# A Cohomology Theory for Commutative Monoids

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## Abstract

**:**

## 1. Introduction and Summary

“There is a one-to-one correspondence between equivalence classes of braided monoidal abelian groupoids $(\mathcal{M},\otimes ,\mathit{c})$ and iso-classes of triples $(M,\mathcal{A},k)$, with $k\in {H}_{\mathrm{c}}^{3}(M,\mathcal{A})$.”

## 2. Preliminaries on Cohomology of Monoids and Simplicial Sets

#### 2.1. Grillet Cohomology of Commutative Monoids: Symmetric Cocycles

#### 2.2. Cohomology of Categories and Simplicial Sets: Leech Cohomology of Monoids

**Example 2.1**(Leech cohomology of monoids). Any monoid M gives rise to a category $\mathbb{D}M$, whose set of objects is M and set of arrows $M\times M\times M$, with $(a,b,c):b\to abc$. Composition is given by $(d,abc,e)(a,b,c)=(da,b,ce)$, and the identity morphism of any object a is $(1,a,1)$. If we say that an abelian group valued functor $\mathcal{A}:\mathbb{D}M\to \mathbf{Ab}$ carries the morphism $(a,b,c)$ to the group homomorphism ${a}_{*}{c}^{*}:{\mathcal{A}}_{b}\to {\mathcal{A}}_{abc}$, then we see that such a functor is a system of data consisting of abelian groups ${\mathcal{A}}_{a}$, $a\in M$, and homomorphisms ${\mathcal{A}}_{b}\stackrel{{a}_{*}}{\u27f6}{\mathcal{A}}_{ab}\stackrel{{b}^{*}}{\u27f5}{\mathcal{A}}_{a}$, $a,b\phantom{\rule{0.166667em}{0ex}}\in M$, such that, for any $a,b,c\in M$,

**Fact 2.2.**Let X be a simplicial set. In order to define a functor $\pi :\Delta /X\to \mathbb{C}$, it suffices to give objects $\pi x\in \mathbb{C}$, $x\in {X}_{n}$, $n\ge 0$, together with morphisms:

**Fact 2.3.**Let $\mathcal{A}:\Delta /X\to \mathbf{Ab}$ be a coefficient system on a simplicial set X. A n-cochain of X with coefficients in $\mathcal{A}$ is a map $\lambda :{X}_{n}\to {\u2a06}_{x\in {X}_{n}}\phantom{\rule{-2.0pt}{0ex}}{\mathcal{A}}_{x}$, such that $\lambda \left(x\right)\in {\mathcal{A}}_{x}$ for each $x\in {X}_{n}$. Thus, ${\prod}_{x\in {X}_{n}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\mathcal{A}}_{x}$ is the abelian group of such n-cochains. As $n\ge 0$ varies, these define a cosimplicial abelian group $\Delta \to \mathbf{Ab}$, $\left[n\right]\mapsto {\prod}_{x\in {X}_{n}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{\mathcal{A}}_{x}$, whose cosimplicial operators

## 3. A Cohomology Theory for Commutative Monoids

**Definition 3.1.**Let M be a commutative monoid. For any abelian group valued functor $\mathcal{A}:\mathbb{H}M\to \mathbf{Ab}$, the commutative cohomology groups of M with coefficients in $\mathcal{A}$, denoted ${H}_{\mathrm{c}}^{n}(M,\mathcal{A})$, are defined by

**Example 3.2.**Let $M=G$ be an abelian group. Then, the simplicial set ${\overline{W}}^{2}\phantom{\rule{-0.166667em}{0ex}}G$ is an Eilenberg–Mac Lane minimal complex $K(G,2)$ [17,24] (Theorem 17.4), [24] (Theorem 23.2). For any abelian group A, regarded as a constant functor $A:\mathbb{H}G\to \mathbf{Ab}$, the commutative cohomology groups ${H}_{\mathrm{c}}^{n}(G,A)={H}^{n+1}(K(G,2),A)$ 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 ${H}_{\mathrm{ab}}^{n}(G,A)$ in [18,19] and by ${H}_{1}^{n}(G,A)$ 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))

