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nonabelian group cohomology

This entry largely discusses Schreier theory of nonabelian group extensions – but from the nPOV.

Idea and Definition

By the general nonsense of cohomology, the abelian group cohomology in degree $k\in \mathbb{N}$ of a group $G$ with coefficients in an abelian group $K$ is the set of equivalence classes of morphisms

$$H^n(G,K)=\{BG\to B^{n}K{\}}_{\sim }$$H^n(G,K) = \{ \mathbf{B}G \to \mathbf{B}^n K \}_\sim

in the (∞,1)-category ∞ Grpd, from the delooping $BG$ of $G$ to the $n$-fold delooping $B^{n}K$ of $K$.

However, if the group $K$ is not abelian, then its $n$-fold delooping does not exist for $n\ge 2$, so accordingly the above does not give a prescription for cohomology of $G$ with coeffiecients in a nonabelian group $K$ in degree greater than 1 (and in degree 1 group cohomology it is not very interesting).

But for nonabelian $K$ there are higher groupoids that approximate the non-existing higher deloopings. Nonabelian group cohomology is the cohomology of $BG$ with coefficients in such approximations.

More precisely, notice that for $n=2$ and $K$ abelian, the $n$-fold delooping $B^{2}K$ is the strict 2-groupoid whose corresponding crossed complex is

$$[B^{2}K]=(K\to *\overrightarrow{\to }*).$$[\mathbf{B}^2 K] = \left( K \to {*} \stackrel{\to}{\to} {*} \right) \,.

But for every group $K$ there is also its automorphism 2-group $\mathrm{AUT}(K)$. Its delooping corresponds to the crossed complex

$$[B\mathrm{AUT}(K)]=(K\stackrel{\delta =\mathrm{Ad}}{\to }\mathrm{Aut}(K)\overrightarrow{\to }*),$$[\mathbf{B} AUT(K)] = \left( K \stackrel{\delta = Ad}{\to} Aut(K) \stackrel{\to}{\to} {*} \right) \,,

where the boundary map $\delta$ is the one that sends an element $k\in K$ to the automorphism $\mathrm{Ad}(k):k\prime \mapsto kk\prime k^{-1}$.

So this looks much like $B^{2}K$ (when that exists) only that it has more elements in degree 1.

Accordingly, what is called nonabelian group cohomology of $G$ with coefficients in $K$ is the set of equivalence classes of morphisms

$$H_{\mathrm{nonab}}^2(G,K):=\{BG\to B\mathrm{AUT}(K){\}}_{\sim }.$$H^2_{nonab}(G,K) := \{ \mathbf{B}G \to \mathbf{B}AUT(K) \}_\sim \,.

Notice that when $K$ has nontrivial automorphisms, this differs in general from the ordinary degree 2 abelian group cohomology even if $K$ is abelian.

It is a familiar fact that abelian group cohomology classifies (shifted) central group extensions. This is really nothing but the statement that to a morphism $BG\to B^{n}K$ we may associate its fibration sequence

$$\begin{array}{ccccc}{B}^{n-1}K& \to & \hat{G}& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ *& \to & BG& \to & B^{n}K\end{array}$$\array{ \mathbf{B}^{n-1} K& \to&\hat G &\to& {*} \\ \downarrow &&\downarrow && \downarrow \\ {*}& \to& \mathbf{B}G &\to& \mathbf{B}^n K }

(where both squares are homotopy pullback squares). In particular, for $n=2$ we get ordinary central extensions

$$BK\to B\hat{G}\to BB.$$\mathbf{B}K \to \mathbf{B}\hat G \to \mathbf{B}B \,.

which may be looped to yield exact sequences of morphisms of groups

$$K\to \hat{G}\to B.$$K \to \hat G \to B \,.

In Schreier theory one notices that similarly nonabelian group cohomology in degree 2 classifies nonabelian group extensions, i.e. sequences

$$K\to \hat{G}\to G.$$K \to \hat G \to G \,.

As we shall discuss below, by following the abstract nonsense as described above, nonabelian degree 2 cocycles really classify something slightly richer, namely exact sequences of groupoids

$$\mathrm{Aut}(K)//K\to \mathrm{Aut}(K)//\hat{G}\to *//G,$$Aut(K)//K \to Aut(K)//\hat G \to {*}//G \,,

where the double slashes denote action groupoids (and $*//G=BG$).

