Extending Eilenberg–Mac Lane’s cohomology of abelian groups, a cohomology theory is introduced for commutative monoids. The cohomology groups in this theory agree with the pre-existing ones by Grillet in low dimensions, but they differ beyond dimension two. A natural interpretation is given for the three-cohomology classes in terms of braided monoidal groupoids.
The lower Leech cohomology groups of monoids , denoted here by , have been proven useful for the classification of interesting monoidal structures. Thus, abelian-group co-extensions of monoids are classified by means of Leech two-cohomology classes  (§2.4.9), whereas Leech three-cohomology classes classify monoidal abelian groupoids  (Theorem 4.2), that is (Brandt) groupoids , whose vertex groups are all abelian, endowed with a monoidal structure by a tensor functor , a unit object I and coherent associativity and unit constraints , and and [3,4].
On commutative monoids, nevertheless, Leech cohomology groups do not properly take into account their commutativity, in contrast to what happens with Grillet’s symmetric cohomology groups [5,6,7,8], which we denote by . For instance, symmetric two-cohomology classes classify abelian-group commutative co-extensions of commutative monoids  (§V.4), whereas symmetric three-cohomology classes classify strictly symmetric monoidal abelian groupoids  (Theorem 3.1), that is monoidal abelian groupoids , as above, but now endowed with coherent symmetry constraints , satisfying and [3,4,10,11].
To some extent, however, Grillet’s symmetric cohomology theory at degrees greater than two seems to be a little too “strict” (for example, when is any abelian group, its symmetric three-cohomology groups are all zero). Therefore, in this paper, we present a different approach for a cohomology theory of commutative monoids, which is inspired in the (first-level) cohomology of abelian groups by Eilenberg and Mac Lane [12,13,14,15] and based on the cohomology theory of simplicial sets by Gabriel and Zisman  (Appendix II).
In the same manner that every monoid M, regarded as a constant simplicial monoid, has associated a classifying simplicial set  satisfying that, for any Leech system of coefficients on M,  (§4.1.1), when the monoid M is commutative, it also has associated an iterated classifying simplicial set . Gabriel–Zisman’s cohomology groups of this simplicial set are used to define, for any Grillet system of coefficients on M (or, equivalently, any abelian group object in the comma category of commutative monoids over M), the commutative cohomology groups of M, denoted , by
For instance, when is an abelian group, as the simplicial set is an Eilenberg–Mac Lane minimal complex , for any abelian group A (regarded as a constant coefficient system on G), the commutative cohomology groups are precisely the Eilenberg–Mac Lane cohomology groups of the abelian group G with coefficients in A [12,13,14,15] (also denoted by in [18,19]).
In this paper, we are mainly interested in the lower cohomology groups , for . Hence, in Section 2, most of our work is dedicated to showing how these commutative cohomology groups can be defined “concretely” by manageable and computable commutative cocycles, such as Grillet did for the cohomology groups by using symmetric cocycles. Thus, for any Grillet system of coefficients on a commutative monoid M, we exhibit a four-truncated complex of commutative cochains , such that
whose construction is based on the construction of the reduced complexes by Eilenberg and Mac Lane  to compute the (co)homology groups of the spaces . Furthermore, the existence of a cochain complex monomorphism , where the first is Grillet’s four-truncated complex of symmetric cochains, easily allows one to state the relationships among the symmetric, commutative and Leech low-dimensional cohomology groups of commutative monoids (see Theorem 3.5):
where, in general, the inclusions and are strict, whereas the homomorphism is neither injective nor surjective.
For , because of the the isomorphisms , there is nothing new to say about how to interpret these latter ones: elements of are derivations, and elements of are iso-classes of (abelian-group) commutative monoid co-extensions.
Then, in Section 4 of the paper, we focus our attention on the commutative cohomology groups , to whose elements we give a natural interpretation in terms of equivalence classes of braided monoidal abelian groupoids , that is monoidal abelian groupoids endowed with coherent and natural isomorphisms (the braidings) , defined as for strictly-symmetric abelian monoids, but now not necessarily satisfying the symmetry condition nor the strictness condition . The result, which was in fact our main motivation to seek the cohomology theory we present, can be summarized as follows (see Theorem 4.5 for details): stating that two triples and , where and , are isomorphic whenever there are isomorphisms and , such that , then
“There is a one-to-one correspondence between equivalence classes of braided monoidal abelian groupoids and iso-classes of triples , with .”
This classification theorem, which extends that given by Joyal and Street in  (§3) for braided categorical groups, leads to bijections
expressing a natural interpretation of commutative three-cohomology classes as equivalence classes of certain commutative two-dimensional co-extensions of M by .
