}, …, v_{n}, u) and (v_{1}, …, v′_{j}, …, v_{n}, u′) of P such that

p_{i}(v_{i}) = (z_{i,1}, …, z_{i,j}, …, z_{i,m_{i}}), p_{i}(v′_{i}) = (z_{i,1}, …, z′_{i,j}, …, z_{i,m_{i}}),

in K_{i,j,0}, for fixed i and j ∈ {1, …, m_{i}}, everything else being equal. Define:

(11.3) (v_{1}, …, v_{j}, …, v_{n}, u)×_{i,j} (v_{1}, …, v′_{j}, …, v_{n}, u) := (v_{1}, …, v_{j} ×_{i,j} v′_{j}, …, v_{n}, u×_{i}u′).

The symbol ×_{i,j} on the right hand side of (11.3) denotes the j^{th} product structure (within j = 1, …, m_{i}) of F_{i}, and the resulting point of P projects onto the point

(z_{1,1}, …, z_{i,j}z′_{i,j}, …, z_{n,m_{n}}).

In general, let us consider points e = [v_{1}, …, v_{n}, u] and e′ = [v′_{1}, …, v′_{n}, u′] of P/(H_{1,1}× ⋯ × H_{n,1}) above (z_{1,1}, …, z_{i,j}, …, z_{n,m_{n}}) and (z_{1,1}, …, z′_{i,j}, …, z_{n,m_{n}}), respectively. (As before, only the z_{i,j} and z′_{i,j} coordinates are different.) For j = 1, …, n, j ≠ i, there exist unique h_{j} ∈ H_{j,1} such that v′_{j} = v_{j} h_{j}. As a result,

[v′_{1}, …, v′_{n}, u′] = [v_{1}, …, v′_{i}, …, v_{n}, u′′],

where u′′ is related to u′ by an application of (11.2). Then we define e ×_{i,j} e′ as the class

[v_{1}, …, v_{j} ×_{i,j} v′_{j}, …, v_{n}, u ×_{i} u′′],

which is computed using (11.3) above. We must show that this is independent of the various choices involved through the use of (11.2). The computation is quite elaborate, but otherwise not illuminating nor eventful, therefore we omit it. We also omit the easy verification that each of these partial multiplication laws is associative.

To claim that we have constructed a genuine multi-extension, we must prove the partial multiplication laws just defined obey pairwise compatibility (interchange) laws. This we prove explicitly. Like in the proof of Proposition 10.2, there are two distinct cases to address, depending on whether the two partial product laws have the same first index. The easiest is when they do not, so we treat it first. Thus, let i ≠ k ∈ {1, … n}. For brevity let Q = P/(H_{1,1} × ⋯ × H_{n,1}) and consider:

e_{i,j} = [v_{1}, …, v_{i}, …, v_{k}, …, v_{n}, u_{i}] ∈ Q_{z_{1,1},...,z_{i,j},...,z_{k,l},...,z_{n,mn}}
e′_{i,j} = [v_{1}, …, v′_{i}, …, v_{k}, …, v_{n}, u′_{i}] ∈ Q_{z_{1,1},...,z′_{i,j},...,z_{k,l},...,z_{n,mn}}
e_{k,l} = [v_{1}, …, v_{i}, …, v′_{k}, …, v_{n}, u_{k}] ∈ Q_{z_{1,1},...,z_{i,j},...,z′_{k,l},...,z_{n,mn}}
e′_{k,l} = [v_{1}, …, v′_{i}, …, v′_{k}, …, v_{n}, u′_{k}] ∈ Q_{z_{1,1},...,z′_{i,j},...,z′_{k,l},...,z_{n,mn}}.

We first compute the products with ×_{i,j}:

e_{i,j} ×_{i,j} e′_{i,j} = [v_{1}, …, v_{i} ×_{i,j} v′_{i}, …, v_{k}, …, v_{n}, u_{i} ×_{i} u′_{i}],
e_{k,l} ×_{i,j} e′_{k,l} = [v_{1}, …, v_{i} ×_{i,j} v′_{i}, …, v′_{k}, …, v_{n}, u_{k} ×_{i} u′_{k}].

Then

(e_{i,j} ×_{i,j} e′_{i,j}) ×_{k,l} (e_{k,l} ×_{i,j} e′_{k,l})

= [v_{1}, …, v_{i} ×_{i,j} v′_{i}, …, v_{k} ×_{k,l} v′_{k}, …, v_{n}, (u_{i} ×_{i} u′_{i}) ×_{k} (u_{k} ×_{i} u′_{k})]

= \begin{matrix}[v_{1}, …, v_{i} ×_{i,j} v′_{i}, …, v_{k} ×_{k,l} v′_{k}, …, v_{n}, (u_{i} ×_{k} u_{k}) ×_{i} (u′_{i} ×_{k} u′_{k}) \\ \{ȷ(u′_{i}), ȷ(u_{k})\}^{−ȷ(u′_{k})}] \end{matrix}

= (e_{i,j} ×_{k,l} e_{k,l}) ×_{i,j} (e′_{i,j} ×_{k,l} e′_{k,l}) {ȷ(u′_{i}), ȷ(u_{k})}^{−ȷ(u′_{k})},

where in the third line we have used the interchange law for the i^{th} and k^{th} product laws in E, Proposition 4.2.

The other case is the one with the same first index, say ×_{i,j} and ×_{i,l}, for j ≠ l. In this situation we consider instead:

e_{i,j} = [v_{1}, …, v_{i}, …, v_{n}, u_{i}] ∈ Q_{z_{1,1},...,z_{i,j},...,z_{i,l},...,z_{n,mn}}
e′_{i,j} = [v_{1}, …, v′_{i}, …, v_{n}, u′_{i}] ∈ Q_{z_{1,1},...,z′_{i,j},...,z_{i,l},...,z_{n,mn}}
e_{i,l} = [v_{1}, …, w_{i}, …, v_{n}, t_{i}] ∈ Q_{z_{1,1},...,z_{i,j},...,z′_{i,l},...,z_{n,mn}}
e′_{i,l} = [v_{1}, …, w′_{i}, …, v_{n}, t′_{i}] ∈ Q_{z_{1,1},...,z′_{i,j},...,z′_{i,l},...,z_{n,mn}}.

Proceeding in an analogous way we first get:

e_{i,j} ×_{i,j} e′_{i,j} = [v_{1}, …, v_{i} ×_{i,j} v′_{i}, …, v_{k}, …, v_{n}, u_{i} ×_{i} u′_{i}],
e_{k,l} ×_{i,j} e′_{k,l} = [v_{1}, …, w_{i} ×_{i,j} w′_{i}, …, v_{n}, t_{k} ×_{i} t′_{k}].

And then:

(e_{i,j} ×_{i,j} e′_{i,j}) ×_{i,l} (e_{i,l} ×_{i,j} e′_{i,l})

= [v_{1}, …, (v_{i} ×_{i,j} v′_{i}) ×_{i,l} (w_{i} ×_{i,j} w′_{i}), …, v_{n}, (u_{i} ×_{i} u′_{i}) ×_{i} (t_{i} ×_{i} t′_{i})]

= \begin{matrix}[v_{1}, …, (v_{i} ×_{i,j} v′_{i}) ×_{i,l} (w_{i} ×_{i,j} w′_{i}) \{ȷ_{i}(w_{i}), ȷ_{i}(v′_{i})\}_{H_{i}}^{−ȷ_{i}(w′_{i})}, …, v_{n}, \\ (u_{i} ×_{i} u′_{i}) ×_{i} (t_{i} ×_{i} t′_{i}) ×_{i} s_{i}(y_{1}, …, \{ȷ_{i}(w_{i}), ȷ_{i}(v′_{i})\}_{H_{i}}^{−ȷ_{i}(w′_{i})}, …, y_{n})] \end{matrix}

= \begin{matrix}[v_{1}, …, (v_{i} ×_{i,l} w_{i}) ×_{i,j} (v′_{i} ×_{i,l} w′_{i}), …, v_{n}, \\ (u_{i} ×_{i} t_{i}) ×_{i} (u′_{i} ×_{i} t′_{i}) \{ȷ(u′_{i}), ȷ(t_{i})\}_{G}^{−ȷ(t′_{i})}], \end{matrix}

where in the intermediate equality we used the action by an element of H_{i,1}, whereas in the last one the interchange law (within the multi-extension F_{i}) and (9.2) were used. But the last line equals

(e_{i,j} ×_{i,l} e_{i,l}) ×_{i,j} (e′_{i,j} ×_{i,l} e′_{i,l}) {ȷ(u′_{i}), ȷ(t_{i})}_{G}^{−ȷ(t′_{i})},

as wanted.

The final step is to show that Q, in addition to being a multi-extension of K_{1,1,0} × ⋯ × K_{n,m_{n},0} by G_{•}, is in fact a butterfly, that is, it carries a trivialization for each of its pullbacks to K_{1,1,0} × ⋯ × K_{i,j,1} × ⋯ × K_{n,m_{n},0}. Such trivializations can be defined as follows.

For each i = 1, …, n and for each j = 1, …, m_{i}, let s_{i,j} be the trivialization s_{i,j}: K_{i,1,0}× ⋯ × K_{i,j,1} × ⋯ × K_{i,m_{i},0} → F_{i}. Observe that the fiber of P above

(v_{1}, …, s_{i,j}(z_{i,1}, …, k_{i,j}, …, z_{i,m_{i}}), …, v_{n}) ∈ F_{1} × ⋯ × F_{i} × ⋯ × F_{n}

(where v_{k} is a point of F_{k}, for k ≠ i) can be identified with G_{1}. This is because that point maps via (ȷ_{1}, …, ȷ_{n}) to (y_{1}, …, 1, …, y_{n}), and the fiber of E above it is identified with G_{1}. (In turn, this follows from the fact that biextensions in particular, and multi-extensions in general, are canonically trivialized over identity sections, see sect. 4.) It follows that the fiber of Q over (z_{1,1}, …, ∂k_{i,j}, …, z_{n,m_{n}}) is similarly identified with G_{1}. Thus, we obtain a trivializing section ŝ_{i,j} of the pullback of Q to K_{1,1,0} × ⋯ × K_{i,j,1} × ⋯ × K_{n,m_{n},0} by sending the point (z_{1,1}, …, k_{i,j}, …, z_{n,m_{n}}) to [v_{1}, …, s_{i,j}(z_{i,1}, …, k_{i,j}, …, z_{i,m_{i}}), …, v_{n}, e_{i}], where e_{i} is the unit (central) section of the restriction of E to H_{1,0} × … {1} × ⋯ × H_{n,0}. Observe this does not depend on the choice of v_{j} ∈ F_{j}, for j ≠ i, since F_{j}/H_{j} is just (z_{j,1}, …, z_{j,m_{j}}). Note that ȷ: Q → G_{0} composed with any of ŝ_{i,j} is identically equal to 1, since so is the result of applying ȷ_{E}: E → G_{0} to e_{i}. The conditions (Compatibility and Restriction) of Definition 7.1 are easy to verify and so this task is left to the reader.

This ends the proof.

11.3. Proposition. The composition in Definition 11.2 gives rise to a well-defined composition functor

MExt(K_{1,1,•}, …; H_{1,•}) × ⋯ × MExt(…, K_{n,m_{n},•}; H_{n,•}) × MExt(H_{1,•}, …, H_{n,•}; G_{•}) ⟶ MExt(K_{1,1,•}, …, K_{n,m_{n},•}; G_{•}),

where we are using the same convention for the indices as above and in sect. A. Same with MExt in place of MExt.

Proof. We need only check that if f: E → E′ and g_{i}: F_{i} → F′_{i} are butterfly isomorphisms, then we get a well defined isomorphism of k = m_{1} + ⋯ + m_{n}-butterflies

f(g_{1}, …, g_{n}): E(F_{1}, …, F_{n}) ⟶ E′(F′_{1}, …, F′_{n}).

This is easy to check from the construction of the juxtaposition product provided above.

Associativity only holds up to isomorphism. This is due to the same phenomenon, ultimately a manifestation of the same lack of strict associativity for fiber products, observed in the unary case (see [AN09, Noo05]). We omit the proof.

11.4. Proposition. The composition in Definition 11.2 is associative up to coherent isomorphism.

12. The bi-multicategories of group-like stacks and crossed modules

By using Proposition 10.2 we can rely on the framework recalled in the technical section A to promote the 2-category of braided group-like stacks to a 2-multicategory. In particular, using Definition A.1, we define the categories {ℋ , 𝒢 }, where to each n we assign

Hom(ℋ ^{n}; 𝒢) = Hom(\underbrace{ℋ , …, ℋ}_{n−times}; 𝒢).

We ambiguously use the same symbol to denote the stack obtained using Hom in place of Hom, as which version will be in use will be clear from the context. From Definition A.3 we immediately obtain:

12.1. Proposition. For group-like stacks 𝒦 , ℋ , and 𝒢, we have the assembly map

(12.1)  ∘ : {ℋ , 𝒢 } ≀ {𝒦 , ℋ } ⟶ {𝒦 , 𝒢 }

which to the object (F; G_{1}, …, G_{n}) assigns the composition F (G_{1}, …, G_{n}).

In particular, M𝒢 := {𝒢 , 𝒢 } is a monoid for this composition, whenever 𝒢 is braided, with identity object given by (id; ) (the empty string in the second slot).

12.2. Remark. As alluded in sect. A, the formalism includes the symmetry structure, which makes it slightly more general than needed in the sequel. All objects of interest shall have augmentations factoring through the discrete subcategory N of S, which amounts to ignoring the permutation structure and hence symmetry conditions.

From Propositions 11.3 and 11.4 and the formalism of sect. A the bicategory of braided crossed modules of T equipped with butterflies as morphisms (see [AN09, Theorem 5.3.6]) is promoted to a genuine bi-multicategory. In particular, we obtain objects {H_{•}, G_{•}} and an analog of Proposition 12.1 with assembly map

(12.2)  ∘ : {H_{•}, G_{•}} ≀ {K_{•}, H_{•}} ⟶ {K_{•}, G_{•}}

for braided crossed modules K_{•}, H_{•}, and G_{•}.

The equivalence between the 2-category 𝔅𝔊𝔯𝔖𝔱 of (braided) group-like stacks and the bicategory 𝔅𝔛𝔐𝔬𝔡 of (braided) crossed modules (cf. [AN09, Theorem 5.3.6]) lifts to the present case. Let M𝔅𝔊𝔯𝔖𝔱 (resp. M𝔅𝔛𝔐𝔬𝔡) the bi-multicategory with morphisms Hom(ℋ_{1}, …, ℋ_{n}; 𝒢) (resp. MExt(K_{1,1,•}, …, K_{n,m_{n},•}; G_{•})).

12.3. Theorem. There is an equivalence of bi-multicategories

Ma: M𝔅𝔛𝔐𝔬𝔡 ⟶^{∼} M𝔅𝔊𝔯𝔖𝔱

induced by the associated stack functor.