In the existing literature – which does not usually present the picture quite in the way we are doing here – nonabelian group cohomology is rarely considered beyond degree. But the picture does straightforwardly generalize. For instance degree 3 nonabelian cohomology of $G$ with coefficients in $K$ may be taken to be the cohomology of $BG$ with coefficients in the 3-groupoid $B\mathrm{AUT}(\mathrm{AUT}(K))$.

$$H_{\mathrm{nonab}}^3(G,K)=\{BG\to B\mathrm{AUT}(\mathrm{AUT}(K)){\}}_{\sim }.$$H^3_{nonab}(G,K) = \{\mathbf{B}G \to \mathbf{B}AUT(AUT(K))\}_\sim \,.

And so on.

Details

We work out in detail what nonabelian group cocycles, such as morphisms

$$BG\to B\mathrm{AUT}(K)$$\mathbf{B}G \to \mathbf{B}AUT(K)

correspond to in terms of claassical group data, using the relation between strict 2-groups and crossed modules that is spelled out in detail at strict 2-group – in terms of crossed modules.

For making the translation we follow the convention LB there.

Degree 2 cocycles

Proposition

Degree 2 cocycles of nonabelian group cohomology on $G$ with coefficients in $K$ are given by the following data:

• a map $\psi :G\to \mathrm{Aut}(K)$;

• a map $\chi :G\times G\to K$

• subject to the constraint that for all $g_1,g_2\in G$ we have

  $$\psi (g_1{g}_2)=\mathrm{Ad}(\chi (g_1,g_2))\psi (g_2)\psi (g_1).$$\psi(g_1 g_2) = Ad(\chi(g_1, g_2)) \psi(g_2) \psi(g_1) \,.

• and subject to the cocycle condition that for all $g_1,g_2,g_3\in G$ we have

  $$\chi (g_1{g}_2,g_3)\psi (g_3)(\xi (g_1,g_2))=\chi (g_1,g_2{g}_3)\chi (g_2,g_3)$$\chi(g_1 g_2, g_3) \psi(g_3)(\xi(g_1,g_2)) = \chi(g_1, g_2 g_3) \chi(g_2, g_3)

Proof

Use the identification of $B\mathrm{AUT}(K)$ with its crossed module $(A\stackrel{\mathrm{Ad}}{\to }\mathrm{Aut}(K))$ in the convention L B as described at strict 2-group – in terms of crossed modules to translate the relevant diagrams – which are of the sort spelled out in great detail at group cohomology: the first three items of the above describe the maps

$$(\psi ,\chi ):\left(\begin{array}{ccc}& & •\\ & {}^{g_1}\nearrow & {\Downarrow }^{=}& {\searrow }^{g_2}\\ •& & \stackrel{g_1{g}_2}{\to }& & •\end{array}\right)\mapsto \left(\begin{array}{ccc}& & •\\ & {}^{\psi (g_1)}\nearrow & {\Downarrow }^{\chi (g_1,g_2)}& {\searrow }^{\psi (g_2)}\\ •& & \stackrel{\psi (g_1{g}_2)}{\to }& & •\end{array}\right).$$(\psi, \chi) : \left( \array{ && \bullet \\ & {}^{\mathllap{g_1}}\nearrow & \Downarrow^{\mathrlap{=}}& \searrow^{\mathrlap{g_2}} \\ \bullet &&\stackrel{g_1 g_2}{\to} && \bullet } \right) \;\;\; \mapsto \;\;\; \left( \array{ && \bullet \\ & {}^{\mathllap{\psi(g_1)}}\nearrow & \Downarrow^{\mathrlap{\chi(g_1,g_2)}}& \searrow^{\mathrlap{\psi(g_2)}} \\ \bullet &&\stackrel{\psi(g_1 g_2)}{\to} && \bullet } \right) \,.

The cocycle condition is the fact that this assignment has to make all tetrahedras commute (since there are only trivial k-morphisms with $k\ge 3$ in $B\mathrm{AUT}(K)$):