2. Preliminaries on Cohomology of Monoids and Simplicial Sets
This section aims to make this paper as self-contained as possible; hence, at the same time as fixing notations and terminology, we also review some necessary aspects and results about the cohomology of monoids and simplicial sets used throughout the paper. However, the material in this preliminary section is perfectly standard by now, so the expert reader may skip most of it.
2.1. Grillet Cohomology of Commutative Monoids: Symmetric Cocycles
The category of commutative monoids is monadic (or tripleable) over the category of sets , and so, it is natural to specialize Barr–Beck cotriple cohomology  to define a cohomology theory for commutative monoids. This was done in the 1990s by Grillet, to whose papers [5,6,7] and book  (Chapters XII, XIII, XIV) we refer the reader interested in a detailed study of these symmetric cohomology groups for commutative monoids M, which we denote here by . For the needs of this paper, it suffices to point out the following basic facts about how to compute them.
For any given commutative monoid M, the coefficients for its cohomology, that is the abelian group objects in the comma category of commutative monoids over M, are provided by abelian group valued functors on the Leech category . This is the category with object set M and arrow set , where ; the composition is given by , and the identity of an object a is . An abelian group valued functor, , thus consists of abelian groups , and homomorphisms , , such that, for any , , and for any , . To compute the lower cohomology groups , there is a truncated cochain complex
called the complex of (normalized on ) symmetric cochains on M with values in , which is defined as follows:
A symmetric one-cochain, , is a function with , such that .
A symmetric two-cochain, , is a function , with , such that
A symmetric three-cochain, , is a function with , such that
A symmetric four-cochain, , is a function with , such that
Under pointwise addition, these symmetric n-cochains constitute the abelian groups , . The coboundary homomorphisms are defined by
are respectively called the groups of symmetric n-cocycles and symmetric n-coboundaries on M with values in . By  (Theorems 1.3 and 2.11), there are natural isomorphisms
2.2. Cohomology of Categories and Simplicial Sets: Leech Cohomology of Monoids
If is any small category, the category of abelian group valued functors is abelian, and it has enough injective and projective objects. There is a “global sections” functor given by
where we write and for . Then, we can form the right derived functors of . These are the cohomology groups of the category with coefficients in ,
Example 2.1 (Leech cohomology of monoids). Any monoid M gives rise to a category , whose set of objects is M and set of arrows , with . Composition is given by , and the identity morphism of any object a is . If we say that an abelian group valued functor carries the morphism to the group homomorphism , then we see that such a functor is a system of data consisting of abelian groups , , and homomorphisms , , such that, for any ,
and for any , . Leech cohomology groups of a monoid M with coefficients in an abelian group valued functor , denoted here by , are defined to be those of its associated category , that is,
Let us remark that the category of monoids is monadic over the category of sets. In , Wells proves that, for any monoid M, a functor can be identified with an abelian group object in the comma category of monoids over M and that, with a dimension shift, both the Barr–Beck cotriple cohomology theory  and the Leech cohomology theory of monoids are the same.
The cohomology theory of small categories is in itself a basis for other cohomology theories, in particular for the cohomology theory of simplicial sets with twisted coefficients defined by Gabriel and Zisman in . Briefly, recall that the simplicial category, Δ, consists of the finite ordered sets , , with weakly order-preserving maps between them, and that the category of simplicial sets is the category of functors , where is the category of sets, with morphisms the natural transformations. The category Δ is generated by the injections (cofaces), which omit the i-th element, and the surjections (codegeneracies), which repeat the i-th element, , subject to the well-known cosimplicial identities: if , etc. (see ). Hence, in order to define a simplicial set, it suffices to give the sets of its n-simplices together with maps
satisfying the well-known basic simplicial identities: if , etc. The category of simplices of a simplicial set X, , has as objects the pairs with , and a morphism consists of a map in Δ together with a simplex . A coefficient system on X is a functor , and the cohomology groups of the simplicial set X with coefficients in are, by definition,
We point out below two useful facts. The first of them is an easy consequence of being the maps , and the cosimplicial identities a set of generators and relations for Δ, and the second one is the dual of Theorem 4.2 in  (Appendix II) and takes into account the normalization theorem.
Fact 2.2. Let X be a simplicial set. In order to define a functor , it suffices to give objects , , , together with morphisms:
satisfying the equations
If is any coefficient system on a simplicial set X, then, for any simplex , we denote by the abelian group and by the homomorphism associated with any morphism in .