Proof. Recall that a functor F: Mℭ → M𝔇 between bi-multicategories is an equivalence if for each tuple (x; y_{1}, …, y_{n}) of objects of Mℭ the functor

F_{(x;y_{1},...,y_{n})} : Hom_{ℭ}(y_{1}, …, y_{n}; x) → Hom_{𝔇}(F (y_{1}), …, F (y_{n}); F (x))

is an equivalence of categories, and the underlying ordinary homomorphism [F]_{1} : ℭ → 𝔇 is essentially surjective.^{9}

The associate stack functor at the level of the underlying bicategories a: 𝔅𝔛𝔐𝔬𝔡 → 𝔅𝔊𝔯𝔖𝔱 is an equivalence, hence, in particular, essentially surjective. The equivalence is proved in [AN09, Theorem 5.3.6] without the braiding hypothesis, however Theorem 4.3.1 in loc. cit. applies to the braided case as well by §7, ibid., and presentations by braided crossed modules are easily obtained by using Lemma 2.4. Thus essential surjectivity follows.

The equivalence at the level of the multi-arrow categories directly follows from Theorem 10.1. Finally we check that the functor u of Theorem 10.1 preserves the (horizontal) compositions up to coherent isomorphism. Like the single variable case, this follows from the fiber product construction of the quasi-inverse to u.

9[F]_{1} is the homomorphism we obtain by restricting F to arrows of arity equal to one.

12.4. Remark. The situation becomes decidedly more pleasant if we confine ourselves to symmetric objects. We can consider a symmetric variant of the above based on the bicategories 𝔖𝔛𝔐𝔬𝔡 and 𝔖𝔊𝔯𝔖𝔱. The equivalence in Theorem 12.3 restricts to between these new entities:

Ma: M𝔖𝔛𝔐𝔬𝔡 ⟶^{∼} M𝔖𝔊𝔯𝔖𝔱,

which, by Theorem 10.1, both ought to be regarded as multicategories enriched over symmetric group-like groupoids. Furthermore, using the version of {−, −} based on Hom, 𝔖𝔊𝔯𝔖𝔱 becomes closed. Indeed one has the analog of [Kel72, Theorem 2], in that

Hom_{ℭ}(𝒦 ≀ ℋ , 𝒢) ≃ Hom_{ℭ}(𝒦 , {ℋ , 𝒢 }),

where ℭ = 𝔖𝔊𝔯𝔖𝔱 (or even ℭ = 𝔖𝔊𝔯𝔖𝔱/S using extraordinary structure briefly discussed in sect. A—see [Kel72] for all the details). A proof can be obtained along the same lines, proceeding from Theorem 10.1 and sect. A. We will not pursue this further.

Part III

13. Bimonoidal structures and weakly categorical rings

For a braided stack ℛ, a biadditive bifunctor m: ℛ × ℛ → ℛ gives a binary law which is automatically distributive with respect to the monoidal structure of ℛ by virtue of (3.1) in Definition 3.1. If m is itself part of a monoidal structure, that is, there exists μ: m(m, id) ⇒ m(id, m) satisfying the standard pentagon identity, then ℛ is said to be bimonoidal. In fact it satisfies the axioms of a categorical ring in the usual sense [JP07], except possibly the requirement for the underlying group-like structure of ℛ to be symmetric. (One axiom was apparently missing in [JP07], and the gap was filled in [Qua13], thus completing the classification.)

From sections 10 and A, the binary operation m and id are objects of Mℛ = {ℛ, ℛ}, and μ: m(m, id) ⇒ m(id, m) is a morphism of Mℛ, with the pentagon identity expressing the equality between two morphisms thereof. Rather than using the classical, “biased,” definition of the monoidal structure, it is more convenient to exploit the multi-categorical structure comprising all the multilinear functors. Now, biased and unbiased definitions yield equivalent structures [Lei04], hence we can simply define it in terms of the multicategory M𝔅𝔊𝔯𝔖𝔱.

13.1. Definition. [See Definition B.1] Let ℛ be a braided stack. A (weak) categorical ring structure on ℛ is a lax monoidal functor (t, θ): N → Mℛ. A (weak) ring-like stack is a braided stack ℛ equipped with such a structure, namely a pseudo-monoid in the 2-multicategory M𝔅𝔊𝔯𝔖𝔱.^{10}

10In the sequel we shall use “weak”, when referring to bimonoidal structures, in this sense; it will not refer to the laxness of the monoidal structures.

In plainer language, N represents all the n-ary operations in a “disembodied” form. Thus, an object n of N corresponds to an operation with n abstract variables. The functor t assigns to n an actual t(n): ℛ × ⋯ × ℛ → ℛ in MExt(ℛ^{n}; ℛ). Given n operations of various arities, they can be substituted into one of arity n, as hinted above. This set of operations is represented by the object (n; m_{1}, …, m_{n}) of N ≀ N (see Definitions A.3 and B.1) and the substitution, resulting in an operation of arity m_{1} + ⋯ + m_{n}, gives a morphism N ≀ N → N which has the properties of a monoidal structure. This is mapped by t into the composition of actual multi-additive multi-functors of ℛ, satisfying the same combinatorial structure. In summary, the functor (t, θ) gives a collection of multi-additive t(n) subject to isomorphisms of functors of the form

θ_{n;m_{1},...,m_{n}}: t(m_{1} + ⋯ + m_{n}) → t(n)(t(m_{1}), …, t(m_{n})),

controlling the compositions.

13.2. Remark. The 2-multicategory M𝔅𝔊𝔯𝔖𝔱 has of course more structure than the abstract situation discussed in sections A and B. In particular, the objects of 𝔅𝔊𝔯𝔖𝔱, being themselves categories, have an internal structure, which we can use to define the wreath product N ≀ 𝒢 for any 𝒢 following sect. A, or, literally, the arguments of [Kel72, §2.1]. Briefly, an object of N ≀ 𝒢 consists of a string (n; x_{1}, …, x_{n}) (where the label n is in effect redundant) with x_{i} an object of 𝒢, i = 1, …, n. There are no morphisms between objects with different integer labels, and a morphism (n; x_{1}, …, x_{n}) → (n; x′_{1}, …, x′_{n}) is simply an n-tuple of morphisms in 𝒢.

It follows that by a special case of the adjunction discussed in [Kel72], which in the present case can be verified by hand, the lax monoidal functor providing the exterior monoidal structure on ℛ corresponds to

(t̃, θ̃): N ≀ ℛ ⟶ ℛ,

which is required to satisfy the diagrams displayed in Remark B.3, which define the structure of a pseudo algebra over N.

We drop the “weak” attribute is the underlying braiding is symmetric. In such a case, we will see the fibers of ℛ are categorical rings in the usual sense, so that ℛ becomes the stack analog of the categorical rings described in e.g. [JP07].

13.3. Remark. For a symmetric ℛ, we can define the ring-like structure in the same way, but working in 𝔖𝔊𝔯𝔖𝔱. In such case the pseudo-algebra structure of ℛ can be verified directly from the adjunction mentioned in Remark 12.4 above.

A pseudo monoid ℛ in M𝔅𝔊𝔯𝔖𝔱 ought to correspond to one in M𝔅𝔛𝔐𝔬𝔡. More precisely, any pseudo monoid ℛ ought to be equivalent to one whose underlying stack is the one associated to a presentation by a braided crossed module, i.e. an object of 𝔅𝔛𝔐𝔬𝔡. This will be made precise below. First, we can write the analog of Definition 13.1, namely:

13.4. Definition. A (weakly) ring-like crossed module is a pseudo monoid in the bimulticategory M𝔅𝔛𝔐𝔬𝔡, that is, a braided crossed module R_{•} : R_{1} →^{∂} R_{0} equipped with a pseudo monoidal functor

(t, θ): N ⟶ MR_{•}.

We drop the “weak” attribute if the underlying braiding is symmetric.

An operation of arity n is realized in this case by an n-butterfly E_{n} = t(n), an object of Hom(R_{•}^{n}; R_{•}), composed according to (B.2), realized by the juxtaposition product

(13.1)  θ_{n;m_{1},...,m_{n}}: E_{m_{1}+···+m_{k}} ⟶ E_{n}(E_{m_{1}}, …, E_{m_{n}}),

plus the coherence condition expressed by the full diagram (B.4), which reads (13.2)

\xymatrix{E_{k}(E_{l_{1}}, …, E_{l_{k}}) \ar[d] & E_{h} \ar[r] \ar[l] & E_{n}(E_{j_{1}}, …, E_{j_{n}}) \ar[d] \\ E_{n}(E_{m_{1}}, …, E_{m_{n}})(E_{l_{1}}, …, E_{l_{k}}) \ar[rr] &  & E_{n}(E_{m_{1}}(E_{l_{1}}, …), …, E_{m_{n}}(…, E_{l_{k}}))}

The bottom arrow is the association isomorphism in the butterfly juxtaposition. For n ≤ 4, with all the sequences emanating from n = 4 as in sect. B, we obtain the “generalized pentagon” diagram (C.2).

Theorem 12.3 and the fact that pseudo monoids are preserved by pseudo functors (Definition B.5) guarantee that a weakly ring-like R_{•} gives rise, through the associated stack construction, to a weakly ring-like stack. More interesting is the converse direction, namely:

13.5. Proposition. If (ℛ, t, θ) is a weakly categorical ring and R_{1} → R_{0} → ℛ a presentation, then R_{•} : R_{1} → R_{0} is a pseudo monoid in M𝔅𝔛𝔐𝔬𝔡.

Proof. From Theorem 12.3 we get an equivalence Ma_{R}: {R_{•}, R_{•}} →^{∼} {ℛ, ℛ}.

Here is an explicit construction of a quasi-inverse to Ma_{R}. By Theorem 10.1, or rather the generalization of the proof of Theorem 8.1, the functor u has a quasi-inverse v computed by the fiber product construction which yields an n-butterfly E_{n} = v(t(n)); for a composition t(n)(t(m_{1}), …, t(m_{n})) we obtain a morphism

v(t(n)(t(m_{1}), …, t(m_{n}))) ⟶^{∼} E_{n}(E_{m_{1}}, …, E_{m_{n}}).

Thus, for any

θ_{n;m_{1},...,m_{n}}: t(m_{1} + ⋯ + m_{n}) → t(n)(t(m_{1}), …, t(m_{n}))

in Mℛ, we obtain

E_{m_{1}+···+m_{n}} ⟶^{∼} v(t(n)(t(m_{1}), …, t(m_{n}))) ⟶^{∼} E_{n}(E_{m_{1}}, …, E_{m_{n}}),

eventually arriving at (13.2).

The notion of morphism for both weakly categorical rings and crossed modules is straightforward. In each instance, the notion is a special case of that of morphism of pseudo-monoid examined in sect. B.

13.6. Notation. From now on it will be convenient to refer to the monoidal structure of ℛ as a group-like object as the “intrinsic” or “internal” one, and denote it by a plus. Correspondingly, the relative unit object will be denoted by 0_{ℛ} , or simply 0, if no confusion is bound to arise. (This retroactively justifies the choice of the attribute “biadditive” for the functor m.) The second monoidal structure (m, μ) will be referred to as the “extrinsic” or “external” one. The result of the application of m will be denoted by a juxtaposition. The corresponding unit object, if it exists, will be denoted by I_{ℛ} or simply I.

14. The presentation of a categorical ring

In this section we show that the presentation of a (weak) categorical ring or ring-like stack is a crossed module with certain additional properties. In particular, its zeroth and first homotopy sheaves are, respectively, a ring and a bimodule over that ring, as in the standard case.

Let ℛ be as above. Let A = π_{0}(ℛ) and M = π_{1}(ℛ) be the sheaves of connected components and of automorphisms of the (additive) identity object. Since the definition of π_{i} applies to the underlying (group-like) stack, by the usual arguments A and M are also zeroth and first homotopy sheaves of any crossed module R_{•} used to present ℛ. In fact the homotopy kernel of the projection ϖ: ℛ → A is identified with the Picard stack ℳ = Tors(M) of M-torsors. One has the familiar diagram

M ı R_{1}

∂

0

0 R_{0}

π

q A

ℳ ℛ ϖ A

with homotopy-exact columns and where the top two levels of each column (the part within the box above) provide a presentation of the item immediately below. The colored sequence in it is the standard extension of A by M by way of the crossed module R_{•}. Of course, the top two rows, considered as a sequence of complexes of length one, is exact only in the sense that the group-like categories or stacks they determine form a short exact sequence.

For objects X_{1}, X_{2}, …, X_{n} of ℛ define the result of applying the n-ary operation t(n) simply by X_{1}X_{2} ⋯ X_{n}. Then, by defining [X_{1}][X_{2}] ⋯ [X_{n}] := [X_{1}X_{2} ⋯ X_{n}], we immediately get that ϖ: ℛ → A is a morphism of pseudo monoids, so that A is a ring—unital if ℛ possesses a unit object I_{ℛ} for the external monoidal structure. (It follows that ℳ behaves like a bilateral ideal in ℛ, namely the monoidal structure on ℛ, and more precisely its binary operation, restricts to a pair of actions:

ℛ × ℳ ⟶ ℳ and ℳ × ℛ ⟶ ℳ .)

While there is no direct map R_{0} × R_{0} → R_{0}, the exterior monoidal structure m_{2} = t(2): ℛ × ℛ → ℛ is “covered” by the diagram

(14.1)  [figure: R_{0} × R_{0}; E; R_{0}; p; ȷ]

which is part of the structure of the biextension. Given a pair (x, y) ∈ R_{0} × R_{0}, and the choice of a point in the fiber e ∈ E_{x,y}, we get a “value” ȷ(e) ∈ R_{0}. The latter is of course only defined up to shifting e by the action of R_{1}, namely ȷ(e r) = ȷ(e) + ∂r, where r ∈ R_{1}. The connected component q(ȷ(e))) ∈ A is well-defined and independent of all choices. Moreover, by Theorem 8.1, or rather its proof, we have E_{x,y} = m_{2}(π(x), π(y)), and therefore ϖ(E_{x,y}) = ϖ(π(x))ϖ(π(y)) = q(x)q(y). We conclude that, with e ∈ E_{x,y} as above, q(ȷ(e)) = q(x)q(y). Put differently, the diagram displayed above, if interpreted as a “virtual ring structure” for R_{0}, covers the multiplication map of A. This consideration extends to any number of variables.

Hence the following standard fact holds:

14.1. Proposition. M is an A-bimodule.

We recall the main idea of the proof, as it will be needed to translate the above fact in terms of the presentation of ℛ and the multi-extensions associated to its second monoidal structure.

Proof Proof of 14.1 (Sketch). Recall that the standard way to relate the automorphisms of 0_{ℛ} to those of any other object X of ℛ is to use the diagram

\xymatrix{X + 0_{ℛ} \ar[r]^{id_{X} +α} \ar[d] & X + 0_{ℛ} \ar[d] \\ X \ar[r]_{α̃} & X}

Left and right multiplications by an object Y (i.e. the additive functors m_{2}(Y, −) and m_{2}(−, Y)) translate the above diagram into the corresponding ones for automorphisms Y α and αY. If β: Y → Z, so that [Y] = [Z] ∈ A, then we arrive at the square

\xymatrix{XY + 0_{ℛ} \ar[r]^{id +αY} \ar[d]_{id_{X} β+id_{0}} & XY + 0_{ℛ} \ar[d]^{id_{X} β+id_{0}} \\ XZ + 0_{ℛ} \ar[r]_{id +αZ} & XZ + 0_{ℛ}}

from which we conclude that αY = αZ. The situations for Y α, Zα is analogous.