**Theorem 3.3.**Let M be any commutative monoid, and let $\mathcal{A}:\mathbb{H}M\to \mathbf{Ab}$ be a functor. For each $n\le 3$, there is a natural isomorphism:

**Lemma 3.4.**Let $\mathcal{A}:\mathbb{H}M\to \mathbf{Ab}$ be a functor, where M is any commutative monoid, and let $h:{M}^{3}\to {\u2a06}_{a\in M}{\mathcal{A}}_{a}$ be a function with $h(a,b,c)\in {\mathcal{A}}_{abc}$. Then, h satisfies the symmetry conditions

**Proof.**The implication (10)⇒(11) and (10)⇒(12) are easily seen. To see that (11)⇒(10), observe that, making the permutation $(a,b,c)\mapsto (c,b,a)$, equation (11) is written as $h(b,c,a)=h(c,b,a)+h(b,a,c)$. If we carry this to equation (11), we obtain

**Theorem 3.5.**For any commutative monoid M and any functor $\mathcal{A}:\mathbb{H}M\to \mathbf{Ab}$, there are natural isomorphisms

**Proof.**The equalities ${Z}_{\mathrm{s}}^{1}(M,\mathcal{A})={Z}_{\mathrm{c}}^{1}(M,\mathcal{A})$ and ${B}_{\mathrm{s}}^{2}(M,\mathcal{A})={B}_{\mathrm{c}}^{2}(M,\mathcal{A})$ are clear. Further ${Z}_{\mathrm{s}}^{2}(M,\mathcal{A})={Z}_{\mathrm{c}}^{2}(M,\mathcal{A})$, since the cocycle condition on a commutative two-cochain g implies the symmetry condition $g(a,b)=g(b,a)$. Hence, the isomorphisms (13) and (14) follow from those in (1) and (8), for $n=1$ and $n=2$, respectively.

**Remark 3.6.**The inclusion ${H}_{\mathrm{s}}^{3}(M,A)\subseteq {H}_{\mathrm{c}}^{3}(M,A)$ is, in general, strict. Let G be any abelian group, and let $A:\mathbb{H}G\to \mathbf{Ab}$ 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 ${H}_{\mathrm{c}}^{3}(G,A)=0$. However, for instance, it holds that ${H}_{\mathrm{s}}^{3}(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/2\mathbb{Z})\cong \mathbb{Z}/2\mathbb{Z}\ne 0$.

**Corollary 3.7.**For any commutative monoid M and any functor $\mathcal{A}:\mathbb{H}M\to \mathbf{Ab}$, there is a natural isomorphism

**Proof.**The equality ${Z}_{\mathrm{c}}^{1}(M,\mathcal{A})=\mathrm{Der}(M,\mathcal{A})$ holds, since any derivation $f:M\to \mathcal{A}$ satisfies the normalization condition $f\left(1\right)=0$. Hence, the result follows from the isomorphisms (7) in Theorem 3.3 for $n=1$. □

**Corollary 3.8.**For any commutative monoid M and any functor $\mathcal{A}:\mathbb{H}M\to \mathbf{Ab}$, 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 $g\in {Z}_{\mathrm{c}}^{2}(M,\mathcal{A})$ is taken to the commutative coextension $\pi :\mathcal{A}{\u22ca}_{g}M\twoheadrightarrow M$, where