Fact 2.3. Let be a coefficient system on a simplicial set X. A n-cochain of X with coefficients in is a map , such that for each . Thus, is the abelian group of such n-cochains. As varies, these define a cosimplicial abelian group , , whose cosimplicial operators
, are respectively given by the formulas
denotes its associated normalized cochain complex, where
is the abelian group of normalized n-cochains, with coboundary ; there is a natural isomorphism
Many cohomology theories for algebraic systems find fundament in the cohomology of simplicial sets; in particular, Leech cohomology theory for monoids, as we explain below. Previously, recall that a simplicial monoid is a contravariant functor from the simplicial category to the category of monoids, . Thus, each is a monoid and the face and degeneracy operators in (2) are homomorphisms. Every simplicial monoid X has associated a classifying simplicial set
which is defined as follows (this is in ): , the unitary set, and
Write the elements of in the form . The face and degeneracy maps are defined by , by , and for by
For example, given any monoid M, let denote the constant M simplicial monoid, that is the simplicial monoid given by , , and by letting each and on be the identity map on M. Then, the -construction on it produces the so-called classifying simplicial set of the monoid
whose face and degeneracy maps are given by the familiar formulas
There is a functor , such that , and
Then, by composition with π, any functor defines a coefficient system on , also denoted by , and therefore, the cohomology groups are defined. By Fact 2.3, these cohomology groups can be computed from the cochain complex , which is given in degree by
and . The coboundary is given, for , by , while, for ,
As Leech proved in  (Chapter II, 2.3, 2-9) that the cohomology groups can be just computed as those of this cochain complex , it follows that there are natural isomorphisms
3. A Cohomology Theory for Commutative Monoids
Let us return now to the case where M is a commutative monoid. Under this hypothesis, the simplicial set in (4) is again a simplicial monoid, with the product monoid structure on each . We can then perform the -construction (3) on it, which gives the simplicial set (actually, a commutative simplicial monoid)
whose set of n-simplices is
Writing an -simplex x of in the form
where each is a k-simplex of , its faces and degeneracies are respectively defined by and , where
Recall now, from Subsection 2.1, that abelian group valued functors on the Leech category provide the coefficients for Grillet’s cohomology groups of a commutative monoid M. There is a functor , which, taking into account Fact 2.2, is determined by , for each -simplex of as in (5), where the product is in the monoid M over all , together with the homomorphisms
Therefore, by composition with π, any functor gives rise to a coefficient system on the simplicial set , equally denoted by
whence the cohomology groups of with coefficients in are defined. Note that these cohomology groups are trivial at dimensions zero and one. Then, making a dimensional shift, we state the following definition.
Definition 3.1. Let M be a commutative monoid. For any abelian group valued functor , the commutative cohomology groups of M with coefficients in , denoted , are defined by
Example 3.2. Let be an abelian group. Then, the simplicial set is an Eilenberg–Mac Lane minimal complex [17,24] (Theorem 17.4),  (Theorem 23.2). For any abelian group A, regarded as a constant functor , the commutative cohomology groups define the first level or abelian Eilenberg–Mac Lane cohomology theory of the abelian group G [12,13,14,15,17] (these are denoted also by in [18,19] and by in ). Although these cohomology groups arise from algebraic topology, they come with algebraic interest. Briefly, recall that there are natural isomorphisms  (26.1), (26.3), (26.4))
where is the group of homomorphisms from G to A, is the group of abelian group extensions of G by A and is the abelian group of quadratic maps from G to A, that is functions , such that is a bilinear function of . A precise classification theorem for braided categorical groups  (Definition 3.1)in terms of cohomology classes was proven by Joyal and Street in  (Theorem 3.3) (see Corollary 4.6 for an approach here to that issue).
Let us stress that, among the groups in the category of abelian groups, only and are relevant, since all groups vanish for . However, for example, it holds that .
In this paper, we are only interested in the cohomology groups for . Both for theoretical and computational interests, it is appropriate to have a more manageable cochain complex than to compute the lower commutative cohomology groups , such as Grillet did for computing the cohomology groups by means of symmetric cochains (see Subsection 2.1). We shall exhibit below such a (truncated) complex, denoted by
and referred to as the complex of (normalized) commutative cochains on M with values in . The construction of this complex is heavily inspired by that given by Eilenberg and Mac Lane of the complexes  for computing the (co)homology groups of the spaces , and it is as follows:
A commutative one-cochain is a function with , such that .
A commutative two-cochain is a function with , such that if a or b are equal to one.
A commutative three-cochain is a pair of functions
with and , such that whenever some of or c are equal to one and if a or b are equal to one.
A commutative four-cochain is a triple of functions
with and , such that whenever some of or d are equal to one and if some of or c are equal to one.
Under pointwise addition, these commutative n-cochains form the abelian groups in (6), . The coboundary homomorphisms are defined by
A quite straightforward verification shows that (6) is actually a truncated cochain complex, that is the equalities and hold.
A basic result here is the following, whose proof is quite long and technical, and we give it in Subsection 3.1, so as not to obstruct the natural flow of the paper.
