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), [
24] (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 [
25]). Although these cohomology groups arise from algebraic topology, they come with algebraic interest. Briefly, recall that there are natural isomorphisms [
26] (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 [
19] (Definition 3.1)in terms of cohomology classes
was proven by Joyal and Street in [
19] (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
[
17] 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
, where
, where
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
where
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 [
15] (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 [
8] (§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
where
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 ;
, where:
, where
, 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 [
2], where their classification was solved by means of Leech’s three-cohomology classes of monoids. Strictly symmetric monoidal abelian groupoids have been classified in [
9], 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 [
27] (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 [
4] and Joyal and Street [
19] 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 [
19]).
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 [
28] and Pirashvili [
29], endowed with a braided structure. Porter [
30] and Joyal-Street [
31] (§3, Example 4) (a preliminary manuscript of [
19])) 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 [
32]. 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 [
16] (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 [
4] (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 [
33] (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 [
19] (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
the equalities:
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 [
4] (I, 4.4.5) and Joyal-Street [
19] (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 [
33] (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 [
18], 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 [
19] (§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 [
2] (Proposition 3). The cohomological classification of these braided categorical groups was stated and proven by Joyal and Street [
19] (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 [
18]. 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. □