nLab
group cohomology

Idea

As described at cohomology, a notion of cohomology exists for every (infinity,1)-topos $H$: for $X$ and $A$ two objects of $H$,

• an $A$-valued cocycle on $X$ is an object in the ∞-groupoid $H(X,A)$;

• a coboundary between two such cocycles is a morphism in $H(X,A)$

• the cohomology classes are the equivalence classes of $H(X,A)$, so that the cohomology set of $A$-valued cohomology on $X$ is

  $$H(X,A):={\Pi }_{0}H(X,A)={\mathrm{Ho}}_H(X,A),$$H(X,A) := \Pi_0 \mathbf{H}(X,A) = Ho_{\mathbf{H}}(X,A) \,,

  where $H$ is the homotopy category of the (∞,1)-category $H$.

Now, ordinary groupoid nonabelian cohomology is the cohomology obtained for $H=$ ∞Grpd $\simeq$ Top: cohomology on ∞-groupoids (or topological spaces) with coefficients in $\mathrm{\infty }$-groupoids.

The various notions of group cohomology are special cases of this:

• Group cohomology with coefficients in a trivial module is the cohomology in $H=$ ∞Grpd for the case that

  • the domain $X=BG$ is a one-object groupoid being the deloopinga group $G$;

  • the coefficient object is $A=B^{n}K$, the strict ω-groupoid coming from the crossed complex that is concentrated in degree $n$, where it is the abelian group $K$ (see Eilenberg-MacLane object for more details).

  For $n\in \mathbb{N}$ One writes

  $$H_{\mathrm{Grp}}^n(G,K):=H(BG,B^{n}K)={\mathrm{Ho}}_{\mathrm{\infty }\mathrm{Grpd}}(BG,B^{n}K)$$H_{Grp}^n(G,K) := H(\mathbf{B}G, \mathbf{B}^n K) = Ho_{\infty Grpd}(\mathbf{B}G , \mathbf{B}^n K)

  for the degree-$n$ group cohomology of $G$ with values in $K$.

• Nonabelian group cohomology is obtained from this by allowing the coefficient object to be of the form

  • $A=B{K}_{(n)}$, for $K_{(n)}$ an arbitrary n-group

  For instance for $K=\mathrm{AUT}(H)$ the automorphism 2-group of a possibly nonabelian group $H$, nonabelian group cohomology classified $H$-extensions of $G$ (see also gerbe (general idea)).

  Details on this case are at nonabelian group cohomology

• Group cohomology with coefficients in a nontrivial module is in turn twisted cohomology version of nonabelian group cohomology:

  • let $A:=B_{\rho }^{n}K$ be a strict ω-groupoid coming from a crossed complex of the form

    $$[B_{\rho }^{n}K]:=(\cdots \to *\to *\to K\to \cdots \to *\to *\to G\overrightarrow{\to }*)$$[\mathbf{B}^n_\rho K] := ( \cdots \to {*} \to {*} \to K \to \cdots \to {*} \to {}* \to G \stackrel{\to}{\to}{*})

    with the abelian group $K$ in degree $n$ and for

    $$\rho :G\to \mathrm{Aut}(K)$$\rho : G \to Aut(K)

    the action of $G$ on $K$ required by the structure of a crossed complex;

  The $n$th group cohomology of $G$ with coefficients in the module $(K,\rho )$ is the connected components of the $\mathrm{\infty }$-groupoid of sections $\sigma$

  $$\begin{array}{ccc}& & B_{\rho }^{n}K\\ & {}^{\sigma }\nearrow & \downarrow \\ BG& \to & BG\end{array}.$$\array{ && \mathbf{B}^n_\rho K \\ & {}^{\sigma}\nearrow & \downarrow \\ \mathbf{B}G &\to& \mathbf{B}G } \,.

  This is an example of twisted cohomology, as explained there.

Examples

We spell out in detail how the above reproduces the ordinary definition of group cohomology.

For the case of ordinary abelian group cohomology, the context of strict omega-groupoids is in principle fully sufficient, since the domain object $BG$ in that case is a 1-groupoid, clearly a strict infinity-groupoid, as are the abelian coefficient $n$-groupoids $B^{n}K$, manifestly so as images of crossed complexes under the equivalence of crossed complexes with strict omega-groupoids.