Combining this with Proposition 13.5 we have:

14.2. Corollary. Every categorical ring ℛ has a presentation

\xymatrix{0 \ar[r] & M \ar[r] & R_{1} \ar[r]^{∂} & R_{0} \ar[r] & A \ar[r] & 0,}

where R_{•} : R_{1} → R_{0} is a braided crossed module, A = π_{0}(ℛ) a ring and M = π_{1}(ℛ) an A-bimodule.

In fact, something more can be said about the presentation, despite R_{0} having the weaker structure given by (14.1). In addition, we will need to describe the A-bimodule structure of M in terms of (14.1) and the biextension E.

Consider the morphism examined in Remark 8.5: for an arrow x → x′ = x + ∂r of [R_{1} → R_{0}]^{∼}, with x ∈ R_{0} and r ∈ R_{1}, we have the morphism of (R_{1}, R_{0})-torsors ρ_{y}: E_{x,y} → E_{x′,y}, e ↦ e ×_{1} s_{1}(r, y). If ∂r = 0, so r ∈ π_{1}(ℛ) = M, then ρ_{y} is an automorphism of E_{x,y}, hence ρ ∈ M as well (depending on x, y), since in general Aut_{Tors(R_{1},R_{0})}(E) ≃ M_{R_{0}×R_{0}}. Similarly, we construct an automorphism λ_{x}: E_{x,y} → E_{x,y} corresponding to the automorphism of y ∈ R_{0} given by r ∈ R_{1}.

14.3. Lemma-Definition. The automorphism ρ_{y} (resp. λ_{x}) is identified with a section −ry (resp. −xr) of M such that if y, y′ ∈ R_{0} (resp. x, x′ ∈ R_{0}) and q(y) = q(y′) ∈ A (resp. q(x) = q(x′) ∈ A), then ry = ry′ (resp. xr = x′r).

With the above notations, define ar := xr and rb := ry, where a = q(x), b = q(y) ∈ A.

14.4. Remark. The apparently bizarre choice of inserting an inverse (denoted additively) is due to the point raised in Remark 2.2.

Proof Proof of the Lemma. The identification between the automorphisms of E and M is standard: for any section e ∈ E_{x,y} we have ρ_{y}(e) = e r_{ρ_{y}}(e), with r_{ρ_{y}}(e) ∈ M. On the other hand, this must be equal to e×_{1} s_{1}(r, y), the latter coming from the composition

E_{x,y} ⟶ E_{x,y} ∧^{R_{1}} E_{0,y} ⟶ E_{x,y}.

Recall from sect. 4 that, analogously to the standard case of abelian biextensions [Gro72], for x, y ∈ R_{0}, the fibers E_{x,0} and E_{0,y} are canonically identified with the unit (R_{1}, R_{0})- torsor: the identification

R_{1} ⟶^{∼} E_{0,y}, 0_{R_{1}} ⟼ e_{y},

sends the unit section of R_{1} (which we write as 0 according to the current additive convention) to the central section e_{y} ∈ E_{0,y}. Note that as a result ȷ(e_{y}) = 0_{R_{0}}. By the above, we have

(14.2)  s_{1}(r, y) = e_{y} (−ry),

where ry := −r_{ρ_{y}}(e_{y}) ∈ R_{1}, and the juxtaposition stands for the right R_{1}-action. In fact, we have ry ∈ M, since ȷ ∘ s_{1} is trivial (cf. sect. 7).

Now, assume we have y → y′ = y + ∂u, u ∈ R_{1}. We have

\xymatrix{ & E_{0,y} \ar[dd]^{u} \\ R_{1} \ar[ur]^{e_{y}} \ar[dr]_{e_{y′}} \\ & E_{0,y′}} e ⟼ e ×_{2} s_{2}(0, u)

Using the identities (7.3) and the restriction condition 1 in Definition 7.1 we calculate:

s_{1}(r, y) ×_{2} s_{2}(0, u) = s_{1}(r, y) ×_{2} s_{2}(∂r, u)
= s_{1}(r, y) ×_{2} s_{1}(r, ∂u)
= s_{1}(r, y + ∂u)
= s_{2}(r, y′).

Using (14.2) the above calculation gives

e_{y} (ry) ×_{2} s_{2}(0, u) = e_{y} ×_{2} (ry) s_{2}(0, u)
= e_{y} ×_{2} s_{2}(0, u) (ry)
= e_{y′} (ry′) .

The fact that the identification is canonical gives e_{y} ×_{2} s_{2}(0, u) = e_{y′} and hence ry = ry′. The situation with e_{x} ∈ E_{x,0}, xr, and x′r is entirely analogous.

14.5. Remark. In the extension of A by M above, R_{1} and R_{0} are only groups, in general, even though the crossed module R_{•} they form comes equipped with a braiding. In particular, unless certain strong triviality conditions on the biextension E_{2} = t(2) ∈ Biext(R_{•}, R_{•}; R_{•}) hold, R_{0} is not a ring, in general. This remark applies to stock categorical rings as well, which therefore admit a presentation of the above type with R_{•} a braided symmetric crossed module. This is in sharp contrast with the so-called Ann-Categories, see [Qua03, QHT08, Qua13], whose underlying categories are Picard groupoids. In a companion paper [Ald15] we prove that Picard stacks with a ring-like structure of this type admit presentations given by crossed bimodules, namely crossed modules such as in the presentation above where R_{0} is a ring and R_{1} is an R_{0}-bimodule, with ∂ a bimodule morphism satisfying an appropriate version of the Pfeiffer identity. The general comparison between these structures is intricate and it will be addressed elsewhere.

15. Decomposition of categorical rings and the cohomology of rings

Let ℛ, A, and M be as above. In general, a decomposition [Bre94a] of ℛ consists in the choice of local data (objects, morphisms) subordinate to a hypercovering of A leading to the calculation of cohomological invariants of ℛ. We use the pseudo-monoidal structure carried by a presentation of ℛ for this purpose.

We let L_{i}^{•}(A), i = 2, 3 be the sharp truncations of the iterated bar complexes B(A, i), i = 2, 3 of A as an abelian group. By [ML58, §11] it carries a product structure, defined explicitly up to cells in B(A, 2). For i = 2, 3 we let H_{i}^{3}(A, M) denote the (hyper)cohomology groups computed using the multiplicative bar construction over L_{•}^{i}(A). From the previous reference we have the H_{3}^{3}(A, M) ≃ HML^{3}(A, M), the third Mac Lane cohomology of A with values in M. Furthermore, we let H̃_{2}^{3}(A, M) denote the group of twisted classes satisfying (15.2) below.

15.1. Theorem. There is a bijective correspondence between equivalence classes of weak ring-like stacks ℛ, with A = π_{0}(ℛ) and M = π_{1}(ℛ), and twisted classes in H̃_{2}^{3}(A, M) defined below (cf. Definition 15.9). In the unital case, i.e. when the external monoidal structure of ℛ has a unit element, A is likewise unital and the underlying braiding of ℛ is necessarily symmetric. Hence the weak ring-like structure of ℛ is fully ring-like, and [ℛ] ∈ H̃_{2}^{3}(A, M) ≃ HML^{3}(A, M).

The statement follows from Proposition 15.15, Corollary 15.16, and Proposition 15.17 below. The rest of this section is devoted to the details of the proof of the theorem.

15.2. The bar complex. We establish the notation for the iterated bar complex of the abelian group A. We quote, with minor changes in the notation, from [Dug14, §7]. The part of B^{∞}(A) of interest is the following complex, written homologically:

L_{•}^{3}(A) : \xymatrix{Z[A] & Z[A^{2}] \ar[l]_{∂_{1}} & Z[A^{3}] ⊕ Z[A^{2}] \ar[l]_{∂_{2}} & Z[A^{4}] ⊕ Z[A^{3}] ⊕ Z[A^{3}] ⊕ Z[A^{2}] \ar[l]_{∂_{3}}}

We place the complex in homological degrees [0, 3], with generators:

Degree  Generators
0  [a]
1  [a |_{1} b]
2 [a |_{1} b |_{1} c] [a |_{2} b]
3  [a |_{1} b |_{1} c |_{1} d] [a |_{1} b |_{2} c] [a |_{2} b |_{1} c] [a |_{3} b]

The differential is given by:

∂_{1}[a |_{1} b] = [b] − [a + b] + [a]
∂_{2}[a |_{1} b |_{1} c] = [b |_{1} c] − [a + b |_{1} c] + [a |_{1} b + c] − [a |_{1} b]
∂_{2}[a |_{2} b] = [a |_{1} b] − [b |_{1} a]
∂_{3}[a |_{1} b |_{1} c |_{1} d] = \begin{matrix}[b |_{1} c |_{1} d] − [a + b |_{1} c |_{1} d] \\ + [a |_{1} b + c |_{1} d] − [a |_{1} b |_{1} c + d] + [a |_{1} b |_{1} c] \end{matrix}
∂_{3}[a |_{1} b |_{2} c] = [a |_{1} b |_{1} c] − [a |_{1} c |_{1} b] + [c |_{1} a |_{1} b] − [b |_{2} c] + [a + b |_{2} c] − [a |_{2} c]
∂_{3}[a |_{2} b |_{1} c] = [a |_{1} b |_{1} c] − [b |_{1} a |_{1} c] + [b |_{1} c |_{1} a] + [a |_{2} c] − [a |_{2} b + c] + [a |_{2} b]
∂_{3}[a |_{3} b] = [a |_{2} b] + [b |_{2} a].

The subcomplex L_{•}^{2}(A) is the one obtained by dropping the component Z[A^{2}] of L_{3}^{3}(A), namely the one generated by symbols [a |_{3} b]. The degrees are shifted in a way compatible with the reset needed to form the infinite bar construction, so in effect we have L_{•}^{i}(A) = B(A, i)[−i], for i = 2, 3, and therefore

H_{2}(L_{•}^{2}(A)) ≃ H_{4}(K(A, 2)), H_{2}(L_{•}^{3}(A)) ≃ H_{5}(K(A, 3)).

In the stable situation, this shift is compatible with the degrees in the Q construction.

15.3. Product structure. Let now A be considered with its ring structure. We describe the product structure on L_{•}^{2}(A) and L_{•}^{3}(A) (see [ML58, §11]). The non zero products among the generators, up to products taking values in degrees ≤ 2,^{11} are the following:

[a][b] = [ab]
[a][b |_{1} c] = [ab |_{1} ac]  [a |_{1} b][c] = [ac |_{1} bc]
[a][b |_{1} c |_{1} d] = [ab |_{1} ac |_{1} ad] [a |_{1} b |_{1} c][d] = [ad |_{1} bd |_{1} cd]
[a][b |_{2} c] = [ab |_{2} ac]  [a |_{2} b][c] = [ac |_{2} bc]

and the most interesting one^{12}:

[a|_{1} b][c|_{1} d] = [ac|_{1} bc|_{1} ad+bd]−[ac|_{1} ad|_{1} bc+bd]+[ad|_{1} bc|_{1} bd]−[bc|_{1} ad|_{1} bd]−[bc|_{2} ad].

With this product L_{•}^{2}(A) and L_{•}^{3}(A) become DGAs. Furthermore, both are equipped with the augmentation η: L_{•}^{k}(A) → A given by η([a]) = a, for a ∈ A, and zero in all other degrees. Thus, they become augmented DGAs to which we can apply the (reduced) bar construction B̄_{i,•}(A) := B̄_{N}(L_{•}^{i}(A), η) of loc. cit. (more details below). We are interested in the resulting cohomology groups H^{3}(Hom(B̄_{i,•}(A), M)), where M is an A-bimodule as above. As remarked, for i = 3 we have H^{3}(Hom(B̄_{3,•}(A), M)) ≃ HML^{3}(A, M).

15.4. The multiplicative bar construction and third cohomology of rings. To avoid typographical clutter, the generators of B̄_{i,•}(A) are denoted ⟦u_{1}, …, u_{n}⟧, where the u_{k}s are homogeneous elements of L_{•}^{i}(A). The degree is

deg⟦u_{1}, …, u_{n}⟧ = n + |u_{1}| + ⋯ + |u_{n}|,

11The remaining part of the product structure does not appear in [ML58] and it is unknown, to the author, at least, whether it is explicitly available.

12There appears to be a discrepancy in the second slot of the second term of the right hand side, presumably a misprint in [ML58].

where we let |u_{k}| = deg u_{k}. We quote [ML58, p. 323] the expression for the differential in the bar complex; it is ∂_{tot} = ∂′ + ∂′′, where

∂′⟦u_{1}, …, u_{n}⟧ = − ∑_{i=1}^{n}(−1)^{ε_{i−1}}⟦u_{1}, …, ∂u_{i}, …, u_{n}⟧

\begin{matrix}∂′′⟦u_{1}, …, u_{n}⟧ = η(u_{1})⟦u_{2}, …, u_{n}⟧ + ∑_{i=1}^{n−1}(−1)^{ε_{i}}⟦u_{1}, …, u_{i}u_{i+1}, …, u_{n}⟧ \\ + (−1)^{ε_{n}}⟦u_{1}, …, u_{n−1}⟧η(u_{n}), \end{matrix}

where we have set ε_{i} = deg⟦u_{1}, …, u_{i}⟧. On the right hand side of the expression for ∂′ the inner ∂u_{i} indicates the differential in the complex L_{•}^{i}(A).

The cells of total dimension (=degree) 3 are:

n  Generators
3 ⟦[a], [b], [c]⟧
2 ⟦[a |_{1} b], [c]⟧ ⟦[a], [b |_{1} c]⟧
1 ⟦[a |_{1} b |_{1} c]⟧ ⟦[a |_{2} b]⟧

Therefore a 3-dimensional cochain over B̄_{i,•}(A) with values in M is a 5-tuple ξ = (f, α_{1}, α_{2}, f_{+}, g_{+}), where

f: A^{3} ⟶ M
α_{1} : A^{2} × A ⟶ M, α_{2} : A × A^{2} ⟶ M
f_{+} : A^{3} ⟶ M, g_{+} : A^{2} ⟶ M.

