Crossed modules and symmetric cohomology of groups

TL;DR

This paper explores the relationship between third symmetric cohomology of groups and crossed modules, revealing conditions under which group extensions correspond to elements of symmetric cohomology, with implications for 2-group structures.

Contribution

It establishes a connection between symmetric cohomology and crossed modules, extending known results to 2-groups and clarifying the role of inverse-preserving sections.

Findings

Extension of 2-groups corresponds to symmetric cohomology elements with inverse-preserving sections

Links symmetric cohomology to properties of crossed modules and 2-groups

Generalizes Staic and Zarelua's results on symmetric cohomology and abelian extensions

Abstract

This paper links the third symmetric cohomology (introduced by Staic and Zarelua ) to crossed modules with certain properties. The equivalent result in the language of 2-groups states that an extension of 2-groups corresponds to an element of $HS^3$ iff it possesses a section which preserves inverses in the 2-categorical sense. This ties in with Staic's (and Zarelua's) result regarding $HS^2$ and abelian extensions of groups.

Equations

$$H{S}^{\ast }(G,M)=H_{\lambda }^{\ast }(G,M)\oplus H_{\delta }^{\ast }(G,M),$$

$$d^n(\varphi )(g_0,g_1,\cdots ,g_n)$$

$$+\sum _{i=1}^n(-1{)}^i\varphi (g_0,\cdots ,g_{i-2},g_{i-1}{g}_i,g_{i+1},\cdots ,g_n)$$

$$+(-1{)}^{n+1}\varphi (g_0,\cdots ,g_{n-1}).$$

$$({\tau }_i\varphi )(g_1,g_2,g_3,\cdots ,g_n)=\{\begin{array}{ll}-g_1\varphi (g_1^{-1},g_1{g}_2,g_3,\cdots ,g_n),\mathrm{if}i=1,& \\ -\varphi (g_1,\cdots ,g_{i-2},g_{i-1}{g}_i,g_i^{-1},g_i{g}_{i+1},\cdots ,g_n),& 1<i<n,\\ -\varphi (g_1,g_2,g_3,\cdots ,g_{n-1}{g}_n,g_n^{-1}),\mathrm{if}i=n.& \end{array}$$

$${\alpha }^n:H{S}^n(G,M)\to H^n(G,M),n\ge 0.$$

$${\gamma }_n:H_{\lambda }^n(G,M)\to H{S}^n(G,M)$$

$$0\to M\stackrel{i}{\to }K\stackrel{p}{\to }G\to 0$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$s(x)s(y)=i(f(x,y))s(xy).$$

$$\partial {(}^{r}t)=r\partial (t)r^{-1}\mathrm{and}{}^{\partial t}s=ts{t}^{-1},r\in R,t,s\in T.$$

$$\psi \circ \partial ={\partial }^{\prime }\circ \varphi ,\varphi {(}^{r}t)={}^{\psi (r)}\varphi (t),r\in R,t\in T.$$

$$0\to M\to T\stackrel{\partial }{\to }R\to G\to 0,$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\xi :\mathsf{Xext}(G,M)\to H^3(G,M).$$

$$ps(x)=x,s(x)s(y)=\partial (\sigma (x,y))s(xy),x,y\in G.$$

$$f(x,y,z)={}^{s(x)}\sigma (y,z)\sigma (x,yz)\sigma (xy,z{)}^{-1}\sigma (x,y{)}^{-1}$$

$$x_1\varphi (x_2,\cdots ,x_{n+1})+\sum _{k=1}^n(-1{)}^k\varphi (x_1,\cdots x_k{x}_{k+1},\cdots ,x_{n+1})$$

$$+(-1{)}^{n+1}\varphi (x_1,\cdots ,x_n)=0.$$

$$x_k=\{\begin{array}{ll}{g}_1,& k=1,\\ g_1^{-1},& k=2,\\ g_1{g}_2,& k=3,\\ g_{k-1},& k\ge 4.\end{array}$$

$$({\tau }_1\varphi +\varphi )(g_1,\cdots ,g_n)=0.$$

$$x_k=\{\begin{array}{ll}{g}_k,& k\le i,\\ g_i^{-1},& k=i+1,\\ g_i{g}_{i+1},& k=i+2,\\ g_{k-1},& k\ge i+3.\end{array}$$