#### Proof of Theorem 3.3

- ${\varphi}_{1}={\psi}_{1}=id$;
- ${\varphi}_{2}\left(\lambda \right)=g$, where $g(a,b)=\lambda (a,b,1)$;
- ${\psi}_{2}\left(g\right)=\lambda $, where $\lambda ({b}_{1},{b}_{2},{a}_{1})={\left({a}_{1}\right)}_{*}g({b}_{1},{b}_{2})-{\left({b}_{1}\right)}_{*}g({b}_{2},{a}_{1})$;
- ${\Gamma}_{2}\left(\lambda \right)={\lambda}^{\prime}$, where ${\lambda}^{\prime}({b}_{1},{b}_{2},{a}_{1})=\lambda ({b}_{1},{b}_{2},1,1,{a}_{1},1)-\lambda ({b}_{1}{b}_{2},1,1,1,1,{a}_{1})$;
- ${\varphi}_{3}\left(\lambda \right)=(h,\mu )$, where:$$h(a,b,c)=\lambda (a,b,c,1,1,1),\phantom{\rule{8.5359pt}{0ex}}\mu (a,b)=\lambda (a,1,1,1,1,b)-\lambda (1,a,1,1,b,1)+\lambda (1,1,a,b,1,1);$$
- ${\psi}_{3}(h,\mu )=\lambda $, where$$\begin{array}{cc}\hfill \lambda ({c}_{1},{c}_{2},{c}_{3},{b}_{1},{b}_{2},{a}_{1})& ={\left({b}_{1}{b}_{2}{a}_{1}\right)}_{*}h({c}_{1},{c}_{2},{c}_{3})+{\left({c}_{1}{c}_{2}{b}_{1}\right)}_{*}h({c}_{3},{b}_{2},{a}_{1})-{\left({c}_{1}{c}_{2}{a}_{1}\right)}_{*}h({c}_{3},{b}_{1},{b}_{2})\hfill \\ & +{\left({c}_{1}{c}_{2}{a}_{1}\right)}_{*}h({b}_{1},{c}_{3},{b}_{2})-{\left({c}_{1}{a}_{1}\right)}_{*}h({c}_{2},{b}_{1},{c}_{3}{b}_{2})+{\left({c}_{1}{a}_{1}\right)}_{*}h({c}_{2},{c}_{3},{b}_{1}{b}_{2})\hfill \\ & +{\left({c}_{1}{c}_{2}{b}_{2}{a}_{1}\right)}_{*}\mu ({c}_{3},{b}_{1});\hfill \end{array}$$
- ${\Gamma}_{3}\left(\lambda \right)={\lambda}^{\prime}$, where$$\begin{array}{cc}\hfill {\lambda}^{\prime}({c}_{1},{c}_{2},{c}_{3},{b}_{1}& ,{b}_{2},{a}_{1})=-\lambda ({c}_{1}{c}_{2},1,1,{c}_{3},1,1,1,{b}_{1},{b}_{2},{a}_{1})+\lambda ({c}_{1},{c}_{2},1,{c}_{3},1,{b}_{1},{b}_{2},1,1,{a}_{1})\hfill \\ & -{\left({a}_{1}\right)}_{*}\lambda ({c}_{1},{c}_{2},{c}_{3},1,1,1,{b}_{1}{b}_{2},1,1,1)+{\left({a}_{1}\right)}_{*}\lambda ({c}_{1}{c}_{2},1,{c}_{3},1,1,1,1,1,{b}_{1}{b}_{2},1)\hfill \\ & -{\left({a}_{1}\right)}_{*}\lambda ({c}_{1}{c}_{2},{c}_{3},1,1,1,1,1,1,1,{b}_{1}{b}_{2})+{\left({b}_{1}\right)}_{*}\lambda ({c}_{1}{c}_{2},{c}_{3},1,1,1,1,1,1,1,{b}_{2}{a}_{1})\hfill \\ & -{\left({b}_{1}\right)}_{*}\lambda ({c}_{1}{c}_{2},1,{c}_{3},1,1,1,1,1,{b}_{2}{a}_{1},1)+{\left({c}_{1}\right)}_{*}\lambda ({c}_{2},{b}_{1},{c}_{3}{b}_{2},1,1,1,1,1,{a}_{1},1)\hfill \\ & -{\left({c}_{1}\right)}_{*}\lambda ({c}_{2},{c}_{3}{b}_{1}{b}_{2},1,1,1,1,1,1,1,{a}_{1})+{\left({c}_{1}{c}_{2}\right)}_{*}\lambda (1,{c}_{3},1,1,{b}_{1}{b}_{2},1,1,1,1,{a}_{1})\hfill \\ & -{\left({c}_{1}{c}_{2}\right)}_{*}\lambda (1,1,{c}_{3},1,{b}_{1},{b}_{2},1,1,{a}_{1},1)+{\left({c}_{1}{c}_{2}{b}_{1}\right)}_{*}\lambda ({c}_{3},1,1,1,1,1,1,{b}_{2},{a}_{1},1)\hfill \\ & -{\left({c}_{1}{c}_{2}{b}_{1}\right)}_{*}\lambda (1,{c}_{3},1,1,1,{b}_{2},{a}_{1},1,1,1)+{\left({c}_{1}{c}_{2}{a}_{1}\right)}_{*}\lambda (1,{c}_{3},1,1,1,{b}_{1},{b}_{2},1,1,1)\hfill \\ & -{\left({c}_{1}{c}_{2}{a}_{1}\right)}_{*}\lambda ({c}_{3},1,1,1,1,1,1,{b}_{1},{b}_{2},1)-{\left({c}_{1}{c}_{2}{a}_{1}\right)}_{*}\lambda (1,1,{c}_{3},1,{b}_{1},1,{b}_{2},1,1,1);\hfill \end{array}$$
- ${\varphi}_{4}\left(\lambda \right)=(t,\gamma ,\delta )$, where$$\begin{array}{cc}\hfill t(a,b,c,d)& =\lambda (a,b,c,d,1,1,1,1,1,1),\hfill \\ \hfill \gamma (a,b,c)& =\lambda (a,1,1,1,1,1,1,b,c,1)-\lambda (1,a,1,1,1,b,c,1,1,1)+\lambda (1,1,a,1,b,1,c,1,1,1)\hfill \\ & -\lambda (1,1,1,a,b,c,1,1,1,1),\hfill \\ \hfill \delta (a,b,c)& =\lambda (a,b,1,1,1,1,1,1,1,c)-\lambda (a,1,b,1,1,1,1,1,c,1)+\lambda (a,1,1,b,1,1,1,c,1,1)\hfill \\ & +\lambda (1,a,b,1,1,1,c,1,1,1)-\lambda (1,a,1,b,1,c,1,1,1,1)+\lambda (1,1,a,b,c,1,1,1,1,1);\hfill \end{array}$$
- ${\psi}_{4}(t,\gamma ,\delta )=\lambda $, where$$\begin{array}{cc}\hfill \lambda ({d}_{1},{d}_{2},{d}_{3},{d}_{4},{c}_{1},{c}_{2},& {c}_{3},{b}_{1},{b}_{2},{a}_{1})={\left({c}_{1}{c}_{2}{c}_{3}{b}_{1}{b}_{2}{a}_{1}\right)}_{*}t({d}_{1},{d}_{2},{d}_{3},{d}_{4})\hfill \\ & -{\left({d}_{1}{d}_{2}{d}_{3}{c}_{1}{a}_{1}\right)}_{*}t({c}_{2},{d}_{4}{c}_{3},{b}_{1},{b}_{2})-{\left({d}_{1}{d}_{2}{d}_{3}{c}_{1}{b}_{2}{a}_{1}\right)}_{*}t({d}_{4},{c}_{3},{c}_{2},{b}_{1})\hfill \\ & +{\left({d}_{1}{d}_{2}{d}_{3}{c}_{1}{a}_{1}\right)}_{*}t({d}_{4},{c}_{2}{c}_{3},{b}_{1},{b}_{2})+{\left({d}_{1}{d}_{2}{d}_{3}{c}_{1}{a}_{1}\right)}_{*}t({c}_{2},{b}_{1},{d}_{4}{c}_{3},{b}_{2})\hfill \\ & -{\left({d}_{1}{d}_{2}{d}_{3}{c}_{1}{a}_{1}\right)}_{*}t({d}_{4},{c}_{2}{b}_{1},{c}_{3},{b}_{2})+{\left({d}_{1}{d}_{2}{d}_{3}{c}_{1}{a}_{1}\right)}_{*}t({c}_{2}{b}_{1},{d}_{4},{c}_{3},{b}_{2})\hfill \\ & +{\left({d}_{1}{d}_{2}{c}_{1}{a}_{1}\right)}_{*}t({d}_{3},{c}_{2},{b}_{1},{d}_{4}{c}_{3}{b}_{2})-{\left({d}_{1}{d}_{2}{c}_{1}{a}_{1}\right)}_{*}t({d}_{3},{c}_{2},{d}_{4}{c}_{3},{b}_{1}{b}_{2})\hfill \\ & -{\left({d}_{1}{b}_{1}{b}_{2}{a}_{1}\right)}_{*}t({d}_{2},{c}_{1},{d}_{3}{c}_{2},{d}_{4}{c}_{3})+{\left({d}_{1}{d}_{2}{c}_{1}{a}_{1}\right)}_{*}t({d}_{3},{d}_{4},{c}_{2}{c}_{3},{b}_{1}{b}_{2})\hfill \\ & -{\left({d}_{1}{d}_{2}{b}_{1}{b}_{2}{a}_{1}\right)}_{*}t({d}_{3},{d}_{4},{c}_{1},{c}_{2}{c}_{3})-{\left({d}_{1}{b}_{1}{b}_{2}{a}_{1}\right)}_{*}t({d}_{2},{d}_{3},{d}_{4},{c}_{1}{c}_{2}{c}_{3})\hfill \\ & +{\left({d}_{1}{b}_{1}{b}_{2}{a}_{1}\right)}_{*}t({d}_{2},{d}_{3},{c}_{1}{c}_{2},{d}_{4}{c}_{3})-{\left({d}_{1}{d}_{2}{b}_{1}{b}_{2}{a}_{1}\right)}_{*}t({c}_{1},{d}_{3},{d}_{4},{c}_{2}{c}_{3})\hfill \\ & -{\left({d}_{1}{d}_{2}{b}_{1}{b}_{2}{a}_{1}\right)}_{*}t({d}_{3},{c}_{1},{c}_{2},{d}_{4}{c}_{3})+{\left({d}_{1}{d}_{2}{b}_{1}{b}_{2}{a}_{1}\right)}_{*}t({c}_{1},{d}_{3},{c}_{2},{d}_{4}{c}_{3})\hfill \\ & +{\left({d}_{1}{d}_{2}{b}_{1}{b}_{2}{a}_{1}\right)}_{*}t({d}_{3},{c}_{1},{d}_{4},{c}_{2}{c}_{3})-{\left({d}_{1}{d}_{2}{d}_{3}{b}_{1}{b}_{2}{a}_{1}\right)}_{*}t({d}_{4},{c}_{1},{c}_{2},{c}_{3})\hfill \\ & +{\left({d}_{1}{d}_{2}{d}_{3}{b}_{1}{b}_{2}{a}_{1}\right)}_{*}t({c}_{1},{d}_{4},{c}_{2},{c}_{3})-{\left({d}_{1}{d}_{2}{d}_{3}{b}_{1}{b}_{2}{a}_{1}\right)}_{*}t({c}_{1},{c}_{2},{d}_{4},{c}_{3})\hfill \\ & +{\left({d}_{1}{d}_{2}{d}_{3}{d}_{4}{c}_{1}{a}_{1}\right)}_{*}t({c}_{2},{c}_{3},{b}_{1},{b}_{2})-{\left({d}_{1}{d}_{2}{d}_{3}{d}_{4}{c}_{1}{a}_{1}\right)}_{*}t({c}_{2},{b}_{1},{c}_{3},{b}_{2})\hfill \\ & -{\left({d}_{1}{d}_{2}{d}_{3}{c}_{1}{c}_{2}{b}_{1}\right)}_{*}t({d}_{4},{c}_{3},{b}_{2},{a}_{1})+{\left({d}_{1}{d}_{2}{c}_{2}{c}_{3}{b}_{1}{b}_{2}{a}_{1}\right)}_{*}\delta ({d}_{3},{d}_{4},{c}_{1})\hfill \\ & -{\left({d}_{1}{d}_{2}{d}_{3}{c}_{1}{b}_{2}{a}_{1}\right)}_{*}\delta ({d}_{4},{c}_{3},{c}_{2}{b}_{1})+{\left({d}_{1}{d}_{2}{d}_{3}{c}_{1}{b}_{1}{b}_{2}{a}_{1}\right)}_{*}\delta ({d}_{4},{c}_{3},{c}_{2})\hfill \\ & -{\left({d}_{1}{d}_{2}{d}_{3}{c}_{3}{b}_{1}{b}_{2}{a}_{1}\right)}_{*}\gamma ({d}_{4},{c}_{1},{c}_{2})-{\left({d}_{1}{d}_{2}{d}_{3}{d}_{4}{c}_{1}{b}_{2}{a}_{1}\right)}_{*}\gamma ({c}_{3},{c}_{2},{b}_{1})\hfill \\ & +{\left({d}_{1}{d}_{2}{d}_{3}{c}_{1}{b}_{2}{a}_{1}\right)}_{*}\gamma ({d}_{4}{c}_{3},{c}_{2},{b}_{1}).\hfill \end{array}$$