To write the cocycle condition δξ = ξ ∘ (∂′ + ∂′′) = 0 in explicit form we need the cells of dimension four:

n  Generators
4 ⟦[a], [b], [c], [d]⟧
3 ⟦[a |_{1} b], [c], [d]⟧ ⟦[a], [b |_{1} c], [d]⟧ ⟦[a], [b], [c |_{1} d]⟧
2 ⟦[a |_{1} b], [c |_{1} d]⟧ ⟦[a |_{1} b |_{1} c], [d]⟧ ⟦[a |_{2} b], [c]⟧ ⟦[a], [b |_{1} c |_{1} d]⟧ ⟦[a], [b |_{2} c]⟧
1 ⟦[a |_{1} b |_{1} c |_{1} d]⟧ ⟦[a |_{1} b |_{2} c]⟧ ⟦[a |_{2} b |_{1} c]⟧ ⟦[a |_{3} b]⟧†

The element marked with a (†) would only figure in the bar complex B̄_{3,•}(A). In explicit form, the cocycle condition on the five components of ξ is quite complex, consisting of several equations. For ease of reading we arrange them in different groups we have (see below for specific comments):

(15.1a)  a f(b, c, d) − f(ab, c, d) + f(a, bc, d) − f(a, b, cd) + f(a, b, c) d = 0

(15.1b) \begin{aligned}f(b, c, d) − f(a + b, c, d) + f(a, c, d) & = α_{1}(ac, bc; d) − α_{1}(a, b; cd) + α_{1}(a, b; c) d \\ −f(a, c, d) + f(a, b + c, d) − f(a, b, d) & = \begin{matrix}a α_{1}(b, c; d) − α_{1}(ab, ac; d) \\ − α_{2}(a; bd, cd) + α_{2}(a; b, c) d \end{matrix} \\ f(a, b, d) − f(a, b, c + d) + f(a, b, c) & = a α_{2}(b; c, d) − α_{2}(ab; c, d) + α_{2}(a; bc, bd) \end{aligned}

(15.1c) f_{+}(ac, bc, ad + bd) − f_{+}(ac, ad, bc + bd)

+ f_{+}(ad, bc, bd) − f_{+}(bc, ad, bd) − g_{+}(bc, ad)

= α_{1}(a, b; d) − α_{1}(a, b; c + d)
+ α_{1}(a, b; c) + α_{2}(b; c, d) − α_{2}(a + b; c, d) + α_{2}(a; c, d)

(15.1d)  \begin{aligned}f_{+}(ad, bd, cd) − f_{+}(a, b, c) d & = \begin{matrix}− α_{1}(b, c; d) + α_{1}(a + b, c; d) \\ − α_{1}(a, b + c; d) + α_{1}(a, b; d) \end{matrix} \\ f_{+}(ab, ac, ad) − a f_{+}(b, c, d) & = \begin{matrix}α_{2}(a; c, d) − α_{2}(a; b + c, d) \\ + α_{2}(a; b, c + d) − α_{1}(a; b, c) \end{matrix} \\ g_{+}(ac, bc) − g_{+}(a, b) c & = −α_{1}(a, b; c) + α_{1}(b, a; c) \\ g_{+}(ab, ac) − a g_{+}(b, c) & = α_{2}(a; b, c) − α_{2}(a; c, b) \end{aligned}

(15.1e) f_{+}(b, c, d) − f_{+}(a + b, c, d) + f_{+}(a, b + c, d) − f_{+}(a, b, c + d) + f_{+}(a, b, c) = 0 ; \begin{aligned}f_{+}(a, b, c) − f_{+}(a, c, b) + f_{+}(c, a, b) & = g_{+}(b, c) − g_{+}(a + b, c) + g_{+}(a, c) \\ f_{+}(a, b, c) − f_{+}(b, a, c) + f_{+}(b, c, a) & = −g_{+}(a, c) + g_{+}(a, b + c) − g_{+}(a, b) \end{aligned} ; g_{+}(a, b) + g_{+}(b, a) = 0 .

15.5. Remarks.

1. The last group of equations (15.1e) is closed and it is the condition for the pair (f_{+}, g_{+}) to be an Eilenberg Mac Lane cocycle. Thus (f_{+}, g_{+}) represents a class in H^{4}(K(A, 2), M) or H^{5}(K(A, 3), M), depending on whether the last term is included.

2. The block (15.1d) can be re-written as follows. Let λ_{a} and ρ_{a} denote the left and right multiplications by a ∈ A. Also, write a f_{+} and f_{+} a for the action of a ∈ A on the values of f_{+}. Same for g_{+}. Then we can rewrite the block as

\begin{aligned}ρ_{d}^{∗}f_{+} − f_{+} d & = −α_{1}(−, −; d) ∘ ∂_{3} \\ λ_{a}^{∗}f_{+} − a f_{+} & = α_{2}(a; −, −) ∘ ∂_{3} \\ ρ_{c}^{∗}g_{+} − g_{+} c & = −α_{1}(−, −; c) ∘ ∂_{2} \\ λ_{a}^{∗}g_{+} − a g_{+} & = α_{2}(a; −, −) ∘ ∂_{2} \end{aligned}

The block (15.1d) expresses the invariance of the class of (f_{+}, g_{+}) under left and right multiplication in A.

3. Equation (15.1a) has the familiar form of a Hochschild cocycle. (As it might be expected, it arises from the associativity constraint on the exterior monoidal structure of ℛ, as it will be shown below.) The behavior of f with respect to the additivity, i.e. its failure to be multilinear, is expressed by the equation block (15.1b).

4. As it will be explained below, the meaning of the somewhat more intricate relation (15.1c), is that it expresses the compatibility (interchange law) between the partial composition laws of the exterior monoidal structure of ℛ. It relates to the last of the relations in the Mac Lane’s product structure of sect. 15.3.

It is well known that H_{5}(K(A, 3)) ≃ A/2A and that H^{5}(K(A, 3), M) ≃ Hom(A/2A, M) ≃ Hom(A,_{2}M) [EML54, §23] (see also [Dug14, §7.3]). The former isomorphism is given by a ↦ [a |_{2} a], the latter sends the class of [(f_{+}, g_{+})] to the (linear) map a ↦ g_{+}(a, a). In fact, dropping the last of equations (15.1e), the same assignment gives a quadratic map from A to M. (Recall that q: A → M is quadratic if q(na) = n^{2}q(a) for all n ∈ Z, and if the associated symmetric function Δq(a, b) = q(a + b) − q(a) − q(b), a, b ∈ A, is bilinear.) Thus H^{5}(K(A, 3), M) ≃ Hom(Γ_{2}(A), M), where Γ_{2}(A) is the Whitehead functor, i.e. the degree four component of the divided power algebra Γ_{•}(A) over A.

Recall that m ∈ M is central if am = ma for all a ∈ A. Let M^{A} be the A-module of central elements of M. Assume that ξ represents a class in H^{3}(Hom(B̄_{2,•}(A), M)), that is, the pair (f_{+}, g_{+}) satisfies, as part of the full set ξ = (f, α_{1}, α_{2}, f_{+}, g_{+}), equations (15.1d) and (15.1e), except the last one. We have the following observation.

15.6. Proposition. Let A be unital. We have an isomorphism

H^{3}(Hom(B̄_{2,•}(A), M)) ≃ H^{3}(Hom(B̄_{3,•}(A), M)) ≃ HML^{3}(A, M).

Moreover, the map HML^{3}(A, M) → Hom(A,_{2}M) [ML58, §11] factors through _{2}M^{A} ≃ Hom_{A}(A,_{2}M^{A}).

Proof. Let q be the map a ↦ g_{+}(a, a). The last two equations of (15.1d) imply that q is an A-bimodule homomorphism. As such, it is determined by a central element in M. This proves the first isomorphism. The second follows from [ML58].

15.7. Remark. Because the complexes B̄_{2,•}(A) and B̄_{3,•}(A) are equal below degree 3, their cohomologies coincide for n ≤ 2. Combined with [Bre99, §6], we have that the statement of Proposition 15.6 holds in general when H^{•} is interpreted as hypercohomology.

The following variant of the previous constructions will be useful below. Given a 5tuple ξ as above, consider the map β_{ξ}: B_{2,4}(A)_{2} → M given by (the subscript 2 refers to the elements in B_{2,4}(A) with n = 2):

β_{ξ}(⟦u_{1}, u_{2}⟧) = \begin{cases}g_{+}(ab, ac) + g_{+}(ac, ab) & ⟦u_{1}, u_{2}⟧ = ⟦[a], [b |_{2} c]⟧, \\ g_{+}(ac, bc) + g_{+}(bc, ac) & ⟦u_{1}, u_{2}⟧ = ⟦[a |_{2} b], [c]⟧, \\ 0 & all other cases. \end{cases}

Further, we define β_{ξ} to be zero on the other components of B_{2,4}(A). Consider the twisted cocycle equation condition:

(15.2)  Dξ := δξ + β_{ξ} = 0.

This replaces the last two equations in the block (15.1d) with

(15.3)  \begin{aligned}−g_{+}(bc, ac) − g_{+}(a, b) c & = −α_{1}(a, b; c) + α_{1}(b, a; c) \\ −g_{+}(ac, ab) − a g_{+}(b, c) & = α_{2}(a; b, c) − α_{2}(a; c, b) \end{aligned}

and it still drops the last one from (15.1e). Observe that for i = 3 equation (15.2) is vacuously equal to the set (15.1).

15.8. Remark. Note that replacing ξ with ξ′ = ξ + δν has the effect of adding to g_{+} the alternation of a map h_{+} : A × A → M, so that β_{ξ} does not depend on the particular choice of g_{+} within its class modulo coboundaries. Hence we get a well defined class of solutions of equation (15.2) modulo adding coboundaries.

15.9. Definition. Let us denote by H̃_{2}^{3}(A, M) the hypercohomology group of classes satisfying (15.2) in degree three, relative to A (cf. Remark 15.7).

There is an evident map HML^{3}(A, M) → H̃_{2}^{3}(A, M), which, at least in the cases we like to consider, is an isomorphism. Indeed we have

15.10. Lemma. Let A be unital. Then (15.2) and (15.1) are equivalent. Therefore H̃_{2}^{3}(A, M) ≃ HML^{3}(A, M).

Proof. Choosing a or c = 1 in (15.3) shows that the last equation (15.1e) holds. (Again, use Remark 15.7 for the general situation.)

15.11. Decomposition of ℛ. For convenience of notation, let us write [∂: R_{1} → R_{0}] ≡ [∂: R → Λ]. Recall that since we denote the interior monoidal structure of ℛ additively, we do the same for R and Λ in the presentation.

15.12. Remark. [Warning] In the following we use a set theoretic-type notation. However, in this sheaf theoretic context, the notation a ∈ A, or a_{1}, …, a_{n} ∈ A means these are generalized points of A of the form, say, a: U → A, etc. where U is an object of the topos. In fact, in order to properly carry out the hypercohomology calculations we will have to choose hypercovers of A and of various simplicial objects associated to it, such as the various Eilenberg-Mac Lane objects K(A, 2) and K(A, 3). We will proceed formally as in the pointwise case, and systematically appeal to the hypercohomology spectral sequence as in Remark 15.7. The actual hypercohomology arguments based are made precise in Appendix E.

For a point a ∈ A, we let Λ_{a} = a^{∗}Λ be the fiber. For any multi-extension E_{n} in the weak ring-like structure of R → Λ denote by E_{a_{1},...,a_{n}} the pullback of E to Λ_{a_{1}} × ⋯ × Λ_{a_{n}}, where a_{1}, …, a_{n} ∈ A.

There is an obvious morphism Λ_{a_{1}} ×⋯×Λ_{a_{n}} → Λ_{a_{1}+···+a_{n}} covering the n-fold iteration of +: A × A → A. If we assume a choice for a point x_{a} ∈ Λ_{a} has been made for all points a ∈ A, i.e. we have a section x of q: Λ → A, then x_{a} + x_{b} (the image of (x_{a}, x_{b}) under Λ_{a} × Λ_{b} → Λ_{a+b}) and x_{a+b} will in general be different. This gives rise to σ: A × A → R by way of

x_{a+b} = x_{a} + x_{b} + ∂σ_{a,b}.

In fact the objects X_{a} := π(x_{a}) of ℛ provide a decomposition of the sequence ℳ → ℛ → A, where ℛ is considered as a gerbe over A [Bre94a, §7]. Note that the above corresponds to the morphism

σ_{a,b}: X_{a+b} ⟶ X_{a} + X_{b}

of (R, Λ)-torsors. Therefore we have the classical fact that the difference between the two possible comparisons between X_{a} + X_{b} + X_{c} and X_{a+b+c}, and the application of equations (2.2) to x_{a}, x_{b}, and x_{c} determines a pair (f_{+}, g_{+}) satisfying the cocycle equations (15.1e) above, hence a class in H^{4}(K(A, 2), M) or H^{5}(K(A, 3), M) (see, e.g. [JS93, Bre99]). For future reference, the relevant relations are (cf. [Bre94a]):

(15.4)  \begin{aligned}σ_{b,c} + σ_{a,b+c} − f_{+}(a, b, c) & = σ_{a,b}^{x_{c}} + σ_{a+b,c}, \\ −g_{+}(a, b) + σ_{a,b} & = \{x_{a}, x_{b}\} + σ_{b,a}. \end{aligned}

(There is an obvious generalization to n variables, but the classical situation is sufficient to describe the decomposition with respect to the “+” operation of ℛ.)

For the multiplication, i.e. the exterior monoidal structure, the reasoning at the end of sect. 13 implies we have instead

(15.5)  [figure: E_{a_{1},...,a_{n}}; Λ_{a_{1}} × ⋯ × Λ_{a_{n}}; Λ_{a_{1}...a_{n}}; p; ȷ]

covering the multiplication in A. Using the choice of a section of the fibers Λ_{a} as above, we have the isomorphism of (R, Λ)-torsors

(15.6)  e_{a_{1},...,a_{n}}: E_{a_{1},...,a_{n}} ⟶^{∼} X_{a_{1}···a_{n}} = (R, x_{a_{1}···a_{n}})

which follows from (15.5) and again the end of sect. 13. We can assume this isomorphism is realized by the choice of a point e_{a_{1},...,a_{n}} ∈ E_{a_{1},...,a_{n}} such that ȷ(e_{a_{1},...,a_{n}}) = x_{a_{1}···a_{n}}.

We can write the morphism (13.1) with respect to this choice of local data. According to the proof of Theorem 11.1, let a_{1,1}, …, a_{1,m_{1}}, …, a_{n,1}, …, a_{n,m_{n}} ∈ A. For i = 1, …, n define b_{i} = a_{i,1} ⋯ a_{i,m_{i}}. Consider the points e_{i} = e_{a_{i,1},...,a_{i,m_{i}}} ∈ E_{a_{i,1},...,a_{i,m_{i}}} and e_{b_{1},...,b_{n}} ∈ E_{b_{1},...,b_{n}}. Then [e_{1}, …, e_{n}, e_{b_{1},...,b_{n}}] is a point of the composition E_{n}(E_{m_{1}}, …, E_{m_{n}}) over Λ_{a_{1,1}} × ⋯ × Λ_{a_{n,an}}, and ȷ([e_{1}, …, e_{n}, e_{b_{1}...,b_{n}}]) = ȷ(e_{b_{1},...,b_{n}}) = b_{1} ⋯ b_{n}. In other words, we have an isomorphism of (R, Λ)-torsors

E_{n}(E_{m_{1}}, …, E_{m_{n}})_{a_{1,1},...,a_{n,mn}} ⟶^{∼} (R, x_{b_{1}···b_{n}}).

On the other hand, since E_{m_{1}+···+m_{n}} covers the multiplication

(a_{1,1}, …, a_{n,m_{n}}) → a_{1,1} ⋯ a_{n,m_{n}} = b_{1} ⋯ b_{n},

we have another isomorphism

(E_{m_{1}+···+m_{n}})_{a_{1,1},...,a_{n,mn}} ⟶^{∼} (R, x_{b_{1}···b_{n}})

determined by a chosen section of E_{m_{1}+···+m_{n}}. Thus, the morphism (13.1) amounts to an automorphism

(15.7)  f_{n;m_{1},...,m_{n}}(a_{1,1}, …, a_{n,m_{n}}): (R, x_{b_{1}···b_{n}}) ⟶^{∼} (R, x_{b_{1}···b_{n}}),

which, using standard facts about (R, Λ)-torsors, we identify with an element of M.