Theorem 3.3. Let M be any commutative monoid, and let be a functor. For each , there is a natural isomorphism:
From this theorem, for , we have isomorphisms
are referred as the groups of commutative n-cocycles and commutative n-coboundaries on M with values in , respectively.
After Theorem 3.3 and the isomorphisms in (1), Grillet symmetric cohomology groups and the commutative ones are closely related, for through the natural injective cochain map
which is the identity map, , on one-cochains, the inclusion map, , on two-cochains, and on three- and four-cochains is defined by the simple formulas and , respectively. The only non-trivial verification here concerns the equality , that is, , for any , but it easily follows from Lemma 3.4 below.
From now on, we shall regard the complex of symmetric cochains as a subcomplex of the complex of commutative cochains, via the above injective cochain map. Thus,
Lemma 3.4. Let be a functor, where M is any commutative monoid, and let be a function with . Then, h satisfies the symmetry conditions
if and only if it satisfies either (11) or (12) below.
Proof. The implication (10)⇒(11) and (10)⇒(12) are easily seen. To see that (11)⇒(10), observe that, making the permutation , equation (11) is written as . If we carry this to equation (11), we obtain
that is the first condition in (10) holds. However, then, we get also the second one simply by replacing the term with in (11). The proof that (12)⇒(10) is parallel. □
Theorem 3.5. For any commutative monoid M and any functor , there are natural isomorphisms
and a natural monomorphism
Proof. The equalities and are clear. Further , since the cocycle condition on a commutative two-cochain g implies the symmetry condition . Hence, the isomorphisms (13) and (14) follow from those in (1) and (8), for and , respectively.
The homomorphism in (15) is the composite of
so it suffices to prove that the homomorphism induced by (9) on the third cohomology groups is injective. To do so, suppose is a symmetric three-cochain, such that is a commutative three-coboundary, that is for some . This means that the equalities:
hold, whence is a symmetric two-cochain and is actually a symmetric two-coboundary. It follows that the inclusion map induces an injective map in cohomology , as required. □
Remark 3.6. The inclusion is, in general, strict. Let G be any abelian group, and let be the constant functor defined by any other abelian group A, as in Example 3.2. Then, by Lemma 3.4 and a result by Mac Lane  (Theorem 4), we have that . However, for instance, it holds that .
If M is any commutative monoid and is a functor, then a function , such that and , is called a derivation of M in , written as . Let
denote the abelian group, under pointwise addition, of derivations .
Corollary 3.7. For any commutative monoid M and any functor , there is a natural isomorphism
Proof. The equality holds, since any derivation satisfies the normalization condition . Hence, the result follows from the isomorphisms (7) in Theorem 3.3 for . □
For the next corollary, let us recall that a commutative (group) coextension of a commutative monoid M by a functor is a surjective monoid homomorphism , such that, for each , it is given a simply transitive group action of the group on the fiber set , , satisfying the equations below.
Let denote the set of equivalence classes of such commutative co-extensions of M by , where two of them, say and , are equivalent whenever there is a monoid isomorphism , such that and , for any and .
Corollary 3.8. For any commutative monoid M and any functor , there is a natural bijection
Proof. After the isomorphism (14) in Theorem 3.5, this is the classification result by Grillet  (§V.4). We are not going to bring Grillet’s proof here, but we recall that in the correspondence between commutative (= symmetric) two-cohomology classes and iso-classes of co-extensions, each is taken to the commutative coextension , where
is the crossed product commutative monoid whose elements are pairs where and and whose multiplication is given by
This multiplication is unitary ( is the unit) since g is normalized, that is ; and it is associative and commutative due to g being a symmetric two-cocycle, that is because of the equalities and . The homomorphism is the projection, , and for each , the action of on is given by addition in , . □
Proof of Theorem 3.3
We start by specifying the relevant truncation of the cochain complex that, by Fact 2.3, yields cocycles and coboundaries on the commutative monoid M at dimensions . To do so, we need to pay attention to the six-dimensional truncated part of
whose face and degeneracy operators are given by
Hence, (with a dimensional shift) the cochain complex for low degrees is
A one-cochain is a function with , such that .
A two-cochain is a function
with , such that .
A three-cochain is a function
with , such that
A four-cochain is a function
such that and:
The coboundary homomorphisms are given by
Then, the claimed isomorphisms (7) follows from the existence of the following diagram of abelian group homomorphisms
which satisfy the equalities and , for ; , for ; ; ; and .
These homomorphisms are defined as follows
, where ;
, where ;
, where ;
A quite tedious, but totally straightforward, verification shows that these homomorphisms , and satisfy the claimed properties, implying that the truncated cochain complexes in (6) and in (16) are homology-isomorphic.