$$\begin{array}{ccccc}•& & \stackrel{\psi (g_2)}{\to }& & •\\ \uparrow & \Downarrow {}^{\chi (g_1,g_2)}& & & \downarrow \\ {}^{\psi (g_1)}\uparrow & & {}^{\psi (g_1{g}_2)}\nearrow & & {\downarrow }^{\psi (g_3)}\\ \uparrow & & & {}^{\chi (g_1{g}_2,g_2)}\Downarrow & \downarrow \\ •& & \stackrel{\psi (g_3)}{\to }& & •\end{array}=\begin{array}{ccccc}•& & \stackrel{\psi (g_2)}{\to }& & •\\ \uparrow & & & {}^{\chi (g_2,g_3)}\Downarrow & \downarrow \\ {}^{\psi (g_1)}\uparrow & & {\searrow }^{\psi (g_2{g}_3)}& & {\downarrow }^{\psi (g_3)}\\ \uparrow & \Downarrow {}^{\chi (g_1,g_2{g}_3)}& & & \downarrow \\ •& & \stackrel{\psi (g_3)}{\to }& & •\end{array}$$\array{ \bullet &&\stackrel{\psi(g_2)}{\to}&& \bullet \\ \uparrow & \Downarrow{}^{\mathrlap{\chi(g_1, g_2)}} &&& \downarrow \\ {}^{\mathllap{\psi(g_1)}}\uparrow &&{}^{\mathllap{\psi(g_1 g_2)}}\nearrow&& \downarrow^{\mathrlap{\psi(g_3)}} \\ \uparrow &&& {}^{\mathllap{\chi(g_1 g_2, g_2)}}\Downarrow & \downarrow \\ \bullet &&\stackrel{\psi(g_3)}{\to}&& \bullet } \;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\; \array{ \bullet &&\stackrel{\psi(g_2)}{\to}&& \bullet \\ \uparrow &&& {}^{\mathllap{\chi(g_2, g_3)}} \Downarrow & \downarrow \\ {}^{\mathllap{\psi(g_1)}}\uparrow &&\searrow^{\mathrlap{\psi(g_2 g_3)}}&& \downarrow^{\mathrlap{\psi(g_3)}} \\ \uparrow & \Downarrow{}^{\mathrlap{\chi(g_1 , g_2 g_3)}} &&& \downarrow \\ \bullet &&\stackrel{\psi(g_3)}{\to}&& \bullet }

Remark

Precisely the same kind of “twisted” cocycles appear as the cocycles of nonabelian gerbes and principal 2-bundles: for a $K$-gerbe these are cocycles with coefficients in $B\mathrm{AUT}(K)$ but on a domain that is the discrete groupoid given by the given base space.

Extensions classified by degree 2-cocycles

The following statement is classically the central statement of Schreier theory. We state and prove it in the abstract nonsense context of general cohomology, where the things classified by a cocycle are nothing but its homotopy fibers.

Proposition

Cohomology classes of nonabelian 2-cocycles $(\psi ,\chi ):BG\to B\mathrm{AUT}(K)$ are in bijection with equivalence classes of extensions

$$K\to \hat{G}\to G$$K \to \hat G \to G

Proof

In fact, we claim a bit more: we claim that the fibration sequence to the left defined by the cocycle $(\psi ,\chi ):BG\to B\mathrm{AUT}(K)$ is

$$\cdots \to \mathrm{Aut}(K)\to \mathrm{Aut}(K)//K\to \mathrm{Aut}(K)//\hat{G}\to BG\stackrel{(\psi ,\xi )}{\to }B\mathrm{AUT}(K),$$\cdots \to Aut(K) \to Aut(K)//K \to Aut(K)//\hat G \to \mathbf{B}G \stackrel{(\psi,\xi)}{\to} \mathbf{B}AUT(K) \,,

where

$$\hat{G}:=K{\times }_{(\psi ,\chi )}G$$\hat G := K \times_{(\psi,\chi)} G

is the twisted product of $K$ with $G$, using the maps $\chi$ and $\psi$, i.e. the group whose underlying set is the cartesian product $K\times G$ with multiplication given by

$$(k_1,g_1)(k_2,g_2)=(\chi (g_1,g_2)\psi (g_2)(k_1)k_2,g_1{g}_2).$$(k_1, g_1) (k_2, g_2) = \left( \chi(g_1,g_2) \psi(g_2)(k_1) k_2 \;\; , \;\; g_1 g_2 \right) \,.

To see this, we compute the homotopy pullback

$$\begin{array}{ccc}\mathrm{Aut}(K)//\hat{G}& \to & *\\ \downarrow & & \downarrow \\ BG& \stackrel{(\psi ,\chi )}{\to }& B\mathrm{AUT}(K)\end{array}$$\array{ Aut(K)//\hat G & \to & {*} \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{(\psi,\chi)}{\to}& \mathbf{B}AUT(K) }

as the ordinary pullback

$$\begin{array}{ccc}\mathrm{Aut}(K)//\hat{G}& \to & E\mathrm{AUT}(K)\\ \downarrow & & \downarrow \\ BG& \stackrel{(\psi ,\chi )}{\to }& B\mathrm{AUT}(K)\end{array}$$\array{ Aut(K)//\hat G & \to & \mathbf{E}AUT(K) \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{(\psi,\chi)}{\to}& \mathbf{B}AUT(K) }

as described at generalized universal bundle. ($E\mathrm{AUT}(K)$ is the universal $\mathrm{AUT}(K)$-principal 2-bundle).