15.13. Cocycle computations. We compute the full class determined by ℛ from the multi-extension structure of the presentation R → Λ → ℛ and prove that it satisfies the full set of equations (15.1).

15.13.1. Setup. First, consider the following (not necessarily commutative) diagrams of (R, Λ)-torsor morphisms

\xymatrix{E_{a,c} ∧ E_{b,c} \ar[r] \ar[d] & E_{x_{a}+x_{b},x_{c}} \ar[r] & E_{a+b,c} \ar[d] \\ (R, x_{ac}) ∧ (R, x_{bc}) & (R, x_{ac} + x_{bc}) \ar[r]^{σ_{ac,bc}^{−1}} & (R, x_{(a+b)c})} (15.8a)

and

\xymatrix{E_{a,b} ∧ E_{a,c} \ar[r] \ar[d] & E_{x_{a},x_{b}+x_{c}} \ar[r] & E_{a,b+c} \ar[d] \\ (R, x_{ab}) ∧ (R, x_{ac}) & (R, x_{ab} + x_{ac}) \ar[r]^{σ_{ab,ac}^{−1}} & (R, x_{a(b+c)})} (15.8b)

arising from the two partial product laws of E = E_{2}. The vertical arrows are the identification (15.6). Both diagrams in (15.8) can be construed as defining an automorphism of their respective lower right corners, which can be identified with an element of M: let them be −α_{1}(a, b; c) and α_{2}(a; b, c) respectively.

On the one hand the top rows can be written as

(15.9)  e_{a,c} ∧ e_{b,c} ↦ e_{a+b,c} g_{1}(a, b; c),  e_{a,b} ∧ e_{a,c} ↦ e_{a,b+c} g_{2}(a; b; c),

by invoking (4.5). On the other hand, following the bottom part, we have

e_{a,c} ∧ e_{b,c} ↦ σ_{ac,bc}^{−1}, e_{a,b} ∧ e_{a,c} ↦ σ_{ab,ac}^{−1}.

Comparing the two we get the relations

(15.10)  −α_{1}(a, b; c) = g_{1}(a, b; c) + σ_{ac,bc},  α_{2}(a; b, c) = g_{2}(a; b, c) + σ_{ab,ac}.

15.13.2. The relations (15.1d). Consider the diagrams:

\xymatrix{E_{x_{a+b}+x_{c},x_{d}} \ar[r]^{σ_{a+b,c}} & E_{a+b+c,d} \ar[dd] & E_{x_{a},x_{b+c}+x_{d}} \ar[r]^{σ_{b+c,d}} & E_{a,b+c+d} \ar[dd] \\ E_{x_{a}+x_{b}+x_{c},x_{d}} \ar[u]^{σ_{a,b}^{xc}} \ar[d]_{σ_{b,c}} & & E_{x_{a},x_{b}+x_{c}+x_{d}} \ar[u]^{σ_{b,c}^{x_{d}}} \ar[d]_{σ_{c,d}} \\ E_{x_{a}+x_{b+c},x_{d}} \ar[r]_{σ_{a,b+c}} & E_{a+b+c,d} & E_{x_{a},x_{b}+x_{c+d}} \ar[r]_{σ_{b,c+d}} & E_{a,b+c+d}} (15.11a)

and

\xymatrix{E_{x_{a}+x_{b},x_{c}} \ar[r]^{σ_{a,b}} & E_{a+b,c} & E_{x_{a},x_{b}+x_{c}} \ar[r]^{σ_{b,c}} & E_{a,b+c} \\ E_{x_{b}+x_{a},x_{c}} \ar[u]^{η^{1}} \ar[r]_{σ_{b,a}} & E_{a+b,c} \ar[u] & E_{x_{a},x_{c}+x_{b}} \ar[u]^{η^{2}} \ar[r]_{σ_{c,b}} & E_{a,b+c} \ar[u]} (15.11b)

Both diagrams (15.11) are parts of more extended ones, giving rise to relations linking f_{+} and g_{+} to the other quantities comprising a 5-tuple satisfying relations (15.1) as follows.

First observe that by applying Lemma 14.3, the right vertical give the automorphisms corresponding to f_{+}(a, b, c) d (resp. a f_{+}(b, c, d)) for (15.11a), and g_{+}(a, b) c (resp. a g_{+}(b, c)) for (15.11b).

Then from (15.8) and (15.11a) form the obvious associativity diagrams for the morphisms E_{a,d} ∧ E_{b,d} ∧ E_{c,d} → E_{a+b+c,d} and E_{a,b} ∧ E_{a,c} ∧ E_{a,d} → E_{a,b+c+d}. Using the cocycle decomposition (4.6) (and Lemma 14.3 for the right vertical arrows of (15.11)) we arrive at:

(15.12)  \begin{aligned}g_{1}(a + b, c; d) + g_{1}(a, b; d)^{x_{cd}} & = g_{1}(a, b + c; d) + g_{1}(b, c; d) + f_{+}(a, b, c) d, \\ g_{2}(a; b + c, d) + g_{2}(a; b, c)^{x_{ad}} & = g_{2}(a; b, c + d) + g_{2}(a; c, d) + a f_{+}(b, c, d), \end{aligned}

(The difference with equations (4.6a) and (4.6c) arises because the top rows of (15.8), contrary to the actual partial multiplication morphisms, lack associativity.) Using (15.10), (15.12), and the first of (15.4), we obtain the first two of the cocycle relations (15.1d).

The commutativity diagrams obtained from (15.11b) and (15.8) can be analyzed in an analogous manner, utilizing the second of (15.4). However, we do not directly obtain the other two equations in the block (15.1d); instead, we arrive at their “flipped” counterpart (15.3).

15.13.3. The interchange relation (15.1c). We check the compatibility law (2.5) after pulling back to Λ_{a} × Λ_{b} × Λ_{c} × Λ_{d}. Use the diagram (15.13)

\xymatrix{(E_{a,c} ∧ E_{b,c}) ∧ (E_{a,d} ∧ E_{b,d}) \ar[rrrr] \ar[dd] \ar[dr] &  &  &  & (E_{a,c} ∧ E_{a,d}) ∧ (E_{b,c} ∧ E_{b,d}) \ar[dl] \ar[dd] \\  & E_{x_{a}+x_{b},x_{c}} ∧ E_{x_{a}+x_{b},x_{d}} \ar[dl] \ar[dr] & I & E_{x_{a},x_{c}+x_{d}} ∧ E_{x_{b},x_{c}+x_{d}} \ar[dl] \ar[dr] \\ E_{a+b,c} ∧ E_{a+b,d} \ar[dr] \ar[ddrr] & II & E_{x_{a}+x_{b},x_{c}+x_{d}} \ar[dl] \ar[dr] & II & E_{a,c+d} ∧ E_{b,c+d} \ar[dl] \ar[ddll] \\  & E_{x_{a+b},x_{c}+x_{d}} \ar[dr] & III & E_{x_{a}+x_{b},x_{c+d}} \ar[dl] \\  &  & E_{a+b,c+d}}

where the triangles commute by definition of the morphisms determined by the top rows of (15.8); the pentagon (marked I) commutes by the compatibility law; the squares II and the square III are obviously commutative by functoriality (an explicit calculation uses Proposition 4.2 and the equations (7.3)). Thus the diagram commutes, and we can use the cocycle equation (4.7) directly, written additively, which gives:

g_{2}(a + b; c, d) + g_{1}(a, b; c)^{x_{(a+b)d}} + g_{1}(a, b; d) = g_{1}(a, b; c + d) + g_{2}(a; c, d)^{x_{b(c+d)}} + g_{2}(b; c, d) − {x_{bc}, x_{ad}}^{x_{bd}}.

Inserting equations (15.10) and using (15.4) finally gives the third block (15.1c) of the cocycle relation (15.1).

15.13.4. The relations (15.1b) and (15.1a). We specialize the expression (15.7) for the morphism (13.1) to the tuples (2; 2, 1) and (2; 1, 2). The morphism μ given by (C.1) corresponds to the element

f(a, b, c) = −f_{(2;1,2)}(a, b, c) + f_{(2;2,1)}(a, b, c) ∈ M.

More precisely, using the isomorphism with (R, x_{abc}) as a reference trivialization, we can identify f(a, b, c) with an automorphism of E_{3} pulled back to Λ_{a} × Λ_{b} × Λ_{c}, hence with a section of M over it. Explicitly, locally on Λ_{a} × Λ_{b} × Λ_{c}, we have the morphism of (R, Λ)-torsors

(15.14)  μ_{a,b,c}: E_{2}(E_{2}, I)_{a,b,c} ⟶ E_{2}(I, E_{2})_{a,b,c},

which, using the composition of multi-extensions given in section 11, we can write as

(15.15)  μ([e_{a,b}, 0_{c}, e_{ab,c}]) = [0_{a}, e_{b,c}, e_{a,bc}] − f(a, b, c),

where [e_{a,b}, e_{ab,c}] ∈ E_{2}(E_{2}, I)_{a,b,c} and [e_{b,c}, e_{a,bc}] ∈ E_{2}(I, E_{2})_{a,b,c}. (Recall the additivity in the notation; we extend it to the action of R on torsors. The brackets denote the class under the action of R. The identity map is represented by the trivial butterfly, and here 0_{a} represents the unit section of the underlying trivial torsor Λ × R at the point x_{a}, say.) We have ȷ([e_{a,b}, e_{ab,c}]) = ȷ([e_{b,c}, e_{a,bc}]) = x_{abc}, so we can see directly that f(a, b, c) ∈ M.

As an isomorphism of tri-extensions, μ is a homomorphism for each of the three partial laws. Writing these conditions for (15.14), we must compute the maps along the following diagram

\xymatrix{E_{2}(E_{2}, I)_{a,c,d} ∧^{R} E_{2}(E_{2}, I)_{b,c,d} \ar[r]^{×_{1}} \ar[d]_{μ∧μ} & E_{2}(E_{2}, I)_{x_{a}+x_{b},x_{c},x_{d}} \ar[r]^{σ_{a,b}} \ar[d]|{μ} & E_{2}(E_{2}, I)_{a+b,c,d} \ar[d]^{μ} \\ E_{2}(I, E_{2})_{a,c,d} ∧^{R} E_{2}(E_{2}, I)_{b,c,d} \ar[r]_{×_{1}} & E_{2}(I, E_{2})_{x_{a}+x_{b},x_{c},x_{d}} \ar[r]_{σ_{a,b}} & E_{2}(I, E_{2})_{a+b,c,d}}

and the other two expressing the compatibility (or lack thereof) with the second and third partial laws. Because there are some new elements compared to the calculations which have appeared thus far, we sketch some of the details. In particular, to compute the two horizontal maps in the second square above we need the explicit form of the trivializations s_{1} for both tri-extensions. Similarly for the other two diagrams. According to the last part of section 11, the form of s_{1}^{2,1} for E_{2}(E_{2}, I) is:

s_{1}^{2,1}(r, x_{c}, x_{d}) = [s_{1}(r, x_{c}), 0_{x_{d}}, 1_{x_{d}}], r = σ_{a,b},

where 0_{x_{d}} denotes the unit section of Λ × R computed at x_{d} ∈ Λ, and 1_{x_{d}} denotes the unit section of E_{2}|_{{0}×Λ} ≃ Λ × R similarly computed at x_{d}. Thus, the map denoted by σ_{a,b} in the top row is given by

e ⟼ e ×_{1} s_{1}(σ_{a,b}, x_{c}, x_{d}).

Similarly, for the analogous map in the bottom row we must use the expression

s_{1}^{1,2}(r, x_{c}, x_{d}) = [0_{∂r}, e_{c,d}, s_{1}(r, x_{c})], r = σ_{a,b},

with a similar interpretation of the notation.

Thus, the upper path to the lower right corner gives

μ([e_{a,c}, e_{ac,d}] ×_{1} [e_{b,c}, e_{bc,d}] ×_{1} [s_{1}(σ_{a,b}, x_{c}), 0_{d}, 1_{x_{d}}])
= μ[e_{a,c} ×_{1} e_{b,c} ×_{1} s_{1}(σ_{a,b}, x_{c}), 0_{d}, e_{ac,d} ×_{1} e_{bc,d}]
= μ[e_{a+b,c} + g_{1}(a, b; c), 0_{d}, e_{ac,d} ×_{1} e_{bc,d}]
= μ[e_{a+b,c}, 0_{d}, e_{ac,d} ×_{1} e_{bc,d} ×_{1} s_{1}(−g_{1}(a, b; c), x_{d})]
= μ[e_{a+b,c}, 0_{d}, e_{ac,d} ×_{1} e_{bc,d} ×_{1} s_{1}(σ_{ac,bc} + α_{1}(a, b; c), x_{d})]
= μ[e_{a+b,c}, 0_{d}, e_{ac+bc,d}] + g_{1}(ac, bc, d) − α_{1}(a, b; c)d
= [0_{a+b}, e_{c,d}, e_{a+b,cd}] + g_{1}(ac, bc; d) − f(a + b, c, d) − α_{1}(a, b; c)d

where in the next to last we have used (7.3), the first of (15.9), and Lemma 14.3; to obtain the last we have used (15.15).

On the other hand, the lower path to the lower right corner gives, with similar calculations

μ[e_{a,c}, e_{ac,d}] ×_{1} μ[e_{b,c}, e_{bc,d}] ×_{1} [0_{∂σ_{a,b}}, e_{c,d}, s_{1}(σ_{a,b}, x_{cd})] = [0_{a+b}, e_{c,d}, e_{a+b,cd}] + g_{1}(a, b; cd) − f(a, c, d) − f(b, c, d).

Comparing the two expressions and using (15.10) yield the first equation of block (15.1b). The others are obtained via identical means.

The last equation (15.1a) becomes now the easiest to obtain, as a condition satisfied by the automorphism μ upon considering the five possible pullbacks to the quadri-extension E_{4}, as per the pentagon diagram (C.2). We leave the details to the reader.

15.14. The class of a ring-like stack. Assembling the steps in sect. 15.13, we have the following

15.15. Proposition. Let ℛ be a (weakly) ring-like stack with π_{0}(ℛ) = A and π_{1}(ℛ) = M. A decomposition of ℛ determines a twisted cocycle ξ = ξ_{ℛ} satisfying (15.2) with the same A-bimodule structure. An equivalence ℛ → ℛ′ gives rise to two twisted cocycles ξ_{ℛ} and ξ_{ℛ′} differing by a coboundary, hence equivalence classes are in one-to-one correspondence with elements of H̃_{2}^{3}(A, M).

Proof. The first statement is a consequence of the preceding calculations. The statement about the equivalence follows from the definition of morphism of pseudo-monoid and the fact that the structure is preserved across pseudo-functors between multi-bicategories (cf. B.4 and B.5), in particular between 13.5

Since the only difference between all these complexes occurs in degrees 3 and 4, the statements carry over to the hypercohomology situation.

Let us use the notation [ℛ] for the class determined by ℛ. Thus of [ℛ] = [ξ_{ℛ} ]. As we have seen, a consequence of Proposition 15.6 and Lemma 15.10 is that in the unital case the classes in H̃_{2}^{3}(A, M) lift to HML^{3}(A, M).