4. Classifying Braided Abelian ⊗-Groupoids by Cohomology Classes
This section is dedicated to showing a precise cohomological classification of braided monoidal abelian groupoids. The case of monoidal abelian groupoids was dealt with in , where their classification was solved by means of Leech’s three-cohomology classes of monoids. Strictly symmetric monoidal abelian groupoids have been classified in , in this case by Grillet’s three-cohomology classes of commutative monoids. Here, we show how every braided monoidal abelian groupoid invariably has a commutative monoid M, a group valued functor and a commutative three-dimensional cohomology class associated with it. Furthermore, the triple thus obtained is an appropriate system of ‘descent data’ to rebuild the braided abelian groupoid up to braided equivalence.
To fix some terminology and notations needed throughout this section, we start by stating that by a groupoid (or Brandt groupoid), we mean a small category, all of whose morphisms are invertible. A groupoid whose isotropy (or vertex) groups , , are all abelian is termed an abelian groupoid. For instance, any abelian group A can be regarded as an abelian groupoid with only one object a and . For many purposes, it is convenient to distinguish A from the one-object groupoid ; the notation
for is not bad (its nerve or classifying space  (Example 1.4) is precisely the Eilenberg–Mac Lane minimal complex ), and we shall use it below. A groupoid in which there are no morphisms between different objects is termed totally disconnected. It is easily seen that any abelian totally disconnected groupoid is actually a disjoint union of abelian groups or, more precisely, of the form , for some family of abelian groups .
We use additive notation for abelian groupoids; thus, the identity morphism of an object x of an abelian groupoid is denoted by , if , are morphisms, their composite is written as , whereas the inverse of u is .
Monoidal categories, and particularly braided monoidal categories, have been studied extensively in the literature, and we refer to Mac Lane [3,20], Saavedra  and Joyal and Street  for the background. We intend to work with braided abelian ⊗-groupoids (or braided monoidal abelian groupoids)
which consist of an abelian groupoid , a functor (the tensor product), an object I (the unit object) and natural isomorphisms , , (called the associativity, left unit, right unit constraints, respectively) and (the braidings), such that the four coherence conditions below hold.
For further use, we recall that in any braided abelian ⊗-groupoid , the equalities below hold (see ).
Example 4.1 (Two-dimensional crossed products). Every commutative three-cocycle gives rise to a braided abelian ⊗-groupoid
that should be thought of as a two-dimensional crossed product of M by , and it is built as follows: its underlying groupoid is the totally disconnected groupoid
where recall that each denotes the groupoid having a as its unique object and as the automorphism group of a. That is, an object of is an element ; if are different elements of the monoid M, then there are no morphisms in between them, whereas its isotropy group at any is .
The tensor product is given by multiplication in M on objects, so , and on morphisms by the group homomorphisms
The unit object is , the unit of the monoid M, and the structure constraints and the braiding isomorphisms are
which are easily seen to be natural since is an abelian group valued functor. The coherence condition (18), (20) and (21) follow from the three-cocycle condition , while the coherence condition (19) holds due to the normalization condition .
Example 4.2. A braided abelian ⊗-groupoid is called strict if all of its structure constraints , and are identities. Regarding a monoid as a category with only one object, it is easy to identify a braided abelian strict ⊗-groupoid with an abelian track monoid, in the sense of Baues-Jibladze  and Pirashvili , endowed with a braided structure. Porter  and Joyal-Street  (§3, Example 4) (a preliminary manuscript of )) show a natural way to produce braided strict abelian ⊗-groupoids from crossed modules in the category of monoids. We recall that construction in this example.
A crossed module in the category is a triplet consisting of a monoid M, a group G endowed with a M-action by a monoid homomorphism , written , and a homomorphism satisfying
Roughly speaking, these two conditions say that the action of M on G behaves like an abstract conjugation. Note that when the monoid M is a group, we have the ordinary notion of a crossed module by Whitehead . Observe that, if , then for all ; that is, the subgroup is contained in the center of G, and therefore, it is abelian. The crossed module is termed abelian whenever, for any , the subgroup is abelian. If, for example, the group G is abelian, or the monoid M is cancellative (a group, for instance), then the crossed module is abelian.
A bracket operation for a crossed module is a function satisfying
This operation should be thought of as an abstract commutator.
Each abelian crossed module with a bracket operator yields a braided abelian strict ⊗-groupoid as follows. Its objects are the elements of the monoid M, and a morphism in is an element with . The composition of two morphisms is given by multiplication in G, . The tensor product is
and the braiding is provided by the bracket operator via the formula
In the very special case where M and G are commutative, the action of M on G is trivial, and ∂ is the trivial homomorphism (i.e., and , for all , ), then a bracket operator amounts a bilinear map, that is, a function satisfying
Thus, for example, when is the additive monoid of non-negative integers and is the abelian group of integers, a bracket is given by . Furthermore, if G is any multiplicative abelian group, then any defines a bracket by .