$$((-1{)}^i{\tau }_i\varphi +(-1{)}^{i+2}\varphi )(g_1,\cdots ,g_n)=0.$$

$$x_k=\{\begin{array}{ll}{g}_k,& k\le n,\\ g_n^{-1},& k=n+1.\end{array}$$

$$((-1{)}^{n-1}{\tau }_n\varphi +(-1{)}^{n+1}\varphi )(g_1,\cdots ,g_n)=0.$$

$$\varphi (x,x^{-1},y)=\varphi (x,y,y^{-1})$$

$$\varphi (x,y,z)=xg(y,z)-g(xy,z)+g(x,yz)-g(x,y).$$

$$g(x,y)=-xg(x^{-1},xy)=-g(xy,y^{-1}).$$

$$g(x,x^{-1})=-g(1,x)=0,$$

$$0=\varphi (x,y,y^{-1})=xg(y,y^{-1})-g(xy,y^{-1})+g(x,1)-g(x,y)$$

$$0=\varphi (x,x^{-1},z)=xg(x^{-1},z)-g(1,z)+g(x,x^{-1}z)+g(x,x^{-1}).$$

$$g(x,y)+g(xy,y^{-1})=0\mathrm{and}g(x,x^{-1}z)+g(x,x^{-1}z)=0.$$

Full text

1. Introduction

Let $G$ be a group and $M$ be a $G$-module. Symmetric cohomology $HS^{*}(G,M)$ was introduced by M. Staic [8] as a variant of classical group cohomology. A. Zarelua’s prior definition [11] of exterior cohomology $H^{*}_{\lambda}(G,M)$ is very closely related to this, as shown in [6]. Namely, we proved in [6] that the symmetric cohomology has a functorial decomposition

[TABLE]

where $H^{i}_{\delta}(G,M)=0$ when $0\leq i\leq 4$ or $M$ has no elements of order two.

There are natural homomorphisms $\alpha^{n}:HS^{n}(G,M)\to H^{n}(G,M)$ (and $\beta^{n}:H^{n}_{\lambda}(G,M)\to H^{n}(G,M)$), which are isomorphisms for $n=0,1$ and a monomorphism for $n=2$. This was shown by Staic in [9]. According to our results in [6], the map $\alpha^{n}$ is an isomorphism for $n=2$ if $G$ has no elements of order two. In this case, $\alpha^{3}$ is a monomorphism. More generally, $\alpha^{n}$ is an isomorphism if $G$ is torsion free.

Classical $H^{2}(G,M)$ classifies extensions of $G$ by $M$. Staic showed in [9] that $HS^{2}(G,M)$ classifies a subclass of extensions of $G$ by $M$, namely those that have a section preserving inversion.

It is also well-known that $H^{3}(G,M)$ classifies the so-called crossed extensions of $G$ by $M$ [4], or equivalently $H^{3}(G,M)$ classifies $2$-groups $\Gamma$, with $\pi_{0}(\Gamma)=G$ and $\pi_{1}(\Gamma)=M$.

The main result of this paper is that for groups $G$ with no elements of order $2$ the group $HS^{3}(G,M)$ classifies crossed extensions of $G$ by $M$ with a certain condition on the section.

There has been interest in the mathematical community in exploring symmetric cohomology. Several papers have already been published on this topic, e.g. [2],[3], [7], [10].

2. Preliminaries

2.1. Preliminaries on symmetric cohomology of groups

Let $G$ be a group and $M$ be a $G$-module. Recall that the group cohomology $H^{*}(G,M)$ is defined as the cohomology of the cochain complex $C^{*}(G,M)$, where the group of $n$-cochains of $G$ with coefficients in $M$ is the set of functions from $G^{n}$ to $M$: $C^{n}(G,M)=\left\{\phi:G^{n}\to M\right\}$ and coboundary map $d^{n}:C^{n}(G,M)\to C^{n+1}(G,M)$ is defined by

[TABLE]

A cochain $\phi$ is called normalised if $\phi(g_{1},\cdots,g_{n})=0$ whenever some $g_{j}=1$. The collection of normalised cochains is denoted by $C^{*}_{N}(G,M)$ and the classical normalisation theorem claims that the inclusion $C^{*}_{N}(G,M)\to C^{*}(G,M)$ induces an isomorphism on cohomology.