15.16. Corollary. Let ℛ be as above, with in addition a unit object for the exterior monoidal structure. Then A is unital and there exists an equivalence ℛ →^{∼} ℛ′ where the underlying categorical group structure of ℛ′ is braided symmetric. Hence [ℛ] = [ℛ′] ∈ HML^{3}(A, M).

We briefly address the question of recovering ℛ (up to equivalence) from [ℛ]. Consider a class [ξ] ∈ H̃_{2}^{3}(A, M). A portion of ξ will represent a class in H^{4}(K(A, 2), M), possibly lifting into the stable range. Let ξ_{+} denote this projection. Standard techniques [Bre94a, §7.6–7] allow to reconstruct a braided (possibly symmetric) stack ℛ = ℛ_{ξ} from ξ_{+}, equipped with ϖ: ℛ → A fitting into the standard short exact sequence ℳ → ℛ → A, with ℳ = Tors(M). Briefly, ξ_{+} determines a 2-gerbe over a simplicial model of K(A, 2) or K(A, 3), suitably re-scaled so that the relevant class appears in degree three. ℛ is obtained by gluing trivial gerbes with band M → 0 over A along ξ_{+}. (We must supplement the cocycles in loc. cit. with those parts pertaining to the braiding structure.) The class of ℛ is that of ξ_{+}, and therefore it is equipped with a decomposition (15.4).

15.17. Proposition. Let ℛ, ξ, and ξ_{+} be as above. Let R → Λ → ℛ be a presentation by a braided crossed module. Then R → Λ carries a bi-extension whose class is ξ.

Proof Proof (Sketch). The main point is to reverse the computation of the cohomology class, using the equations for the cocycle ξ (either (15.1) or their twisted form (15.2)) to obtain a well-defined biextension E → Λ × Λ providing (R, Λ) with a pseudo-monoid structure.

Starting with the implementation of (15.5), for all a, b ∈ A define E_{a,b} → Λ_{a} × Λ_{b} as E_{a,b} = Λ_{a} × Λ_{b} × R equipped with ȷ: E_{a,b} → Λ given by 0_{R} ↦ x_{ab}. From (15.10), given ξ and the decomposition (15.4), we compute the nonabelian cocycles g_{1} and g_{2} we can use to define partial laws

×_{1} : E_{a,c} ∧^{R} E_{b,c} ⟶ E_{a+b,c} , ×_{2} : E_{a,b} ∧^{R} E_{a,c} ⟶ E_{a,b+c} ,

which are well defined by equations (15.12) and (15.13).

Note that, unless a = 0, the pullback of ∂: R → Λ along Λ_{a} → Λ is trivial, whereas it is isomorphic to Im ∂ for a = 0. This ensures (again from equation (15.5) and choosing a normalized section x) the triviality conditions required by Definition 7.1 to have a full-fledged butterfly are satisfied.

Finally, we use (15.14) and (15.15) to define μ: E_{2}(E_{2}, I) → E_{2}(I, E_{2}); the identity

g_{1}(ac, bc; d) − f(a + b, c, d) − α_{1}(a, b; c)d = g_{1}(a, b; cd) − f(a, c, d) − f(b, c, d),

and its companions found above, together with (15.1a) and (15.1b), ensure E satisfies the required pentagonal structure.

Part IV

A. Multi-variable compositions

In this technical addendum we give a brief treatment of multivariable functor calculus in a bicategory. Our approach is descriptive and explicit, and it is aimed at a definition of pseudo-monoid suitable for the applications in the text to multi-additive functors and multi-linear butterflies (cf. sects. 10 and following).

We resort to multi categorical-based ideas, in fact we borrow some of Kelly’s clubs formalism (see [Kel72, Kel74]), which is convenient in the present context.^{13} We include permutations, even though this is slightly more general than needed in the main part of the text. Permutations can be included at nearly no additional cost, covering the symmetric monoidal case, which is what the formalism was originally designed to do. As a result, the formalism can still be used to symmetrize the (external) monoidal structures described in the text. Unlike [Kel72], we need our objects to inhabit a bicategory, as opposed to a 2-category, due to the fact that crossed modules equipped with butterflies as morphisms form a genuine bicategory equivalent to the 2-category of group-like stacks.

13See also [Tho95] for further applications to (symmetric) monoidal categories.

We define a bi-multicategory ℭ the structure defined by the following data:^{14}

• A class of objects x, y, …

• For each tuple (y_{1}, …, y_{n}; x) of objects, a groupoid of arrows Hom_{ℭ}(y_{1}, …, y_{n}; x). The cells are denoted

[figure: (y_{1}, …, y_{n}); f; g; α; x].

• For each object x, a functor ı_{x}: 1 → Hom_{ℭ}(x; x), where 1 is the singleton category. The resulting distinguished object is the identity arrow id_{x}: (x) → x.

• Compositions functors

Hom_{ℭ}(y_{1}, …, y_{n}; x) × Hom_{ℭ}(z_{1,1}, …, z_{1,m_{1}}; y_{1}) × ⋯ × Hom_{ℭ}(z_{n,1} …, z_{n,m_{n}}; y_{n})

⟶ Hom_{ℭ}(z_{1,1}, …, z_{n,m_{n}}; x)

(f; g_{1}, …, g_{n}) ⟶ f(g_{1}, …, g_{n}).

• Associativity data for the composition and the identities as in a bicategory.

• For each n and for each tuple (y_{1}, …, y_{n}; x), an action by Σ_{n}, that is, a functor

ξ^{∗}: Hom_{ℭ}(y_{ξ(1)}, …, y_{ξ(n)}; x) ⟶ Hom_{ℭ}(y_{1}, …, y_{n}; x)

such that (ξη)^{∗} ≃ η^{∗}ξ^{∗} and the composition functors are equivariant for this action. Moreover, these isomorphisms are subject to appropriate coherence conditions.

The definition of pseudo- (or lax-)functor F: ℭ → 𝔅 between multi-bicategories is mutatis mutandis the same as for bicategories. For the symmetric structure, we add the condition that the functors

F_{y_{1},...,y_{n};x}: Hom_{ℭ}(y_{1}, …, y_{n}; x) → Hom_{𝔅}(F (y_{1}), …, F (y_{n}); F (x))

preserve the Σ_{n}-action, for all n.

The last item in the data list intuitively affords for cells of the form

(A.1)  [figure: (y_{1}, …, y_{n}); x; (y_{ξ(1)}, … y_{ξ(n)})]

14Without the extra structure given by permutations, the name “bi-multicategory” appears in [Pis14], where it denotes the bicategory-analog of a multicategory: a ℭ𝔞𝔱-enriched multicategory with weakly associative composition. For enrichment over simplicial sets, see also [Rob11], which also incorporates permutations.

Dropping all Hom_{ℭ}-groupoids except those of the form Hom_{ℭ}(y; x), i.e. those of valence (or arity) one, we obtain an ordinary bicategory, the underlying bicategory of ℭ. Also, a 2-multicategory is a bi-multicategory in which all the associativity and identity data are strict. If all the groupoids Hom_{ℭ}(−; −) are discrete we obtain an ordinary multicategory.

The following is the analog of the “generalized functor category” in Kelly [Kel72, §2].

A.1. Definition. Let x, y be two objects of ℭ. Then {y, x} denotes the category whose objects are pairs (n, f), where n is a natural number and f is an object of Hom_{ℭ}(y, …, y; x) (y is repeated n times); morphisms (n, f) → (m, g) only exist if n = m and consist of a permutation ξ ∈ Σ_{n} and a morphism α: f → ξ^{∗}(g). The composition of morphisms is dictated by the composition in ℭ. Explicitly, if (η, β): (n, g) → (n, h), then (η, β) ∘ (ξ, α) = (ηξ, ξ^{∗}(β) ∘ α) modulo the (unnamed) coherence morphism ξ^{∗}η^{∗} ≃ (ηξ)^{∗}.

A.2. Remark. Let S be the skeletal category of finite sets with permutations as morphisms. (S can be obtained as the core, i.e. the largest subgroupoid, of the Segal category Γ of finite sets.) Thus an object of S can be identified with a natural number. For any two objects x, y of ℭ there results an obvious projection functor p: {y, x} → S. In fact, {y, x} can be obtained as the Grothendieck construction applied to the functor F: S → ℭ𝔞𝔱 defined by F (n) = Hom_{ℭ}(y, …, y; x) and F (ξ) = ξ^{∗}.

For the following definition is the analog of the operations defined in [Kel72, §2.1, §2.2].

A.3. Definition. For all objects x, y, z of ℭ:

1. {y, x} ≀ {z, y} is the category whose objects are lists (f; g_{1}, …, g_{n}), where n = p(f); morphisms (f; g_{1}, …, g_{n}) → (f′; g′_{1}, …, g′_{n}) (there are no morphisms if p(f) ≠ p(f′)) consist of a morphism α: f → ξ^{∗}(f′) and for 1 ≤ i ≤ n, morphisms β_{i}: g_{ξ(i)} → η_{ξ(i)}^{∗}(g′_{i}), where ξ = p(f) ∈ Σ_{n}, p(g_{i}) η_{i} = p(g_{i}) ∈ Σ_{n_{i}}. Such a morphism is denoted (α; β_{1}, …, β_{n}).

2. There is functor, called the assembly map

∘ : {y, x} ≀ {z, y} ⟶ {z, x},

sending the object (f; g_{1}, …, g_{n}) to the composition f(g_{1}, …, g_{n}) and the morphism (α; β_{1}, …, β_{n}) to the pasting of α and the n-tuple (η_{ξ(1)}, …, η_{ξ(n)}), which is denoted α(β_{1}, …, β_{n}) for simplicity, following Kelly, again.

Note that the details of the constructions stated in a direct fashion in Definitions A.1 and A.3 can be deduced from the combinatorics of the composition in the bi-multicategory ℭ. Similarly, the associativity for the composition given by the assembly map is also inherited from the ambient bi-multicategory, hence, for objects x, y, z, w of ℭ we have:

(A.2)  \xymatrix{(\{y, x\} ≀ \{z, y\}) ≀ \{w, z\} \ar[rr]|{ass.} \ar[d]_{∘ ≀id} &  & \{y, x\} ≀ (\{z, y\} ≀ \{w, z\}) \ar[d]^{id ≀∘} \\ \{z, x\} ≀ \{w, z\} \ar[r] & \{w, x\} \ar@{=>}[ul] & \{y, x\} ≀ \{w, y\} \ar[l]}

A.4. Definition. [Kelly [Kel72, Kel74]] A Club is a bi-multicategory with one object.

Given a club with unique object ∗, set k = {∗, ∗}. By abuse of notation we refer to k itself as a club. Then the assembly map and (A.2) give a monoidal structure ∘ : k ≀ k → k.

Examples of clubs are:

• S itself, with projection id: S → S.

• The natural numbers identified with a discrete subcategory N of S, with projection ι: N ↪ S.

In both cases k = S and k = N, the composition morphism k ≀ k → k sends the object (n; m_{1}, …, m_{n}) to m_{1} + ⋯ + m_{n}. In the case of N, however, we dispense with the symmetric group action.

A.5. Remark. [Notation] Kelly [Kel72] uses the notations ∘ (resp. μ) where we use ≀ (resp. ∘). The symbol ≀ usually denotes the so-called wreath product.

The wreath product F ≀ D (or simply C ≀ D) of two categories C and D, the first equipped with a functor F: C → Γ,^{15} is defined in a way somewhat similar to Definition A.3 (see e.g. [Ber07]). In particular, objects are tuples of the form (c; d_{1}, …, d_{n}), where F (c) has cardinality n and d_{1}, …, d_{n} are objects of D. Morphisms, however are slightly different.

B. (Unbiased) Monoids and monoidal structures

We keep the notations and environment of section A. ℭ is a bicategory identified with the underlying bicategory of a bi-multicategory Mℭ. We add another piece of notation: for an object x, let Mx = {x, x}. Having observed that this is a monoid for the composition, a monoid object in ℭ is defined in the expected way. Recall that N is a club, as discussed in the previous section. Recall that a lax monoidal functor is one that respects the monoidal structures up to coherent natural transformation.^{16}

B.1. Definition. A (pseudo-)monoid in ℭ is an object x of Mℭ equipped with a lax monoidal functor (t, θ): N → Mx. A symmetric (pseudo-)monoid would is the same thing with the club S, instead.

In the next diagrams we spell out the conditions for (t, θ) to be a lax monoidal functor. Explicitly, the lax monoidal functor is given by a diagram

(B.1)  [figure: N ≀ N; ∘; t≀t; θ; N; t; Mx ≀ Mx; ∘; Mx]

15Γ is the Segal category of finite sets.

16The distinction between lax and op-lax, i.e. the direction of the 2-arrows in the natural transformations is immaterial as we work with bicategories whose 2-arrows are isomorphisms.

From it we have the coordinate expression for the natural transformation θ, which assigns to the object (n; m_{1}, …, m_{n}) ∈ N ≀ N the morphism

(B.2)  θ_{n;m_{1},...,m_{n}}: t(m_{1} + ⋯ + m_{k}) ⟶ t(n)(t(m_{1}), …, t(m_{n}))

in Mx. These data are subject to be compatible with the associativity conditions for the assembly maps of both N and Mx, namely they must satisfy the following commutative diagram of natural transformations: (B.3)

[figure: (Mx ≀ Mx) ≀ Mx; ass; ∘ ≀Id; Mx ≀ (Mx ≀ Mx); Id ≀∘; (N ≀ N) ≀ N; θ≀t; (t≀t)≀t; ∘ ≀Id; ass; N ≀ (N ≀ N); t≀(t≀t); Id ≀∘; N ≀ N; ∘; θ; t≀t; N; t; N ≀ N; t≀t; ∘; Mx ≀ Mx; ∘; Mx; θ; Mx ≀ Mx; ∘; t≀θ]

In (B.3) the top quadrangle and the small pentagon are strictly commutative. For the quadrangle, it follows from the functoriality of the associator, whereas for the small pentagon the associativity morphism reads

((n; m_{1}, …, m_{n}); l_{1}, …, l_{k}) ⟼ (n; (m_{1}; l_{1}, …), …, (m_{n}; …, l_{k})),

where k = m_{1} + ⋯ + m_{n}; both paths evaluate to h = l_{1} + ⋯ + l_{k} = j_{1} + ⋯ + j_{n}, where for i = 1, …, n we have j_{i} = l_{m_{i−1}+1} + ⋯ + l_{m_{i−1}+m_{i}}. Lastly, the back face of the diagram, that is the large pentagon, is just (A.2) specialized to the same object. Thus, the commutativity in diagram (B.3), when expressed in coordinates, reads as follows: (B.4)

\xymatrix{t(k)(t(l_{1}), …, t(l_{k})) \ar[d] & t(h) \ar[r] \ar[l] & t(n)(t(j_{1}), …, t(j_{n})) \ar[d] \\ t(n)(t(m_{1}), …, t(m_{n}))(t(l_{1}), …, t(l_{k})) \ar[rr] &  & t(n)(t(m_{1})(t(l_{1}), …), …, t(m_{n})(…, t(l_{k}))}

B.2. Remark. By construction, the pseudo-monoid structure on x given by the club morphism t: N → Mx possesses n-ary operations t(n) for arbitrary values of n. If x has an internal structure (namely objects, as in the main text), then this implements a monoidal structure in “unbiased” form (see e.g. [Lei04]), with the n-ary unprivileged operations of all degrees. In this case diagram (B.3) expresses the coherence condition in unbiased form, namely, that the two possible ways to remove parentheses from an expression must coincide as it is apparent from the coordinate version (B.4).