So one possibility is to model ${\mathrm{Ho}}_{\mathrm{\infty }\mathrm{Grpd}}$ in this case as the homotopy category induced by the model structure on strict omega-groupoids.

This is, more or less implicitly, the route taken in chapter 12 of

• Ronnie Brown, Phil Higgins?, R. Sivera, Nonabelian algebraic topology (pdf, web)

Since every $\mathrm{\infty }$-groupoid is fibrant, this model category category of strict $\mathrm{\infty }$-groupoids is in fact a category of fibrant objects and hence the hom-sets in its homotopy category may be computed as colimits over $\mathrm{\infty }$-anafunctors, namely

$$H(BG,B^{n}K)={\mathrm{colim}}_{Y\stackrel{\simeq }{\to }>BG}\mathrm{Hom}(Y,B^{n}K)$$H(\mathbf{B}G, \mathbf{B}^n K) = colim_{Y \stackrel{\simeq}{\to}\gt \mathbf{B}G} Hom(Y,\mathbf{B}^n K)

where the colimit is over all strict $\omega$-groupoids $Y$ with $Y\to BG$ an acyclic fibration, which here is a k-surjective functor for all $k$.

On the other hand, since also the full model structure is around, this colimit localizes on the cofibrant replacement $Y=F(N(BG))$ of $BG$. But this is nothing but the free strict $\omega$-groupoid on the nerve of $BG$, which is the usual bar resolution of $G$ (see the discusson at nerve):

$$N(BG)=(\cdots \to G\times G\times G\overrightarrow{\overrightarrow{\to }}G\times G\overrightarrow{\to }G\to *)$$N (\mathbf{B}G) = \left( \cdots \to G \times G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \times G \stackrel{\to}{\to} G \to {*} \right)

This is of course nothing but the incarnation of $BG$ as an object in the category of weak infinity-groupoids modeled as Kan complexes.

For instance the 2-cells in $N(BG)$ are of the form

$$N(BG{)}_2=\{\begin{array}{ccc}& & *\\ & {}^{g_1}\nearrow & & {\searrow }^{g_2}\\ *& & \stackrel{g_1{g}_2}{\to }& & \end{array}\mid g_1,g_2\in G\},$$N(\mathbf{B}G)_2 = \left\{ \left. \array{ && {*} \\ & {}^{g_1}\nearrow && \searrow^{g_2} \\ {*} &&\stackrel{g_1 g_2}{\to}&& } \right| g_1, g_2 \in G \right\} \,,

where the diagram indicates what the face maps on $N(BG)=G\times G$ are.

Accordingly, the 3-cells look like

$$N(BG{)}_3=\{\begin{array}{ccccc}*& & \stackrel{g_2}{\to }& & *\\ {\uparrow }^{g_1}& & {}^{g_1{g}_2}\nearrow & & {\downarrow }^{g_3}\\ *& & \stackrel{g_1{g}_2{g}_3}{\to }& & \end{array}\Rightarrow \begin{array}{ccccc}*& & \stackrel{g_2}{\to }& & *\\ {\uparrow }^{g_1}& & {\searrow }^{g_2{g}_3}& & {\downarrow }^{g_3}\\ *& & \stackrel{g_1{g}_2{g}_3}{\to }& & \end{array}\mid g_1,g_2,g_3\in G\}.$$N(\mathbf{B}G)_3 = \left\{ \left. \array{ {*} &&\stackrel{g_2}{\to}&& {*} \\ \uparrow^{g_1} &&{}^{g_1 g_2}\nearrow&& \downarrow^{g_3} \\ {*} &&\stackrel{g_1 g_2 g_3}{\to}&& } \;\;\;\; \Rightarrow \;\;\;\; \array{ {*} &&\stackrel{g_2}{\to}&& {*} \\ \uparrow^{g_1} &&\searrow^{g_2 g_3}&& \downarrow^{g_3} \\ {*} &&\stackrel{g_1 g_2 g_3}{\to}&& } \right| g_1, g_2, g_3 \in G \right\} \,.

The free strict $\omega$-groupoid on $N(BG)$ has as $n$-morphisms the free $n$-groupoids generated from one $n$-oriental per such $n$-simplex in $N(BG)$.