In [8] Staic introduced a subcomplex $CS^{*}(G,M)\subset C^{*}(G,M)$, whose homology is known as the symmetric cohomology of $G$ with coefficients in $M$ and is denoted by $HS^{*}(G,M)$. The definition is based on an action of $\Sigma_{n+1}$ on $C^{n}(G,M)$ (for every $n$) compatible with the differential. In order to define this action, it is enough to define how the transpositions $\tau_{i}=(i,i+1)$, $1\leq i\leq n$ act. For $\phi\in C^{n}(G,M)$ one defines:

[TABLE]

Denote by $CS^{n}(G,M)$ the subgroup of the invariants of this action. That is, $CS^{n}(G,M)=C^{n}(G,M)^{\Sigma_{n+1}}$. Staic proved that $CS^{*}(G,M)$ is a subcomplex of $C^{*}(G,M)$ [8], [9] and hence the groups $HS^{*}(G,M)$ are well-defined. There is an obvious natural transformation

[TABLE]

According to [8],[9], $\alpha^{n}$ is an isomorphism if $n=0,1$ and is a monomorphism for $n=2$. For extensive study of the homomorphism $\alpha^{n}$ for $n\geq 2$ we refer to [6].

Denote by $CS^{*}_{N}(G,M)$ the intersection $CS^{*}(G,M)\cap C^{*}_{N}(G,M)$. Unlike to the classical cohomology the inclusion $CS_{N}^{*}(G,M)\to CS^{*}(G,M)$ does not always induces an isomorphism on cohomology. The groups $H_{\lambda}^{*}(G,M)=H^{*}(CS_{N}^{*}(G,M)$ are isomorphic to the so called exterior cohomology of groups introduced by Zarelua in [11]. According to [6] the canonical map

[TABLE]

induced by the inclusion $CS^{*}_{N}(G,M)\to CS^{*}(G,M)$, is an isomorphism if $n\leq 4$, or $M$ has no elements of order two.

2.2. Symmetric extensions and $HS^{2}$

It is a classical fact, that $H^{2}(G,M)$ classifies the extension of $G$ by $M$ and $H^{3}(G,M)$ classifies the crossed extensions of $G$ by $M$. One can ask what objects classify the symmetric cohomology groups $HS^{2}(G,M)$ and $HS^{3}(G,M)$. The answer to this question in the dimension two was given in [9]. The aim of this work is to prove a similar result in the dimension three.

Let $G$ be a group and $M$ be a $G$-module. Recall that an extension of $G$ by $M$ is a short exact sequence of groups

[TABLE]

such that for any $k\in K$ and $m\in M$, one has $ki(m)k^{-1}=i(p(k)m)$. An $s$-section to this extension is a map $s:G\to K$ such that $p\circ s(x)=x$ for all $x\in G$. Let ${\bf Extgr}(G,M)$ be the category whose objects are extensions of $G$ and $M$ and morphisms are commutative diagrams

[TABLE]

The set of connected components of the category ${\bf Extgr}(G,M)$ is denoted by ${\sf Extgr}(G,M)$. It is well-known that there exists a natural map ${\sf Extgr}(G,M)\to H^{2}(G,M),$ which is a bijection. To construct this map, one needs to choose an $s$-section $s$ and then define $f\in C^{2}(G,M)$ by

[TABLE]

One checks that $f$ is a $2$-cocycle and its class in $H^{2}$ is independent of the chosen $s$-section $s$.

Let $0\to M\xrightarrow[]{i}K\xrightarrow[]{p}G\to 0$ be an extension and $s$ be a $s$-section. Then $s$ is called symmetric if $s(x^{-1})=s(x)^{-1}$ holds for all $x\in G$. An extension is called symmetric if it possesses a symmetric $s$-section. The symmetric extensions form a full subcategory ${\bf ExS}(G,M)$ of the category ${\bf Extgr}(G,M)$. The set of connected components of ${\bf ExS}(G,M)$ is denoted by ${\sf ExS}(G,M)$. The main result of [9] claims that the restriction of the bijection ${\sf Extgr}(G,M)\to H^{2}(G,M)$ on ${\sf ExS}(G,M)$ yields a bijection ${\sf ExS}(G,M)\to HS^{2}(G,M)$.