Recall from the discussion there that a morphism in $\mathrm{AUT}(K)$ is a triangle

$$\begin{array}{ccc}& & •\\ & {}^{\alpha }\swarrow & {}^k⇙& {\searrow }^{\beta }\\ •& & \stackrel{\gamma }{\to }& & •\end{array}$$\array{ && \bullet \\ & {}^{\mathllap{\alpha}}\swarrow &{}^\mathrlap{k}\swArrow& \searrow^{\mathrlap{\beta}} \\ \bullet &&\stackrel{\gamma}{\to}&& \bullet }

in $B\mathrm{AUT}(K)$, and composition of morphisms is pasting of these triangles along their vertical edges. 2-morphisms in $E\mathrm{AUT}(K)$ are given by paper-cup pasting diagrams of such triangles in $B\mathrm{AUT}(K)$

Accordingly, the pullback $BG{\times }_{(\psi ,\xi )}E\mathrm{AUT}(K)$ has

• objects are elements of $\mathrm{Aut}(K)$ (this is the bit not seen in the classical picture of Schreier theory, as that doesn’t know about groupoids);

• morphisms are pairs

  $$(k,g):=\left(\begin{array}{ccc}& & •\\ & {}^{\alpha }\swarrow & {}^k⇙& {\searrow }^{\beta }& & & & & \in \mathrm{AUT}(K)\\ •& & \stackrel{\psi (g)}{\to }& & •\\ \\ •& & \stackrel{g}{\to }& & •& & & & \in BG\end{array}\right)$$(k,g) \;\;\; := \left( \array{ && \bullet \\ & {}^{\mathllap{\alpha}}\swarrow &{}^\mathrlap{k}\swArrow& \searrow^{\mathrlap{\beta}} &&&&& \in \mathbf{AUT(K)} \\ \bullet &&\stackrel{\psi(g)}{\to}&& \bullet \\ \\ \bullet &&\stackrel{g}{\to}&& \bullet &&&& \in \mathbf{B}G } \right)

• 2-morphisms (though of as 2-simplexes) take two such triangles $(k_1,g_1)$ and $(k_2,g_2)$ to the pair $(k\prime ,g_1,g_2)$, where $k\prime$ is given by the pasting diagram

  $$\begin{array}{ccc}& & •\\ & \swarrow & \downarrow & \searrow \\ \downarrow & {\Downarrow }^{k_1}& •& {}^{k_2}\Downarrow & \downarrow \\ \downarrow & \nearrow & {\Downarrow }^{\chi (g_1,g_2)}& \searrow & \downarrow \\ •& & \stackrel{}{\to }& & •\end{array}.$$\array{ && \bullet \\ &\swarrow& \downarrow & \searrow \\ \downarrow &\Downarrow^{\mathrlap{k_1}}& \bullet &{}^{\mathllap{k_2}}\Downarrow& \downarrow \\ \downarrow & \nearrow &\Downarrow^{\mathrlap{\chi(g_1, g_2)}} & \searrow & \downarrow \\ \bullet && \stackrel{}{\to} && \bullet } \,.

Translating these diagrams into forumas using the convention LB as described at strict 2-group – in terms of crossed modules yields the given formulas.

Homotopies between 2-cocycles

Given two 2-cocycles

$$({\psi }_1,{\chi }_1),({\psi }_2,{\chi }_2):BG\to B\mathrm{AUT}(K)$$(\psi_1, \chi_1), (\psi_2, \chi_2) : \mathbf{B}G \to \mathbf{B}AUT(K)

a homotopy (coboundary) between them is a transformation

$$\lambda :({\psi }_1,{\chi }_1)\Rightarrow ({\psi }_2,{\chi }_2).$$\lambda : (\psi_1, \chi_1) \Rightarrow (\psi_2, \chi_2) \,.