## 4. Classifying Braided Abelian ⊗-Groupoids by Cohomology Classes

**Example 4.1**(Two-dimensional crossed products). Every commutative three-cocycle $(h,\mu )\in {Z}_{\mathrm{c}}^{3}(M,\mathcal{A})$ gives rise to a braided abelian ⊗-groupoid

**Example 4.2.**A braided abelian ⊗-groupoid is called strict if all of its structure constraints ${\mathit{a}}_{x,y,z}$, ${\mathit{l}}_{x}$ and ${\mathit{r}}_{x}$ 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.

**Example 4.3.**Let $(h,\mu ),({h}^{\prime},{\mu}^{\prime})\in {Z}_{\mathrm{c}}^{3}(M,\mathcal{A})$ be commutative three-cocycles of a commutative monoid. Then, any commutative cochain $g\in {C}_{\mathrm{c}}^{2}(M,\mathcal{A})$, such that $(h,\mu )=({h}^{\prime},{\mu}^{\prime})+{\partial}^{2}g$ induces a braided ⊗-isomorphism

**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 ${\mathit{a}}_{x,y,z}$, ${\mathit{l}}_{x}$ and ${\mathit{r}}_{x}$ 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 ${\phi}_{x,y}$ and ${\phi}_{0}$ are all identities). Indeed, it is possible to find two braided abelian strict ⊗-groupoids, say $\mathcal{M}$ and ${\mathcal{M}}^{\prime}$, that are related by a braided ⊗-equivalence between them, but there is no strict ⊗-equivalence either from $\mathcal{M}$ to ${\mathcal{M}}^{\prime}$ nor from ${\mathcal{M}}^{\prime}$ to $\mathcal{M}$.

**Theorem 4.5**(Classification of braided abelian ⊗-groupoids). $\left(i\right)$ For any braided abelian ⊗-groupoid $\mathcal{M}$, there exist a commutative monoid M, a functor $\mathcal{A}:\mathbb{H}M\to \mathbf{Ab}$, a commutative three-cocycle $(h,\mu )\in {Z}_{\mathrm{c}}^{3}(M,\mathcal{A})$ and a braided ⊗-equivalence

**Proof.**$\left(i\right)$ Let $\mathcal{M}=(\mathcal{M},\otimes ,\text{I},\mathit{a},\mathit{l},\mathit{r},\mathit{c})$ be any given braided abelian ⊗-groupoid.

**c**of ${\mathcal{M}}^{\prime}$ are those isomorphisms uniquely determined by (26)–(28), respectively. Now, a quick analysis indicates that, for any object $a\in \mathrm{Ob}{\mathcal{M}}^{\prime}=M$,

**Corollary 4.6.**$\left(i\right)$ For any abelian groups G and A and any three-cocycle $(h,\mu )\in {Z}_{\mathrm{c}}^{3}(G,\mathcal{A})$, the braided abelian groupoid $A{\u22ca}_{h,\mu}G$ is a braided categorical group.

**Proof.**$\left(i\right)$ Recall from Example 3.2 that we are here regarding A as the constant abelian group valued functor on $\mathbb{H}G$ it defines. Since G is a group, for any object a of $A{\u22ca}_{h,\mu}G$ (i.e., any element $a\in G$), we have $a\otimes {a}^{-1}=a{a}^{-1}=1=\text{I}$. Thus, $A{\u22ca}_{h,\mu}\phantom{\rule{-0.166667em}{0ex}}G$ is actually a braided categorical group.