Suppose , are braided abelian ⊗-groupoids. A braided ⊗-functor (or braided monoidal functor)
consists of a functor on the underlying groupoids , natural isomorphisms and an isomorphism , such that the following coherence conditions hold
If is another braided ⊗-functor, then an isomorphism is a natural isomorphism on the underlying functors, , such that the coherence conditions below are satisfied.
Example 4.3. Let be commutative three-cocycles of a commutative monoid. Then, any commutative cochain , such that induces a braided ⊗-isomorphism
which is the identity functor on the underlying groupoids and whose structure isomorphisms are given by and , respectively. Since the groups are abelian, these isomorphisms are natural. The coherence condition (26) and (28) follow from the equality , whilst the conditions in (27) trivially hold because of the normalization conditions .
If is any commutative one-cochain and , then an isomorphism of braided ⊗-functors is defined by putting , for each . So defined, θ is natural because of the abelian structure of the groups ; the first condition in (29) holds owing to the equality and the second one thanks to the normalization condition of f.
With compositions defined in a natural way, braided abelian ⊗-groupoids, braided ⊗-functors and isomorphisms form a 2-category  (Chapter V, §1). A braided ⊗-functor is called a braided ⊗-equivalence if it is an equivalence in this 2-category of braided abelian ⊗-groupoids, that is when there exists a braided ⊗-functor and braided isomorphisms and . From  (I, Proposition 4.4.2), it follows that a braided ⊗-functor is a braided ⊗-equivalence if and only if the underlying functor is an equivalence of groupoids, that is if and only if it is full, faithful and essentially surjective on objects or  (Chapter 6, Corollary 2) if and only if the induced map on the sets of iso-classes of objects
is a bijection, and the induced homomorphisms on the automorphism groups
are all isomorphisms.
Remark 4.4. From the coherence theorem for monoidal categories  (Corollary 1.4, Example 2.4), it follows that every braided abelian ⊗-groupoid is braided ⊗-equivalent to a braided strict one, that is to one in which all of the structure constraints , and are identities (see Example 4.2). This suggests that it is relatively harmless to consider braided abelian ⊗-groupoids as strict. However, it is not so harmless when dealing with their homomorphisms, since not every braided ⊗-functor is isomorphic to a strict one (i.e., one as in (25) in which the structure isomorphisms and are all identities). Indeed, it is possible to find two braided abelian strict ⊗-groupoids, say and , that are related by a braided ⊗-equivalence between them, but there is no strict ⊗-equivalence either from to nor from to .
Our goal is to state a classification for braided abelian ⊗-groupoids, where two of them connected by a braided ⊗-equivalence are considered the same. The main result in this section is the following
Theorem 4.5 (Classification of braided abelian ⊗-groupoids). For any braided abelian ⊗-groupoid , there exist a commutative monoid M, a functor , a commutative three-cocycle and a braided ⊗-equivalence
For any two commutative three-cocycles and , there is a braided ⊗-equivalence:
if and and only if there exist an isomorphism of monoids and a natural isomorphism , such that the equality of cohomology classes below holds.
Proof. Let be any given braided abelian ⊗-groupoid.
In a first step, we assume that is totally disconnected and strictly unitary, in the sense that its unit constraints and are all identities. Then, a system of data , such that as braided abelian groupoids, is defined as follows:
• The monoid M. Let be the set of objects of . The function on objects of the tensor functor determines a multiplication on M, simply by making , for any . Because of the strictness of the unit in , this multiplication on M is unitary with , the unit object of . Furthermore, it is associative and commutative since, as is totally disconnected, the existence of the associativity constraints and the braidings forces the equalities and . Thus, M becomes a commutative monoid.
• The functor . For each , let be the vertex group of the underlying groupoid at a. The group homomorphisms have an associative, commutative and unitary behavior in the sense that the equalities
hold. These follow from the abelian nature of the groups of automorphisms in , since the diagrams below commute due to the naturality of the structure constraints and the braiding.
Then, if we write for the homomorphism, such that
show that the assignments , , define an abelian group valued functor on . Note that this functor determines the tensor product ⊗ of , since
• The three-cocycle . The associativity constraint and the braiding of are necessarily written in the form and , for some given lists and . Since is strictly unitary, the equations in (19) and (22) give the normalization conditions for h, while the equations in (23) imply the normalization conditions for μ. Thus, is a commutative three-cochain, which is actually a three-cocycle, since the coherence conditions (18), (20) and (21) are now written as
which amount to the cocycle condition .