Its components

$$\lambda :(•\stackrel{g}{\to }•)\mapsto \left(\begin{array}{ccc}•& \stackrel{{\psi }_1(g)}{\to }& •\\ {}^{\lambda (•)}\downarrow & {}^{\lambda (g)}⇙& {\downarrow }^{\lambda (•)}\\ •& \stackrel{{\psi }_2(g)}{\to }& •\end{array}\right)$$\lambda : (\bullet \stackrel{g}{\to} \bullet) \;\; \mapsto \;\; \left( \array{ \bullet &\stackrel{\psi_1(g)}{\to}& \bullet \\ {}^{\mathllap{\lambda(\bullet)}} \downarrow &{}^{\mathllap{\lambda(g)}}\swArrow& \downarrow^{\mathrlap{\lambda(\bullet)}} \\ \bullet &\stackrel{\psi_2(g)}{\to}& \bullet } \right)

are given in terms of group elements by

• $\lambda (•)\in \mathrm{Aut}(K)$

• $\{\lambda (g)\in K\mid g\in G\}$

such that

(1)$$\lambda (•){\psi }_1(g)=\mathrm{Ad}(\lambda (g)){\psi }_2(g)\lambda (•).$$\lambda(\bullet) \psi_1(g) = Ad(\lambda(g)) \psi_2(g) \lambda(\bullet) \,.

The naturality condition on this datat is that for all $g_1,g_2\in G$ we have

$$\begin{array}{ccc}& & •\\ & {}^{{\psi }_1(g_1)}\nearrow & {\Downarrow }^{{\chi }_1(g_1,g_2)}& {\searrow }^{{\psi }_1(g_2)}\\ •& & \stackrel{{\psi }_1(g_2{g}_1)}{\to }& & •\\ {}^{\lambda (•)}\downarrow & & {}^{\lambda (g_1,g_2)}⇙& & {\downarrow }^{\lambda (•)}\\ •& & \stackrel{{\psi }_2(g_2{g}_1)}{\to }& & •\end{array}=\begin{array}{cccc}& {}^{{\psi }_1(g_1)}\nearrow & \downarrow & {\searrow }^{{\psi }_1(g_1)}\\ \downarrow & {}^{\lambda (g_2)}⇙& {\downarrow }^{\lambda (•)}& {}^{\lambda (g_2)}⇙& \downarrow \\ {}^{\lambda (•)}\downarrow & {}^{{\psi }_2(g_1)}\nearrow & {\Downarrow }^{{\chi }_2(g_1,g_2)}& {\searrow }^{{\psi }_2(g_2)}& {\downarrow }^{\lambda (•)}\\ •& & \stackrel{{\psi }_2(g2g_1)}{\to }& & •\end{array}$$\array{ && \bullet \\ & {}^{\mathllap{\psi_1(g_1)}}\nearrow &\Downarrow^{\chi_1(g_1,g_2)}& \searrow^{\mathrlap{\psi_1(g_2)}} \\ \bullet &&\stackrel{\psi_1(g_2 g_1)}{\to}&& \bullet \\ {}^{\mathllap{\lambda(\bullet)}}\downarrow && {}^{\mathllap{\lambda(g_1, g_2)}}\swArrow && \downarrow^{\mathrlap{\lambda(\bullet)}} \\ \bullet &&\stackrel{\psi_2(g_2 g_1)}{\to}&& \bullet } \;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\; \array{ & {}^{\mathllap{\psi_1(g_1)}}\nearrow &\downarrow& \searrow^{\mathrlap{\psi_1(g_1)}} \\ \downarrow &{}^{\lambda(g_2)}\swArrow & \downarrow^{\mathrlap{\lambda(\bullet)}} &{}^{\lambda(g_2)}\swArrow& \downarrow \\ {}^{\mathllap{\lambda(\bullet)}}\downarrow & {}^{\mathllap{\psi_2(g_1)}} \nearrow & \Downarrow^{\chi_2(g_1,g_2)} & \searrow^{\mathrlap{\psi_2(g_2)}} & \downarrow^{\mathrlap{\lambda(\bullet)}} \\ \bullet &&\stackrel{\psi_2(g2 g_1)}{\to}&& \bullet }

In terms of the conventionl LB at strict 2-group – in terms of crossed modules, this is equivalent to the equation

(2)$$\lambda (g_2{g}_1)\rho (\lambda (•))({\chi }_1(g_1,g_2))={\chi }_2(g_1,g_2)\rho ({\psi }_1(g_2))(\lambda (g_2))\lambda (g_2).$$\lambda(g_2 g_1) \; \rho(\lambda(\bullet))(\chi_1(g_1,g_2)) = \chi_2(g_1, g_2) \; \rho(\psi_1(g_2))(\lambda(g_2)) \; \lambda(g_2) \,.

Compare this to the discussion of 2-coboundaries of extensions at group extension.

Nonabelian Lie algebra cohomology

When the groups in question are Lie groups, there is an infinitesimal version of nonabelian group cohomology:

• nonabelian Lie algebra cohomology

See there for details.