2.3. Crossed Modules

Recall the classical relationship between third cohomology of groups and crossed modules. A crossed module is a group homomorphism $\partial:T\to R$ together with an action of $R$ on $T$ satisfying:

[TABLE]

It follows from the definition that the image $Im(\partial)$ is a normal subgroup of $R$, and the kernel $Ker(\partial)$ is in the center of $T$. Moreover the action of $R$ on $T$ induces an action of $G$ on $Ker(\partial)$, where $G=Coker\ \partial$.

A *morphism * from a crossed module $\partial:T\to R$ to a crossed module $\partial^{\prime}:T^{\prime}\to R^{\prime}$ is a pair of group homomorphisms $(\phi:T\to T^{\prime},\psi:R\to R^{\prime})$ such that

[TABLE]

For a group $G$ and for a $G$-module $M$ one denotes by ${\bf Xext}(G,M)$ the category of exact sequences

[TABLE]

where $\partial:T\to R$ is a crossed module and the action of $G$ on $M$ induced from the crossed module structure coincides with the prescribed one. The morphisms in ${\bf Xext}(G,M)$ are commutative diagrams

[TABLE]

where $(\phi,f)$ is a morphism of crossed modules $(T,G,\partial)\to(T^{\prime},G^{\prime},\partial^{\prime})$. We let ${\sf Xext}(G,M)$ be the class of the connected components of the category ${\bf Xext}(G,M)$. Objects of the category ${\bf Xext}(G,M)$ are called crossed extensions of $G$ by $M$.

It is a classical fact (see for example [4]) that there is a canonical bijection

[TABLE]

The map $\xi$ has the following description. Let $0\to M\to T\xrightarrow[]{\partial}R\xrightarrow[]{p}G\to 0$ be a crossed extension. An $s$-section of it is a pair of maps $(s:G\to R,\sigma:G\times G\to T)$ for which the following hold

[TABLE]

An $s$-section is called normalised if $s(1)=1$ and $\sigma(1,x)=1=\sigma(x,1)$ for all $x\in G$. It is clear that every crossed extension has a normalised s-section. Any (normalised) $s$-section $(s,\sigma)$ gives rise to a (normalised) $3$-cocycle $f\in Z^{3}(G,M)$ defined by

[TABLE]

To make dependence of $f$ on $\sigma$ and $s$ we sometimes write $f_{\sigma}$ or even $f_{s,\sigma}$ instead of $f$. Then the map $\xi$ assigns the class of $f$ in $H^{3}(G,M)$ to the class of $0\to M\to T\xrightarrow[]{\partial}R\xrightarrow[]{p}G\to 0$ in ${\sf Xext}(G,M)$.

3. A characterisation of symmetric cocycles

In this section we prove the following auxilary results, which will be used in the next section.

Lemma 1**.**

i) If $\phi\in CS^{n}_{N}(G,M)$, $n\geq 2$ then $\phi(g_{1},\cdots,g_{n})=0$, whenever $g_{i+1}=g_{i}^{-1}$ for some $1\leq i\leq n-1.$

Proof.

i) By definition we have $(\tau_{i}(\phi)+\phi)(g_{1},\cdots,g_{n})=0$ for any $g_{1},\cdots,g_{n}\in G$. If $g_{i+1}=g_{i}^{-1}$ for some $i$, then $\tau_{i}(\phi)(g_{1},\cdots,g_{n})=0$ by the normalisation condition. Hence $\phi(g_{1},\cdots,g_{i},g_{i}^{-1},\cdots,g_{n})=0$. ∎

The converse in general is not true, however we have the following important fact.

Lemma 2**.**

If $\phi\in C^{n}_{N}(G,M)$ is a cocycle, $n\geq 2$, then $\phi\in CS^{n}_{N}(G,M)$ iff $\phi(g_{1},\cdots,g_{n})=0$, whenever $g_{i+1}=g_{i}^{-1}$ for some $1\leq i\leq n-1.$

Proof.

Thanks to Lemma 1 we need to prove that $\tau_{i}(\phi)+\phi=0$, if $\phi(g_{1},\cdots,g_{n})=0$, whenever $g_{k+1}=g_{k}^{-1}$ for some $1\leq k\leq n-1.$

By assumption

[TABLE]

for any $x_{1},\cdots,x_{n+1}\in G$. First we take

[TABLE]

to obtain

[TABLE]

Next, fix $1<i<n$ and put

[TABLE]

to obtain

[TABLE]

Thus $\tau_{i}\phi+\phi=0$, $1<i<n$.