B.3. Remark. Consider the singletons {x} as a copy of the singleton category 1. Any category with products gives rise to a multicategory in a standard way [Lei04, Example 2.16]. Applied to {x}, there is singleton worth of morphisms from x to x of arbitrary arity. The definition of the ≀-operation introduced in Appendix A can be copied in order to define a category N ≀ x (dropping the braces from the notation for convenience). One could show (for example, by formally checking the adjunction properties of ≀ and {−, −} as in[Kel72]) that (t, θ) in Definition B.1 correspond to a functor

t̃: N ≀ x ⟶ x

and a natural transformation

[figure: (N ≀ N) ≀ x; ∘ ≀x; θ̃; N ≀ (N ≀ x); N≀t̃; N ≀ x; t̃; x; N ≀ x; t̃]

By definition, (t̃, θ̃) equip x with a structure of N-(pseudo-)algebra. We will not work with this variant.

One can consider morphisms of monoids as follows. Denote N ≀ x (cf. Remark B.3) by Tx. Let f: y → x be a (unary) arrow. Then f determines a functor Tf: Ty → Tx and two functors

f_{∗}{y, y} ⟶ {y, x} Tf^{∗}{x, x} ⟶ {y, x}

by post-composing with f or pre-composing with Tf, respectively.

B.4. Definition. Let x, y be (pseudo-)monoids in ℭ. A morphism of monoids is a pair (f: y → x, λ) fitting the diagram

[figure: N; t_{y}; t_{x}; λ; {y, y}; f_{∗}; {x, x}; Tf^{∗}; {y, x}]

and compatible with the pseudo-algebra condition (B.1).

The compatibility between the above diagram and (B.1) means that in the coordinates of (B.2) we have

\xymatrix{f ∘ t_{y}(m_{1} + ⋯ + m_{k}) \ar[r]^{fθ_{y}} \ar[d]_{λ_{m_{1}+···+mn}} & f ∘ (t_{y}(n)(t_{y}(m_{1}), …, t_{y}(m_{n}))) \ar[r]^{1} & (f ∘ t_{y}(n))(t_{y}(m_{1}), …, t_{y}(m_{n})) \ar[d]^{λ_{n}∘Id} \\ t_{x}(m_{1} + ⋯ + m_{k}) ∘ Tf \ar[d]_{θ_{x}Tf} &  & (t_{x}(n) ∘ Tf)(t_{y}(m_{1}), …, t_{y}(m_{n})) \ar[d]^{2} \\ (t_{x}(n)(t_{x}(m_{1}), …, t_{x}(m_{n}))) ∘ Tf \ar[r]_{3} & t_{x}(n)((t_{x}(m_{1}), …, t_{x}(m_{n})) ∘ Tf) & t_{x}(n)(f ∘ t_{y}(m_{1}), …, f ∘ t_{y}(m_{n})) \ar[l]^{Id ∘λ_{m_{1},...,mn}}}

where the numbered arrows result from associativity isomorphism for composition.

Pseudo-monoids behave in the expected manner with respect to pseudo-functors. Specifically, let F: Mℭ → M𝔅 be a pseudo-functor, which by virtue of our identification, we think of as coming from an pseudo-functor F: ℭ → 𝔅 between bicategories. It is clear that for any objects x, y of ℭ, F induces a functor

F_{(y);x}: {y, x} ⟶ {F (y), F (x)}.

B.5. Definition. Let x be a pseudo-monoid in ℭ. Then F (x) acquires the structure of a pseudo-monoid in 𝔅 by virtue of the composition:

\xymatrix{N \ar[r]^{t} & Mx \ar[r]^{F_{•}} & MF (x),}

with the full pseudo-monoid structure for MF (x) results from the composition

[figure: N ≀ N; t≀t; ∘; Mx ≀ Mx; F_{•}≀F_{•}; ∘; MF (x) ≀ MF (x); ∘; N; t; Mx; F_{•}; θ; MF (x); ε]

C. Pentagons

Pentagon diagrams express the coherence condition in a monoidal category. This condition is replaced by a diagram of the form (B.3) or (B.4) if the monoidal structure is given in unbiased form [Lei04]. The actual pentagon can be recovered if these diagrams are specialized to arities equal to 4 and the monoidal category comes from a monoid inhabiting a 2-category. This shows the equivalence of the biased and unbiased definitions. A slight generalization of the pentagon occurs if the monoid inhabits a bicategory, and both these situations arise in the main text. In addition, conditions arising from pentagons are cocycle conditions in appropriate cohomology theories. It is useful to compute them once and for all in the general setting of a monoid object in a bicategory.

To begin with, observe that by applying (B.1) and (B.2) to the objects (2; 2, 1) and (2; 1, 2) ∈ N ≀ N we obtain morphisms θ_{2;2,1} : t(3) → t(2)(t(2), t(1)), θ_{2;1,2} : t(3) → t(2)(t(1), t(2)), and combining these two we obtain

(C.1)  μ: t(2)(t(2), t(1)) ⟶ t(2)(t(1), t(2)).

The pentagons (or the diagrams related to them) arise from the decomposition of the operation t(4), by way of (B.4), down to terms only involving the binary and unary operations t(2) and t(1), respectively. We can assume that the unary operation t(1): x → x coincides with the identity id_{x}. These decompositions can be encoded by working our way along the small pentagon in the diagram (B.3), so for instance one of them corresponds

BIEXTENSIONS, CATEGORICAL RINGS  957

to the sequences

\xymatrix{((2; 2, 1); 2, 1, 1) \ar[rr] \ar[d] &  & (2; (2; 2, 1), 1) \ar[d] \\ (3; 2, 1, 1) \ar[r] & 4 & (2; 3, 1) \ar[l]}

(We have simply written 1 in place of the more accurate but cumbersome expression (1; 1).)

There are six distinct sequences including the one above. Their starting points, counted from the upper left corner of the small pentagon in the diagram (B.3), are:

1. ((2; 2, 1); 2, 1, 1),

2. ((2; 2, 1); 1, 2, 1),

3. ((2; 1, 2); 1, 2, 1),

4. ((2; 1, 2); 1, 1, 2),

5. ((2; 1, 2); 2, 1, 1),

6. ((2; 2, 1); 1, 1, 2). Let us also use the notations m_{i} = t(i), i = 1, …, 4, with the special provision m_{1} =

958  ETTORE ALDROVANDI

id_{x} = id. With these, applying (B.4) and (B.3), we obtain the diagram on page 958: (C.2)

\xymatrix{ &  & m_{2}(id(m_{2}), m_{2}(id, id)) \ar[rr] &  & m_{2}(id, m_{2})(m_{2}, id, id) \\  & m_{2}(m_{2}(id, id), id(m2)) \ar[ddl] &  & 5 &  & m_{2}(m_{2}, id)(m_{2}, id, id) \ar[ul]_{μ} \\  &  & m_{2}(m_{2}, m_{2}) \ar[uu] \ar[ul] \ar[uurr] \ar[dll] &  & m_{3}(m_{2}, id, id) \ar[ll] \ar[uu] \ar[ur] \\ m_{2}(m_{2}, id)(id, id, m_{2}) \ar[dd]_{μ} & 6 &  &  &  & 1 & m_{2}(m_{2}(m_{2}, id), id) \ar[uul] \ar[dd]^{μ} \\  & m_{3}(id, id, m_{2}) \ar[dl] \ar[ul] \ar[uur] &  & m_{4} \ar[uul] \ar[uur] \ar[rr] \ar[ll] \ar[ddl] \ar[ddr] &  & m_{2}(m_{3}, id) \ar[ur] \ar[dr] \\ m_{2}(id, m_{2})(id, id, m_{2}) & 4 &  &  &  & 2 & m_{2}(m_{2}(id, m_{2}), id) \ar[ddl] \\  &  & m_{2}(id, m_{3}) \ar[dd] \ar[dl] &  & m_{3}(id, m_{2}, id) \ar[dd] \ar[dr] \\  & m_{2}(id, m_{2}(id, m_{2})) \ar[uul] &  & 3 &  & m_{2}(m_{2}, id)(id, m_{2}, id) \ar[dl]^{μ} \\  &  & m_{2}(id, m_{2}(m_{2}, id)) \ar[rr] \ar[ul]^{μ} &  & m_{2}(id, m_{2})(id, m_{2}, id)}

The six petals correspond to the indicated sequences and are numbered accordingly. We have marked the (magenta) arrows resulting from the morphism (C.1). Note that there are five of them. In particular:

• If Mℭ is unital, the petals 5 and 6 collapse to the red arrows.

• If furthermore Mℭ is in fact a 2-multicategory, i.e. the 2-categorical analog of a multicategory, so that ℭ itself is a genuine 2-category, the associativity morphisms at the tops of all petals reduce to identities. As a result, the outer perimeter reduces to a standard pentagon

(C.3)

\xymatrix{ & m_{2}(m_{2}, m_{2}) \ar[dddddl]_{μ} \\  &  &  &  &  & m_{2}(m_{2}(m_{2}, id), id) \ar[dd]^{μ} \ar[ullll]_{μ} \\  &  & m_{4} \ar[uul] \ar[rr] \ar[ddl] \ar[ddr] &  & m_{2}(m_{3}, id) \ar[ur] \ar[dr] \\  &  &  &  &  & m_{2}(m_{2}(id, m_{2}), id) \ar[dddllll]^{μ} \\  & m_{2}(id, m_{3}) \ar[dd] \ar[dl] &  & m_{3}(id, m_{2}, id) \ar[ddll] \ar[urr] \\ m_{2}(id, m_{2}(id, m_{2})) \\  & m_{2}(id, m_{2}(m_{2}, id)) \ar[ul]^{μ}}

D. Some lemmas on symmetric braidings

D.1. Lemma. Let (C, ⊗, c) be a strictly associative braided monoidal category. For every x, y, z, w, the diagram

\xymatrix{x ⊗ y ⊗ z ⊗ w \ar[r]^{1⊗c_{y,z}⊗1} \ar[d]_{c_{x,y}⊗c_{z,w}} & x ⊗ z ⊗ y ⊗ w \ar[d]^{c_{x⊗z,y⊗w}} \\ y ⊗ x ⊗ w ⊗ z \ar[r]_{1⊗c_{x,w}⊗1} & y ⊗ w ⊗ x ⊗ z}

commutes if and only if the braiding c is symmetric.

Proof. The naturally occurring diagram relating c_{x⊗z,y⊗w} to c_{x,y} ⊗ c_{z,w} is

[figure: x ⊗ y ⊗ z ⊗ w; 1⊗c_{z,y}⊗1; c_{x,y}⊗c_{z,w}; x ⊗ z ⊗ y ⊗ w; c_{x⊗z,y⊗w}; y ⊗ x ⊗ w ⊗ z; 1⊗c_{x,w}⊗1; y ⊗ w ⊗ x ⊗ z]

which is valid with respect to any braiding. This is equal to the diagram in the statement if and only if c_{y,z} is the inverse of c_{z,y}.

D.2. Lemma. Let (C, ⊗, c) be a strictly associative braided monoidal category. If the braiding c is symmetric, the following diagram

\xymatrix{x ⊗ y ⊗ z ⊗ u ⊗ v ⊗ w \ar[r]^{1⊗c_{yz,uv}⊗1} \ar[d]_{c_{x,y}⊗1⊗1⊗c_{v,w}} & x ⊗ u ⊗ v ⊗ y ⊗ z ⊗ w \ar[d]^{c_{x.u}⊗1⊗1⊗c_{z,w}} \\ y ⊗ x ⊗ z ⊗ u ⊗ w ⊗ v \ar[d]_{1⊗c_{xz,uw}⊗1} & u ⊗ x ⊗ v ⊗ y ⊗ w ⊗ z \ar[d]^{1⊗c_{xv,yw}⊗1} \\ y ⊗ u ⊗ w ⊗ x ⊗ z ⊗ v \ar[r]_{c_{y.u}⊗1⊗1⊗c_{z,v}} & u ⊗ y ⊗ w ⊗ x ⊗ v ⊗ z}

commutes.

Proof. Use Lemma D.1 to replace c_{yz,uv} with c_{y,u} ⊗c_{z,v} and similarly for c_{xz,uw} and c_{xv,yw}. This converts the diagram in the statement into the following one

\xymatrix{x ⊗ y ⊗ u ⊗ z ⊗ v ⊗ w \ar[r]^{1⊗c_{y,u}⊗c_{z,v}⊗1} \ar[d]_{c_{x,y}⊗1⊗1⊗c_{v,w}} & x ⊗ u ⊗ y ⊗ v ⊗ z ⊗ w \ar[d]^{c_{x.u}⊗1⊗1⊗c_{z,w}} \\ y ⊗ x ⊗ u ⊗ z ⊗ w ⊗ v \ar[d]_{1⊗c_{x,u}⊗c_{z,w}⊗1} & u ⊗ x ⊗ y ⊗ v ⊗ w ⊗ z \ar[d]^{1⊗c_{x,y}⊗c_{v,w}⊗1} \\ y ⊗ u ⊗ x ⊗ w ⊗ z ⊗ v \ar[r]_{c_{y.u}⊗1⊗1⊗c_{z,v}} & u ⊗ y ⊗ x ⊗ w ⊗ v ⊗ z}

which consists of two juxtaposed copies of the same kind of hexagon:

\xymatrix{x ⊗ y ⊗ u \ar[r]^{1⊗c_{y,u}} \ar[d]_{c_{x,y}⊗1} & x ⊗ u ⊗ y \ar[d]^{c_{x.u}⊗1} \\ y ⊗ x ⊗ u \ar[d]_{1⊗c_{x,u}} & u ⊗ x ⊗ y \ar[d]^{1⊗c_{x,y}} \\ y ⊗ u ⊗ x \ar[r]_{c_{y.u}⊗1} & u ⊗ y ⊗ x}

The latter commutes in any braided monoidal (strictly associative) category.

A permutative category is a symmetric monoidal category (C, ⊗, c) in which associativity and unitality hold strictly. Thus the diagrams in Lemmas D.1 and D.2 commute if and only if the category (C, ⊗, c) is permutative. On the other hand, the statements of Lemma D.1 and Lemma D.2 hold in general. Requiring that C be strictly associative merely simplifies the diagrams and overall legibility. For example, in the general case the diagram in the statement of Lemma D.1 would need replacing with:

\xymatrix{x ⊗ (z ⊗ (y ⊗ w)) & (x ⊗ z) ⊗ (y ⊗ w) \ar[l] \ar[r]^{c_{x⊗z,y⊗w}} & (y ⊗ w) ⊗ (x ⊗ z) \ar[r] & y ⊗ (w ⊗ (x ⊗ z)) \\ x ⊗ ((z ⊗ y) ⊗ w) \ar[u] &  &  & y ⊗ ((w ⊗ x) ⊗ z) \ar[u] \\ x ⊗ ((y ⊗ z) ⊗ w) \ar[u]^{1⊗c_{y,z}⊗1} \ar[d] &  &  & y ⊗ ((x ⊗ w) ⊗ z) \ar[u]_{1⊗c_{x,w}⊗1} \ar[d] \\ x ⊗ (y ⊗ (z ⊗ w)) & (x ⊗ y) ⊗ (z ⊗ w) \ar[l] \ar[r]_{c_{x,y}⊗c_{z,w}} & (y ⊗ x) ⊗ (w ⊗ z) \ar[r] & y ⊗ (x ⊗ (w ⊗ z))}

with a similarly modified proof. A similar, but more complicated modifications apply to the diagram in the proof of Lemma D.2.