Since an easy comparison (see Example 4.1) shows that , the proof of this part is complete, under the hypothesis of being totally disconnected and strictly unitary.
It remains to prove that the braided abelian ⊗-groupoid is braided ⊗-equivalent to another one that is totally disconnected and strictly unitary. To do that, we combine the transport process by Saavedra  (I, 4.4.5) and Joyal-Street  (Example 2.4), which shows how to transport the braided monoidal structure on an abelian ⊗-groupoid along an equivalence on its underlying groupoid, with the generalized Brandt’s theorem, which asserts that every groupoid is equivalent (as a category) to a totally disconnected groupoid  (Chapter 6, Theorem 2). Since every braided abelian ⊗-groupoid is braided ⊗-equivalent to a braided abelian strict ⊗-groupoid (see Remark 4.4), there is no loss of generality in assuming that is itself strictly unitary.
Then, let be the set of isomorphism classes of the objects of ; let us choose, for each , any representative object , with ; and let us form the totally disconnected abelian groupoid
whose set of objects is M and whose vertex group at any object is .
This groupoid is equivalent to the underlying groupoid . To give a particular equivalence , let us select for each and each an isomorphism in . In particular, for every , we take , the identity morphism of . Then, let be the functor that acts on objects by and on morphisms by . We also have the more obvious functor , which is defined on objects by and on morphisms by . Clearly, , and the natural equivalence satisfies the equalities and . Therefore, the given braided monoidal structure on can be transported to one on , such that the functors F and underlie braided ⊗-functors, and the natural equivalences and turn out to be ⊗-isomorphisms. In the transported structure, the tensor product is the dotted functor in the commutative square
and the unit object is . The functors F and are endowed with the isomorphisms
and the structure constraints and the braiding c of are those isomorphisms uniquely determined by (26)–(28), respectively. Now, a quick analysis indicates that, for any object ,
Similarly, we have , and therefore, is strictly unitary.
We first assume that there exist an isomorphism of monoids and a natural isomorphism , such that . This means that there is a commutative two-cochain , such that the equalities below hold.
Then, a braided isomorphism:
is defined as follows. The underlying functor acts by . The structure isomorphisms of F are given by and . So defined, it is easy to see that F is an isomorphism between the underlying groupoids. Verifying the naturality of the isomorphisms , that is the commutativity of the squares
for , , is equivalent (since the groups are abelian) to verify the equalities
which hold since the naturality of just says that
The coherence conditions (26) and (28) are verified as follows
whereas the conditions in (27) trivially follow from the equalities .
Conversely, suppose that is any braided equivalence. By , there is no loss of generality in assuming that F is strictly unitary in the sense that . As the underlying functor establishes an equivalence between the underlying groupoids,
and these are totally disconnected, it is necessarily an isomorphism.
Let us write for the bijection describing the action of F on objects; that is, such that , for each . Then, i is actually an isomorphism of monoids, since the existence of the structure isomorphisms forces the equality .
Let us write for the isomorphism giving the action of F on automorphisms ; that is, such that , for each and . The naturality of the automorphisms tell us that the equalities (36) hold (see diagram (35)). These, for the case when , give the equalities in (37), which amounts to being a natural isomorphism of abelian group valued functors on .
Writing now , for each , the equations hold due to the coherence (27), and thus, we have a commutative two-cochain
which satisfies (32) and (33) owing to the coherence (26) and (28), as we can see just by retracting our steps in (38) and (39), respectively. This means that , and therefore, we have that , whence . □
A braided categorical group  (§3) is a braided abelian ⊗-groupoid in which, for any object x, there is an object with an arrow . Actually, the hypothesis of being abelian is superfluous here, since every monoidal groupoid in which every object has a quasi-inverse is always abelian  (Proposition 3). The cohomological classification of these braided categorical groups was stated and proven by Joyal and Street  (Theorem 3.3) by means of Eilenberg–Mac Lane’s commutative cohomology groups , of abelian groups G with coefficients in abelian groups A (see Example 3.2). Next, we obtain Joyal–Street’s classification result as a corollary of Theorem 4.5.
Corollary 4.6. For any abelian groups G and A and any three-cocycle , the braided abelian groupoid is a braided categorical group.
For any braided categorical group , there exist abelian groups G and A, a three-cocycle and a braided ⊗-equivalence
For any two commutative three-cocycles and , where and are abelian groups, there is a braided ⊗-equivalence
if and and only if there exist isomorphism of groups and , such that the equality of cohomology classes below holds.
Proof. Recall from Example 3.2 that we are here regarding A as the constant abelian group valued functor on it defines. Since G is a group, for any object a of (i.e., any element ), we have . Thus, is actually a braided categorical group.