Finally, we take

[TABLE]

to obtain

[TABLE]

Thus $\tau_{n}\phi+\phi=0$ and Lemma follows. ∎

In particular a cocycle $\phi\in C^{3}_{N}(G,M)$ is symmetric (and hence defines a class in $HS^{3}(G,M)=H^{3}_{\lambda}(G,M)$ iff

[TABLE]

for all $x,y\in G$.

Next, Lemma helps us to distinguish boundary elements in $CS^{3}_{N}(G,M)$.

Lemma 3**.**

Suppose $\phi(x,y,z)$ is a (normalised) symmetric cocycle. Also suppose that it is a coboundary: so there exists a $g(x,y)\in C^{2}_{N}(G,M)$ such that

[TABLE]

Then $g$ is symmetric iff $g(x,x^{-1})=0$ for all $x\in G$.

Proof.

Recall that $g$ is symmetric iff

[TABLE]

If these conditions hold, we can take $y=x^{-1}$ to obtain

[TABLE]

because $g$ is normalised. Conversely, assume $g(x,x^{-1})=0$ for all $x\in G$. Since $\phi$ is symmetric we have

[TABLE]

and

[TABLE]

Since $g$ is normalized we have $g(x,1)=g(1,z)=0$. By assumption, we also have $g(y,y^{-1})=0=g(x,x^{-1})$. Hence,

[TABLE]

Replacing $z$ by $xy$ in the last equality, one gets symmetric conditions on $g$. ∎

4. Third symmetric cohomology and crossed modules

We start with proving the following result, which links symmetric cocycles and crossed modules. Our notations are the same as at the end of Section 2.3.

Proposition 4**.**

The class of $0\to M\to T\xrightarrow[]{\partial}R\xrightarrow[]{p}G\to 0$ in ${\sf Xext}(G,M)$ lies in the image of the composite map

[TABLE]

iff the crossed extension $0\to M\to T\xrightarrow[]{\partial}R\xrightarrow[]{p}G\to 0$ has a normalised $s$-section $(s,\sigma)$ for which the following two identities hold

[TABLE]

Proof.

In fact $f=f_{\sigma,\tau}$ is symmetric iff

[TABLE]

thanks to Lemma 2. By definition of $f$ these conditions are equivalent to

[TABLE]

and

[TABLE]

Since $\sigma(1,-)=1=\sigma(-,1)$, we obtain

[TABLE]

and

[TABLE]

and we are done.

∎

Definition 5**.**

A normalised $s$-section $(s,\sigma)$ of a crossed extension

[TABLE]

is called weakly symmetric if the following identities hold

• i)

$s(x^{-1})=s(x)^{-1}$,

• ii)

$\sigma(x,x^{-1})=1$, $x,y\in G$.

We have the following easy fact.

Lemma 6**.**

Let $G$ be a group which has no elements of order two. Then any crossed extension $0\to M\to T\xrightarrow[]{\partial}R\xrightarrow[]{p}G\to 0$ has a weakly symmetric $s$-section.

Proof.

In this case $G\setminus\{1\}$ is a disjoint union of two element subsets of the form $\{x,x^{-1}\}$, $x\not=1$. Let us choose a representative in each class. If $x$ is a representative, we set $s(x)$ to be an element in $p^{-1}(x)$. We then extend $s$ to whole $G$ by $s(1)=1$ and $s(x^{-1})=s(x)^{-1}$, where $x$ is a representative. We see that for $y=x^{-1}$, one has $s(x)s(y)s(xy)^{-1}=1$. Thus one can choose $\sigma$ with property $\sigma(x,x^{-1})=1$ and lemma follows. ∎

Definition 7**.**

A weakly symmetric $s$-section $(s,\sigma)$ of a crossed extension

[TABLE]

is called symmetric if the following identities hold

• i)

$\sigma(x,y)\cdot\,^{s(x)}\sigma(x^{-1},xy)=1$, $x,y\in G$.

• ii)

$\sigma(x,y)\cdot\sigma(xy,y^{-1})=1$, $x,y\in G$.

A crossed extension $0\to M\to T\xrightarrow[]{\partial}R\xrightarrow[]{p}G\to 0$ is called symmetric if it has a symmetric $s$-section.