E. Hypercohomology computations

In order to carry out a full, explicit cohomology computation for the class of a ring-like stack ℛ, it is necessary to resolve various simplicial objects related to the ring A, notably the Eilenberg-Mac Lane’s K(A, n) for n = 2, 3, by way of certain bisimplicial objects.

In the main text, and specifically in sect. 15, we simply referred to the spectral sequence for the hypercohomology of the simplicial object K(A, 1)^{17} and focus on its E^{0,3}-term, showing that it is isomorphic to HML^{3}(A, M) proper. While this is appropriate to show the agreement with Mac Lane cohomology, it is important to provide the main outline of the full calculation.

17As done in ref. [Bre99, §6], discussing the passage from a monoidal category to a monoidal stack

It is easier to illustrate the full computation in the (known) cases of H^{3}(K(A, 1), M), or H^{4}(K(A, 2), M) first, that is, by just considering the underlying structure of braided (symmetric) stack of ℛ. The full case just calls for the addition of more data following the same pattern.

As it is well known, ℛ is only a gerbe over A (see e.g. [LMB00]), and the decomposition of ϖ: ℛ → A requires choosing local objects. Thus we need to choose covers ξ_{n}: U_{n,0} → A^{n}, which we complete to an hypercover of K(A, 1). Following ref. [Fri82], this means an augmented bisimplicial object

U_{•,•} ⟶ K(A, 1)_{•},

such that each U_{n,•} → A^{n} is a hypercover. (In fact, and it is convenient to do so, U can be arranged so that its horizontal zero level U_{•,0} → K(A, 1) is an acyclic fibration: this ensures it becomes finer in the horizontal direction so as to guarantee the existence of local choices.) We denote by d_{i,h} (resp. d_{i,v}) the face maps of U in the horizontal (resp. vertical) direction, where the “horizontal direction” is the direction of K(A, 1).

Let us use the conventions:

1. a = ξ_{1} : U_{1,0} → A,

2. (a, b) = ξ_{2} : U_{2,0} → A × A,

3. (a, b, c) = ξ_{3} : U_{3,0} → A × A × A,

etcetera, where of course the meaning of the letters a, b, c, … depends on the level n and it is determined, in effect, by the simplicial face maps. Thus, at level n = 2 we have

a = d_{2}ξ_{2} = ξ_{1}d_{h,2}
b = d_{0}ξ_{2} = ξ_{1}d_{h,0}
a + b = d_{1}ξ_{2} = ξ_{1}d_{h,1},

whereas at level n = 3 we have, with the same conventions:

a = d_{2}d_{3}ξ_{3} = ξ_{1}d_{h,2}d_{h,3}
b = d_{0}d_{3}ξ_{3} = ξ_{1}d_{h,0}d_{h,3}
c = d_{0}d_{1}ξ_{3} = ξ_{1}d_{h,0}d_{h,1}
a + b = d_{1}d_{3}ξ_{3} = ξ_{1}d_{h,1}d_{h,3}
b + c = d_{0}d_{2}ξ_{3} = ξ_{1}d_{h,0}d_{h,2}
a + b + c = d_{1}d_{2}ξ_{3} = ξ_{1}d_{h,1}d_{h,2}.

The choice of an object X of the pullback ℛ_{a} = ξ_{1}^{∗}ℛ determines the class of ℛ, as a gerbe over A with band M, in H^{2}(A, M). This class is obtained by computing it in the standard way along the hypercover U_{1,•} → A. As it is well known, the class itself is represented by an object of α ∈ M(U_{1,2}), which is simply the defect in the commutativity of the triangle formed by the three possible pullbacks to U_{1,2} of the isomorphism φ: d_{1,v}^{∗}X →^{∼} d_{0,v}^{∗}X defined over U_{1,1} (see [Bre94b, Bre94a]). As a result, α is a Čech cocycle relative to the hypercover U_{1,•}:

(E.1)  d_{0,v}^{∗}α + d_{1,v}^{∗}α − d_{2,v}^{∗}α + d_{3,v}^{∗}α = 0 .

In the horizontal direction, by successively pulling back X along the horizontal face maps at vertical level n = 0 we obtain objects X_{a}, X_{a+b}, … (recall the convention above) providing decompositions for the various pullbacks d_{i,h}^{∗} … d_{j,h}^{∗}ℛ over A^{n}.

At level n = 2, over U_{2,0}, we get an isomorphism

σ_{a,b}: X_{a} + X_{b} ⟶^{∼} X_{a+b},

which is only compatible with the various isomorphisms φ_{a}: d_{1,v}^{∗}X_{a} →^{∼} d_{0,v}^{∗}X_{a} up to an automorphism, say δ ∈ M(U_{2,1}), of d_{0,v}^{∗}X_{a+b}. A calculation of the pullbacks to U_{2,2,} yields

(E.2)  d_{0,v}^{∗}δ − d_{1,v}^{∗}δ + d_{2,v}^{∗}δ = −d_{0,h}^{∗}α + d_{1,h}^{∗}α − d_{2,h}^{∗}α .

Observe that modulo α, the element δ gives the class in H^{1}(A × A, M) of the sheaf Hom(X_{a} + X_{b}, X_{a+b}).

Also, still over U_{2,0}, the braiding of ℛ gives the diagram

\xymatrix{X_{a} + X_{b} \ar[d]_{σ_{a,b}} \ar[r] & X_{b} + X_{a} \ar[d]^{σ_{b,a}} \\ X_{a+b} \ar[r]_{g_{+}(a,b)} & X_{a+b}}

defining g_{+}(a, b) ∈ M(U_{2,0}). If the braiding is in addition symmetric, then the previous diagram yields

g_{+}(a, b) + g_{+}(b, a) = 0,

namely g_{+} is antisymmetric under the map determined by the pullback of U_{2,•} by the swap map of A^{2}. If we denote this operation by τ, then the above relation would be written g_{+} + τ^{∗}g_{+} = 0

Similarly, an analysis of the associativity of the monoidal structure of ℛ yields f_{+}(a, b, c) ∈ M(U_{3,0}), as an automorphism of the object X_{a+b+c} of d_{2,h}^{∗}d_{1,h}^{∗}ℛ over A^{3}. Under pullback of both f_{+} and g_{+} to U_{3,1} we obtain:

(E.3)  \begin{aligned}d_{0,v}^{∗}f_{+} − d_{1,v}^{∗}f_{+} & = d_{0,h}^{∗}δ − d_{1,h}^{∗}δ + d_{2,h}^{∗}δ − d_{3,h}^{∗}δ \\ d_{0,v}^{∗}g_{+} − d_{1,v}^{∗}g_{+} & = −δ + τ^{∗}δ. \end{aligned}

Finally, the familiar relations describing a (symmetric) braiding arise by analyzing the behavior of f_{+} when pulled back to U_{4,0} and that of g_{+} when pulled back to U_{3,0}, yielding the standard cocycle relation

d_{0,h}^{∗}f_{+} − d_{1,h}^{∗}f_{+} + d_{2,h}^{∗}f_{+} − d_{3,h}^{∗}f_{+} + d_{4,h}^{∗}f_{+} = 0,

plus the two equations arising from Mac Lane’s hexagonal diagrams (and the antisymmetry condition above, if ℛ is symmetric).

Again, observe that modulo δ, f_{+} and g_{+} define objects in H^{0}(A^{4}, M) and H^{0}(A^{3}, M) satisfying the standard cocycle relations for a class in H^{4}(K(A, 2), M) (or H^{5}(K(A, 3), M)). This is the relevant part in the discussion of the class arising from a braided (or symmetric) monoidal stack as above.

In summary, the quadruplet (f_{+}, g_{+}, δ, α) defines a cocycle of degree 3 with respect to the total complex defined as follows. Let B(A, k) be the iterated bar construction on A, where k = 2, 3. If k = 1, then B(A, 1) is simply the complex Z[K(A, 1)]^{∼}.^{18} With a mild abuse of language, for each (vertical) level n let B(U_{•,n}, k) denote the complex obtained in a way analogous to B(A, k) from the abelian sheaves Z[U_{•,n}]. Thus we get an augmented simplicial object

\xymatrix{⋯ \ar@{->>}[r] \ar[r] & B(U_{•,1}, k) \ar@{->>}[r] & B(U_{•,0}, k) \ar[r]^{ξ} & B(A, k).}

Then we form (neglecting B(A, k)) the total complex with respect to the vertical direction to obtain the necessary complex.

A glance at the complex used in the main text, section 15, namely the bar complex built on top of a DGA structure on the complexes L^{2}(A) (or L^{3}(A)), reveals that the same construction with the hypercover

U_{•,•} → K(A, 1)

would also work in this case, with the difference that in total degree 3 we have the quintuplet used in the paper, namely (f, α_{1}, α_{2}, f_{+}, g_{+}) plus the automorphisms α and δ as above arising from the decomposition of ℛ over A and A × A, supplemented by an additional one, ε ∈ M(U_{2,1}), arising from the descent condition on the pullback of the various biextensions E_{a,b} and their higher analogs.

To accommodate for these changes, we must form simplicial objects corresponding to the bar complexes B̄_{k,•}(A), k = 2, 3, in section 15.3. That is, for each complex B(U_{•,n}, k) above we apply the (normalized) bar construction B̄_{k}(U_{•,n}) := B̄(B(U_{•,n}, k), η) to form the augmented simplicial object

(E.4)  \xymatrix{⋯ \ar@{->>}[r] \ar[r] & B̄_{k}(U_{•,1}) \ar@{->>}[r] & B̄_{k}(U_{•,0}) \ar[r]^{ξ} & B̄_{k,•}(A) ,}

for k = 2, 3. In addition to our notational conventions, we must consider appropriate ones for the multiplicative structures: we consider products ab = (d_{2}ξ_{2})(d_{0}ξ_{2}), abc = (d_{2}d_{3}ξ_{3})(d_{0}d_{3}ξ_{3})(d_{0}d_{1}ξ_{3}), etc. and use the same notation for the corresponding maps at any level in the vertical direction.

To explain the required steps to obtain the full class in some more detail, consider the biexact bifunctors

m_{a,b}: ℛ_{a} × ℛ_{b} ⟶ ℛ_{ab}

18A down-shifted version is the complex denoted L^{2}(A) or L^{3}(A) in the paper.

defined over U_{2,0} (again, recall the above convention about naming the components of ξ_{1}, ξ_{2}, …) corresponding to the biextension E_{a,b} introduced in sect. 15.11, and their higher arity analogs. In terms of local objects, we have an isomorphism

m_{a,b}: X_{a}X_{b} ⟶^{∼} X_{ab} ,

which we can safely indicate with the same name, between objects of ℛ_{ab}. Analogously to the case of the sum operation, m_{a,b} is only compatible with the (horizontal) pullbacks of φ: d_{1,v}^{∗}X →^{∼} d_{0,v}^{∗}X up to an automorphism ε ∈ M(U_{2,1}). For added clarity, let us specify the labels as ε(a, b). Comparing the three possible pullbacks to U_{2,2} yields the relation

(E.5)  d_{0,v}^{∗}ε(a, b) − d_{1,v}^{∗}ε(a, b) + d_{2,v}^{∗}ε(a, b) = −aα(b) + α(ab) − α(a)b ,

where a notation like α(ab) means the pullback of α ∈ M(U_{1,2}) along the analog B̄_{k}(U_{2,2}) → B̄_{k}(U_{1,2}) of the map ab, as mentioned above.

The remaining three relations are found by analyzing the behavior of the diagrams (15.14), (15.8) which define the quantities f(a, b, c), α_{1}(a, b; c), and α_{2}(a; b, c), under pullback from U_{3,0} to U_{3,1}. They can be readily computed by interpreting f(a, b, c), α_{1}(a, b; c), and α_{2}(a; b, c) as 2-arrows in the diagrams

[figure: ℛ_{a} × ℛ_{b} × ℛ_{c}; m_{a,b}×1; 1×m_{b,c}; ℛ_{ab} × ℛ_{c}; m_{ab,c}; ℛ_{a} × ℛ_{bc}; m_{a,bc}; ℛ_{abc}; f(a,b,c)]

and

[figure: ℛ_{a} × ℛ_{b} × ℛ_{c}; m_{a,c}×m_{b,c}; σ_{a,b}×1; ℛ_{ac} × ℛ_{bc}; m_{ac,bc}; ℛ_{a+b} × ℛ_{c}; m_{a+b,c}; ℛ_{(a+b)c}; −α_{1}(a,b;c)] , [figure: ℛ_{a} × ℛ_{b} × ℛ_{c}; m_{a,b}×m_{a,c}; 1×σ_{b,c}; ℛ_{ab} × ℛ_{ac}; m_{ab,ac}; ℛ_{a} × ℛ_{b+c}; m_{a,b+c}; ℛ_{a(b+c)}; α_{2}(a;b,c)] ,

and then comparing, for each diagram, the two possible pullbacks to U_{3,1}. We obtain the following relations: (E.6)

\begin{aligned}d_{0,v}^{∗}f(a, b, c) − d_{1,v}^{∗}f(a, b, c) & = −aε(b, c) + ε(ab, c) − ε(a, bc) + ε(a, b)c \\ d_{0,v}^{∗}α_{1}(a, b; c) − d_{1,v}^{∗}α_{1}(a, b; c) & = −δ(ac, bc) + δ(a, b)c − ε(a, c) + ε(a + b, c) − ε(b, c) \\ d_{0,v}^{∗}α_{2}(a, b; c) − d_{1,v}^{∗}α_{2}(a, b; c) & = δ(ab, ac) − aδ(b, c) + ε(a, b) − ε(a, b + c) + ε(a, c) . \end{aligned}

In summary, the relations (E.1), (E.2) and (E.5), and (E.3) and (E.6), together with the ones found in the main text, sect. 15.13, over U_{4,0}, show the entire collection

((f, α_{1}, α_{2}, f_{+}, g_{+}); (δ, −ε), α)

forms a cocycle of total degree 3 in the double complex obtained from (E.4) for k = 3 (minus the term in degree −1, of course). In particular, the relations (E.3) and (E.6) feature the horizontal differential of (δ, −ε), which has horizontal degree 2, according to sect. 15.4. Similarly, the relations (E.2), (E.5) feature the horizontal differential of α, with horizontal degree 1.

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Department of Mathematics, Florida State University 1017 Academic Way, Tallahassee, FL 32312-4510, USA

Email: aldrovandi@math.fsu.edu
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