**Definition 4.7.**Let M be a commutative monoid, and let $\mathcal{A}$ be any abelian group valued functor on $\mathbb{H}M$. A braided two-coextension of M by $\mathcal{A}$ is a surjective braided ⊗-functor $p:\mathcal{M}\twoheadrightarrow M$, where $\mathcal{M}$ is a braided abelian ⊗-groupoid, such that, for any $a\in M$, it is given an (associative and unitary) action of the groupoid $K({\mathcal{A}}_{a},1)$ on the fiber groupoid ${p}^{-1}\left(a\right)$ by means of a functor

**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 $p:\mathcal{M}\twoheadrightarrow M$, as above, is equivalent to giving a surjective braided ⊗-functor $p:\mathcal{M}\twoheadrightarrow M$ satisfying

**Theorem 4.9**(Classification of braided two-co-extensions). For any commutative monoid M and any functor $\mathcal{A}:\mathbb{H}M\to \mathbf{Ab}$, 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 $(h,\mu )\in {Z}_{\mathrm{c}}^{3}(M,\mathcal{A})$, the braided abelian ⊗-groupoid $\mathcal{A}{\u22ca}_{h,\mu}M$ in (24) comes with a natural structure of braided two-coextension of M by $\mathcal{A}$, in which the surjective braided functor $\pi :\mathcal{A}{\u22ca}_{h,\mu}\phantom{\rule{-0.166667em}{0ex}}M\twoheadrightarrow M$ is given by the identity map on objects, $\pi \left(a\right)=a$. The fiber groupoid over any $a\in M$ is just ${\pi}^{-1}\left(a\right)=K({\mathcal{A}}_{a},1)$, and the action functor $K({\mathcal{A}}_{a},1)\phantom{\rule{-0.166667em}{0ex}}\times {\pi}^{-1}\left(a\right)\to {\pi}^{-1}\left(a\right)$ is given by addition in ${\mathcal{A}}_{a}$, that is $u\xb7v=u+v$. If $({h}^{\prime},{\mu}^{\prime})\in {Z}_{\mathrm{c}}^{3}(M,\mathcal{A})$ in any other three-cocycle, such that $(h,\mu )=({h}^{\prime},{\mu}^{\prime})-{\partial}^{2}g$, for some two-cochain $g\in {C}_{c}^{2}(M,\mathcal{A})$, then the associated braided ⊗-isomorphism in (30), $F\left(g\right):\mathcal{A}{\u22ca}_{h,\mu}M\to \mathcal{A}{\u22ca}_{{h}^{\prime},{\mu}^{\prime}}M$, is easily recognized as an equivalence between the braided co-extensions $\mathcal{A}{\u22ca}_{h,\mu}M\twoheadrightarrow \phantom{\rule{-0.166667em}{0ex}}M$ and $\mathcal{A}{\u22ca}_{{h}^{\prime}\phantom{\rule{-0.166667em}{0ex}},{\mu}^{\prime}}M\twoheadrightarrow M$. Thus, we have a well-defined map

**Lemma 4.10.**Let ${p}^{\prime}:{\mathcal{M}}^{\prime}\twoheadrightarrow M$ be a braided two-coextension of M by $\mathcal{A}$, and suppose that $\mathcal{M}$ is any braided abelian ⊗-groupoid, which is braided ⊗-equivalent to ${\mathcal{M}}^{\prime}$. Then, $\mathcal{M}$ can be endowed with a braided two-coextension structure of M by $\mathcal{A}$, say $p:\mathcal{M}\twoheadrightarrow M$, such that $p:\mathcal{M}\twoheadrightarrow M$ and ${p}^{\prime}:{\mathcal{M}}^{\prime}\twoheadrightarrow M$ are equivalent braided two-co-extensions. □

**Proof.**Let $F=(F,\phi ):\mathcal{M}\to {\mathcal{M}}^{\prime}$ be a braided ⊗-equivalence. Then, a braided two-coextension structure of ${\mathcal{M}}^{\prime}$ is given as follows: let:

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Calvo-Cervera, M.; Cegarra, A.M. A Cohomology Theory for Commutative Monoids. *Mathematics* **2015**, *3*, 1001-1031.
https://doi.org/10.3390/math3041001

**AMA Style**

Calvo-Cervera M, Cegarra AM. A Cohomology Theory for Commutative Monoids. *Mathematics*. 2015; 3(4):1001-1031.
https://doi.org/10.3390/math3041001

**Chicago/Turabian Style**

Calvo-Cervera, María, and Antonio M. Cegarra. 2015. "A Cohomology Theory for Commutative Monoids" *Mathematics* 3, no. 4: 1001-1031.
https://doi.org/10.3390/math3041001