Let be a braided categorical group. By Theorem 4.5 , there are a commutative monoid M, a functor , a commutative three-cocycle and a braided ⊗-equivalence . Then, is a braided categorical group as is, and for any , it must exist another with a morphism in ; this implies that in M, since the groupoid is totally disconnected, whence is an inverse of a in M. Therefore, is actually an abelian group.
Let be the abelian group attached by at the unit of G. Then, a natural isomorphism is defined, such that, for any , . Therefore, if we take , Theorem 4.5 gives the existence of a braided equivalence , whence , and the given are braided ⊗-equivalent.
This follows directly form Theorem 4.5 . □
The classification result in Theorem 4.5 involves an interpretation of the elements of in terms of certain two-dimensional co-extensions of M by , such as the elements of are interpreted as commutative monoid co-extensions in Corollary 3.8. To state this fact, in the next definition, we regard any commutative monoid M as a braided abelian discrete ⊗-groupoid (i.e., whose only morphisms are the identities), on which the tensor product is multiplication in M. Thus, if is any braided abelian ⊗-groupoid, a braided ⊗-functor is the same thing as a map satisfying whenever , and .
Definition 4.7. Let M be a commutative monoid, and let be any abelian group valued functor on . A braided two-coextension of M by is a surjective braided ⊗-functor , where is a braided abelian ⊗-groupoid, such that, for any , it is given an (associative and unitary) action of the groupoid on the fiber groupoid by means of a functor
which is simply transitive, in the sense that the induced functor:
is an equivalence and satisfies
for every , , , and .
Let us point out that if , for some , then since the functor , for , is essentially surjective. Furthermore, the functoriality of the action means that if are composablearrows in , then, for any , we have
Remark 4.8. These braided two-co-extensions can be seen as a sort of (braided, non-strict) linear track extensions in the sense of Baues, Dreckmann and Jibladze [28,34]. Briefly, note that to give a commutative two-coextension , as above, is equivalent to giving a surjective braided ⊗-functor satisfying
together with a family of isomorphisms of groups satisfying:
The family of isomorphisms and the action of on are related to each other by the equations , for any , , and .
Let denote the set of equivalence classes of such braided two-co-extensions of M by , where two of them, say and , are equivalent whenever there is a braided ⊗-equivalence , such that and , for any morphism in and . Then, we have:
Theorem 4.9 (Classification of braided two-co-extensions). For any commutative monoid M and any functor , there is a natural bijection
Proof. This is a consequence of Theorem 4.5 with only a slight adaptation of the arguments used for its proof. For any three-cocycle , the braided abelian ⊗-groupoid in (24) comes with a natural structure of braided two-coextension of M by , in which the surjective braided functor is given by the identity map on objects, . The fiber groupoid over any is just , and the action functor is given by addition in , that is . If in any other three-cocycle, such that , for some two-cochain , then the associated braided ⊗-isomorphism in (30), , is easily recognized as an equivalence between the braided co-extensions and . Thus, we have a well-defined map
To see that it is injective, suppose , such that the associated braided two-co-extensions are made equivalent by a braided ⊗-functor, say , which can be assumed to be strictly unitary . Then, the two-cochain built in (40) satisfies that , whence .
Finally, to prove that the map is surjective, let be any given braided two-coextension of M by . By Theorem 4.5 and Lemma 4.10 below, we can assume that , for some commutative monoid , a functor , and a three-cocycle . Then, a monoid isomorphism and a natural isomorphism become determined by the equations and , for any and . Furthermore, taking , the braided ⊗-isomorphism in (34) for the two-cochain , , is then easily seen as an equivalence between the braided extensions and . □
Lemma 4.10. Let be a braided two-coextension of M by , and suppose that is any braided abelian ⊗-groupoid, which is braided ⊗-equivalent to . Then, can be endowed with a braided two-coextension structure of M by , say , such that and are equivalent braided two-co-extensions. □
Proof. Let be a braided ⊗-equivalence. Then, a braided two-coextension structure of is given as follows: let:
be the braided ⊗-functor composite of and F. This is clearly surjective, since is and F is essentially surjective. For every , let be the action defined by , where is unique arrow in , such that
This is a simply-transitive well-defined action since F is a full, faithful and essentially surjective functor. In order to check (41), we have:
and the result follows since F is faithful and is an isomorphism. Thus, we have defined the braided two-coextension , which is clearly equivalent to the original one by means of F. □
This work has been supported by “Dirección General de Investigación” of Spain, Project: MTM2011-22554; and for the first author also by FPUgrant FPU12-01112.
Both authors have contributed equally to this work and they agree to the final version.
Conflicts of Interest
The authors declare no conflict of interest.
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