Symmetric crossed extensions of $G$ by $M$ form a subset ${\sf XextS}(G,M)$ of the set of ${\sf Xext}(G,M)$.

Theorem 8**.**

Let $G$ be a group which has no elements of order two, then there is a natural bijection

[TABLE]

Proof.

By Lemma 6 any crossed extension has a weakly symmetric $s$-section $(s,\sigma)$. By Proposition 4 the corresponding $3$-cocycle is symmetric if

[TABLE]

Now, if we replace $x$ by $x^{-1}$ in the first identity and then act by $s(x)$, we see that these conditions are exactly ones in the Definirion 7. Hence by Proposition 4, the image of the composite map

[TABLE]

is exactly ${\sf XextS}(G,M)$. On the other hand, since $G$ has no elements of order two, the map $\alpha^{3}$ is injective. This follows from the part ii) of Corollary 4.4 [6], because $HS^{3}=H^{4}_{\lambda}$, thanks to Theorem 3.9 [6]. It follows that the induced map $HS^{3}(G,M)\to{\sf Xext}(G,M)$ is a bijection. ∎

5. Interpretation in terms of 2-groups

For us (strict) 2-groups are group objects in the category of small categories. Thus it is a category $\sf C$ (in fact a groupoid) equipped with a bifunctor $\cdot:\sf C\times C\to C$, $(a,b)\mapsto a\cdot b$ which is strictly associative and satisfies group object axioms. These objects are also known under the name categorical groups and 1-cat-groups see, [5]. Recall the relationship between crossed modules and 2-groups [5]. Let $\partial:T\to R$ be a crossed module. It defines a 2-group $\mathsf{Ca}_{T\to R}$. Objects of $\mathsf{Ca}_{T\to R}$ are elements of $R$. A morphism from $r\in R$ to $r^{\prime}\in R$ is an element $t\in T$ such that

[TABLE]

In this situation we use the notation $r\xrightarrow[]{t}r^{\prime}$. The composite of arrows $r\xrightarrow[]{t}r^{\prime}\xrightarrow[]{t^{\prime}}r^{\prime\prime}$ is $r\xrightarrow[]{t^{\prime}t}r^{\prime\prime}$. It is clear that $r\xrightarrow[]{1}r$ is the identity arrow ${{\sf Id}}_{r}$ in the category $\mathsf{Ca}_{T\to R}$. Any morphism in $\mathsf{Ca}_{T\to R}$ is an isomorphism. The inverse of $r\xrightarrow[]{t}r^{\prime}$ is $r^{\prime}\xrightarrow[]{t^{-1}}r$. As usual we set $M=\mathop{\sf Ker}\nolimits(\partial)$. Observe that any $m\in M$ defines an endomorphism $r\xrightarrow[]{m}r$ of $r\in R$ and conversely, any endomorphism of $r$ has this form.

The bifunctor

[TABLE]

given on objects by the multiplication rule in the group $R$, while on morphisms it is given by

[TABLE]

In particular, we have

[TABLE]

[TABLE]

It is well-known that any 2-group is isomorphic to the 2-group of the form $\mathsf{Ca}_{T\to R}$. For a uniquely defined (up to isomorphism) crossed module $\partial:T\to R$, see for example [5].

In particular crossed modules gives rise to monoidal categories. So, we can consider monoidal functors. We recall the corresponding definition. Let $\sf C$ and $\sf D$ be 2-groups. An $s$-functor $\sf C\to D$ is a pair $(F,\xi)$, where $F:\sf C\to D$ is a functor and $\xi$ is a natural transformation from the composite functor ${\sf C\times C}\xrightarrow[]{\cdot}{\sf C}\xrightarrow[]{F}\sf D$ to the composite funtor ${\sf C\times C}\xrightarrow[]{F\times F}{\sf D\times D}\xrightarrow[]{\cdot}\sf D$ Thus for any objects $x$ and $y$ of $\sf C$ we have a morphism $\xi_{x,y}:F(x\cdot y)\to F(x)\cdot F(y)$, which is natural in $x$ and $y$.

In what follows, we will assume that $(F,\xi)$ is normalised, meaning that $F(1)=1$ and $\xi_{x,y}={\sf Id}$, if $x=1$ or $y=1$. Thus for any object $x$ we have a morphism $\xi(x,x^{-1}):1\to F(x)\cdot F(x^{-1})$, which will be play an important role later.