In chapter 12 of Brown-Higgins-Sivera group cocycles are computed as morphisms out of this cofibrant replacement $F(N(BG))$ of the ordinary 1-groupoid $BG$ in the category of strict omega-groupoids. (Or rather, there the equivalent crossed complexes) are used.

Alternatively, we can pass along the inclusion

$$\mathrm{Str}\omega \mathrm{Grpd}\hookrightarrow \mathrm{\infty }\mathrm{Grpd}\hookrightarrow \mathrm{SSet}$$Str \omega Grpd \hookrightarrow \infty Grpd \hookrightarrow SSet

of strict $\mathrm{\infty }$-groupoids into all $\mathrm{\infty }$-groupoids modeled as Kan complexes and compute the homotopy classes of morphisms there. Every Kan complex is already cofibrant (while of course still also being fibrant), so once the situation is interpreted in SSet we can compute group cohomology in terms of ordinary morphisms $N(BG)\to N(B^{n}K)$ without having to resolve further, without having to resort to anafunctors etc. Of course it is the nerve operation involved both in forming the cofibrant replacement in $\mathrm{Str}\mathrm{\infty }\mathrm{Grpd}$ as well as in passing to $\mathrm{SSet}$ that accomplishes the required resolutions in either case.

The upshot of all this is just that the following illustrative pictures may be interpreted either in $\mathrm{Strct}\mathrm{\infty }\mathrm{Grpod}$ or in $\mathrm{SSet}$:

degree-$1$ group cohomology

A degree-one group cocycle $c$, $[c]\in H_{\mathrm{Grp}}^1(G,K)$ is just a functor $c:BG\to BK$. This is a group homomorphism $G\to K$.

Degree-$2$ group cohomology

A degree-2 group cocycle $c$, $[c]\in H_{\mathrm{Grp}}^2(G,K)$ is on 2-cells a map

$$c_2:\left(\begin{array}{ccc}& & *\\ & {}^{g_1}\nearrow & & {\searrow }^{g_2}\\ *& & \stackrel{g_2{g}_1}{\to }& & \end{array}\right)\mapsto \left(\begin{array}{ccc}& & *\\ & {}^{*}\nearrow & {\Downarrow }^{c(g_1,g_2)}& {\searrow }^{*}\\ *& & \stackrel{*}{\to }& & \end{array}\right)$$c_2 \;\; : \;\; \left( \array{ && {*} \\ & {}^{g_1}\nearrow && \searrow^{g_2} \\ {*} &&\stackrel{g_2 g_1}{\to}&& } \right) \;\;\; \mapsto \;\;\; \left( \array{ && {*} \\ & {}^{{*}}\nearrow &\Downarrow^{c(g_1,g_2)}& \searrow^{{*}} \\ {*} &&\stackrel{{*}}{\to}&& } \right)

i.e. a map $c:G\times G\to K$ such that it extends to a morphism on 3-cells:

$$\begin{array}{rl}{c}_3& :(\begin{array}{ccccc}*& & \stackrel{g_2}{\to }& & *\\ {\uparrow }^{g_1}& & {}^{g_2{g}_1}\nearrow & & {\downarrow }^{g_3}\\ *& & \stackrel{g_3{g}_2{g}_1}{\to }& & \end{array}\Rightarrow \begin{array}{ccccc}*& & \stackrel{g_2}{\to }& & *\\ {\uparrow }^{g_1}& & {\searrow }^{g_3{g}_2}& & {\downarrow }^{g_3}\\ *& & \stackrel{g_3{g}_2{g}_1}{\to }& & \end{array})\\ & \mapsto \left(\begin{array}{ccccc}*& & \stackrel{*}{\to }& & *\\ {\uparrow }^{*}& {\Downarrow }^{c(g_1,g_2)}& {}^{*}\nearrow & {\Downarrow }^{c(g_2,g_3)}& {\downarrow }^{*}\\ *& & \stackrel{*}{\to }& & \end{array}\stackrel{\mathrm{Id}}{\Rightarrow }\begin{array}{ccccc}*& & \stackrel{*}{\to }& & *\\ {\uparrow }^{*}& {\Downarrow }^{c(g_1,g_2{g}_3)}& {\searrow }^{*}& {\Downarrow }^{c(g_2,g_3)}& {\downarrow }^{*}\\ *& & \stackrel{*}{\to }& & \end{array}\right)\end{array}.$$\begin{aligned} c_3 \;\;\; &: \;\;\; \left( \array{ {*} &&\stackrel{g_2}{\to}&& {*} \\ \uparrow^{g_1} &&{}^{g_2 g_1}\nearrow&& \downarrow^{g_3} \\ {*} &&\stackrel{g_3 g_2 g_1}{\to}&& } \;\;\;\; \Rightarrow \;\;\;\; \array{ {*} &&\stackrel{g_2}{\to}&& {*} \\ \uparrow^{g_1} &&\searrow^{g_3 g_2}&& \downarrow^{g_3} \\ {*} &&\stackrel{g_3 g_2 g_1}{\to}&& } \right) \\ & \mapsto \left( \array{ {*} &&\stackrel{{*}}{\to}&& {*} \\ \uparrow^{{*}} &\Downarrow^{c(g_1,g_2)} &{}^{{*}}\nearrow&\Downarrow^{c(g_2,g_3)}& \downarrow^{{*}} \\ {*} &&\stackrel{{*}}{\to}&& } \;\;\;\; \stackrel{Id}{\Rightarrow} \;\;\;\; \array{ {*} &&\stackrel{{*}}{\to}&& {*} \\ \uparrow^{{*}} &\Downarrow^{c(g_1,g_2 g_3)} &\searrow^{{*}}&\Downarrow^{c(g_2, g_3)}& \downarrow^{{*}} \\ {*} &&\stackrel{{*}}{\to}&& } \right) \end{aligned} \,.

Since there are no non-identity 3-morphisms in $B^{2}K$ (non-degenerate 3-cells in $N(B^{2}K)$) the 3-cell on the right is required to be the identity. Since the composition of the 2-cells on the right is their addition (group multiplication in the abelian group $K$) this says that the assignment $c_2:G\times G\to G$ has to be such that

$$c(g_1,g_2)-c(g_1,g_2\cdot g_3)+c(g_1\cdot g_2,g_3)-c(g_2,g_3)=0$$c(g_1, g_2) - c(g_1, g_2 \cdot g_3) + c(g_1 \cdot g_2, g_3) - c(g_2, g_3) = 0

This expresses the commutativity of the above tetrahedra. And it is indeed the ordinary formula for a cocycle in degree-2 group cohomology.

Degree-$3$ group cohomology

similarly…

Generalizations

Structured group cohomology

Group cohomology depends delicately on the internalization context. A survey of central results in topological and smooth group cohomology is

• Jim Stasheff, Continuous cohomology of groups and classifying spaces , Bull. Amer. Math. Soc. Volume 84, Number 4 (1978), 513-530 (web)

An early reference on the equivalence of local Lie group cohomology and Lie algebra cohomology is

• S. Świerczkowski, Cohomology of group germs and Lie algebras Pacific J. Math 39(2) 1971.

This is also discussed in

• Jean-Luc Brylinski, Differentiable Cohomology of Gauge Groups (arXiv)

Nonabelian group cohomology

If the coefficient group $K$ is nonabelian, its higher deloopings $B^{n}K$ to not exist. But n-groupoids approximating this non-existant delooping do exists. Cohomology of $BG$ with coefficients in these is called nonabelian group cohomology or Schreier theory. See there for more details.

References

Aspects of this general point of view on group cohomology is described for instance in chaper 12 of

• R. Brown, P. Higgins, R. Sivera, Nonabelian algebraic topology (pdf, web)

Much of what is called “nonabelian cohomology” in the existing literature concerns the case of nonabelian group cohomology with coefficients in the automorphism 2-group $\mathrm{AUT}(H)$ of some possibly nonabelian group $H$.

This is the topic of Schreier theory.

A random example for this use of terminology would be

• Roggenkamp, Scott, Automorphisms and nonabelian cohomology (pdf)

For a conceptual discussion of nonabelian group cohomology see

• John Baez, Mike Shulman, Lectures on n-Categories and Cohomology (arXiv)