An $s$-functor is monoidal if for any objects $x,y,z$ of the category $\sf C$ the diagram

[TABLE]

commutes.

Let $0\to M\to T\xrightarrow[]{\partial}R\xrightarrow[]{p}G\to 0$ be a crossed extension. In this situation we have two 2-groups ${\sf Ca}_{T\to R}$ and ${\sf Ca}_{G}$. The second one is $G$ considered as a discrete category (equivalently, the $2$-group, corresponding to the crossed module $1\to G$). The homomorphism $p$ yields the strict monoidal functor ${\sf Ca}_{T\to R}\to{\sf Ca}_{G}$, which is still denoted by $p.$

One can consider sections of $p$. We will ask different level of compatibility of sections with monoidal structures. The weakest condition to ask to such a section is to be a functor. Since ${\sf Ca}$ is a discrete category, we see that such a section of the functor $p$ is nothing but a set section of the map $p:R\to G$. Next, is to ask to the functor ${\sf Ca}_{G}\to{\sf Ca}_{T\to R}$ to be an $s$-functor. Call them $s$-sections of $p$. One easily, observes that there is a one-to-one correspondence between $s$-sections $(F,\xi)$ of the functor $p$ and $s$-sections of a crossed extension $0\to M\to T\xrightarrow[]{\partial}R\xrightarrow[]{p}G\to 0$. In fact, if $(s,\sigma)$ is an $s$-section, then $(F,\xi)$ is an $s$-functor ${\sf Ca}_{G}\to{\sf Ca}_{T\to R}$, for which $F\circ p={\sf Id}_{{\sf Ca}_{G}}$. Here the functor $F$ and natural transformation $\xi$ are defined as follows. Since ${\sf Ca}_{G}$ is a discrete category, the functor $F$ is uniquely determined by the rule:

[TABLE]

Next, the natural transformation $\xi$ is uniquely determined by the family of morphisms

[TABLE]

Even stronger assumption is to ask to the pair $(F,\xi)$ to be a monoidal functor. As the following well-known result shows this condition is a 2-dimensional analogue of a splitting in a short exact sequence.

Proposition 9**.**

The class of a crossed extension

[TABLE]

is zero iff the strict monoidal functor $p:{\sf Ca}_{T\to R}\to{\sf Ca}_{G}$ has a section ${\sf Ca}_{G}\to{\sf Ca}_{T\to R}$, which is monoidal. That is there exists an $s$-section $(s,\sigma)$ of $0\to M\to T\xrightarrow[]{\partial}R\xrightarrow[]{p}G\to 0$, for which the corresponding $s$-functor $(F,\xi)$ is monoidal.

Since we did not find appropriate reference we give the proof.

Proof.

Let $(s,\sigma)$ be an $s$-section of $0\to M\to T\xrightarrow[]{\partial}R\xrightarrow[]{p}G\to 0$. Then the diagram in the definition of the monoidal functor for $(F,\xi)$ has the form

[TABLE]

Thus commutativity of this diagram is equivalent to the vanishing of the 3-cocycle $f$ in (2.3.1). So, if part is done. Conversely, if $f_{\sigma}$ defined in (2.3.1) is a coboundary and

[TABLE]

for a function $k:G^{2}\to M$, then $(s,\tau)$ is also an $s$-section for which $f_{\tau}=0$. Here $\tau(g,h)=\sigma(g,h)-k(g,h)$. ∎

We now introcuce another conditions to an $s$-functor $(F,\xi)$, which are weaker then the monoidal functor.

Definition 10**.**

An $s$-functor $(F,\xi)$ is called symmetric, if

[TABLE]

and

[TABLE]

Then we have the following obvious facts:

• •

Any monoidal functor is symmetric $s$-functor.

• •

The $s$-functor corresponding to an $s$-section $(s,\sigma)$ is symmetric, if $(s,\sigma)$ satisfies the condition listed in Proposition 4.

Thus 2-groups, for which the corresponding class in $H^{3}(G,M)$ lies in the image of $HS^{3}(G,M)$ can be characterized, as those, for which there exists a symmetric $s$-functor ${\sf Ca}_{G}\to{\sf Ca}_{T\to R}$, which is a section of $p:{\sf Ca}_{T\to R}\to{\sf Ca}_{G}$.