nLab nonabelian group cohomology

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

Idea and Definition

As group cohomology of a group $G$ is the cohomology of its delooping $\mathbf{B}G$, so nonabelian group cohomology is the corresponding nonabelian cohomology.

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

$$H^n(G,K) \;\simeq\; \big\{ \mathbf{B}G \to \mathbf{B}^n K \big\}_{/\sim}$$

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

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

But for nonabelian $K$ there are higher groupoids that “approximate” the non-existing higher deloopings, in some sense. Nonabelian group cohomology is the cohomology of $\mathbf{B}G$ with coefficients in such approximations.

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

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

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

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

where the boundary map $\delta$ is the one that sends an element $k \in K$ to the adjoint action automorphism $Ad(k) \,\colon\, k' \mapsto k k' k^{-1}$.

So this looks much like $\mathbf{B}^2 K$ (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

$$H^2_{nonab}(G,K) \;\coloneqq\; \big\{ \mathbf{B}G \to \mathbf{B}AUT(K) \big\}_{/\sim} \,.$$

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. Abstractly, this is really nothing but the statement that to a morphism $\mathbf{B}G \to \mathbf{B}^n K$ we may associate its homotopy fiber, making the fiber sequence

$$\array{ \mathbf{B}^{n-1} K &\longrightarrow& \mathbf{B}\hat G &\to& {*} \\ \big\downarrow &&\downarrow && \big\downarrow \\ {*} &\longrightarrow& \mathbf{B}G &\to& \mathbf{B}^n K }$$

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

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

which may be looped to yield exact sequences of groups

$$K \longrightarrow \hat G \longrightarrow B \,.$$

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

$$K \longrightarrow \hat G \longrightarrow G \,.$$

As we shall discuss below, along the lines of the abstract discussion abive, nonabelian degree=2 cocycles really classify something slightly richer, namely exact sequences of groupoids

$$Aut(K) \sslash K \longrightarrow Aut(K) \sslash \hat G \longrightarrow {*}\sslash G \,,$$

where the double slashes denote homotopy quotients (presented as action groupoids, and so ${*}\sslash G = \mathbf{B}G$ is the delooping groupoid).

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=2. 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 $\mathbf{B}G$ with coefficients in the 3-groupoid $\mathbf{B}AUT\big(AUT(K)\big)$.

$$H^3_{nonab}(G,K) \;\coloneqq\; \Big\{ \mathbf{B}G \to \mathbf{B}AUT\big(AUT(K)\big) \Big\}_{/\sim} \,.$$

and so on.

Details

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

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

correspond to in terms of classical 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:

1. a map $\psi \,\colon\, G \to Aut(K)$;

2. a map $\chi \,\colon\, G \times G \to K$

subject to

1. the constraint that for all $g_1, g_2 \in G$ we have

   $$\psi(g_1 g_2) = Ad\big(\chi(g_1, g_2)\big) \psi(g_2) \psi(g_1) \,.$$

2. the cocycle condition that for all $g_1, g_2, g_3 \in G$ we have

$$\chi(g_1 g_2, g_3) \cdot \psi(g_3)\big(\chi(g_1,g_2)\big) \,=\, \chi(g_1, g_2 g_3) \cdot \chi(g_2, g_3) \,.$$

Proof

Use the identification of $\mathbf{B}AUT(K)$ with its crossed module $\big(A \overset{Ad}{\to} Aut(K)\big)$ 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 encode a map of 2-morphisms

and the cocycle condition encodes that this assignment makes all tetrahedra commute (since there are only trivial k-morphisms with $k \geq 3$ in $\mathbf{B}AUT(K)$):

$$\array{ \bullet &&\stackrel{\psi(g_2)}{\to}&& \bullet \\ \uparrow & \Downarrow{}^{\mathrlap{\chi(g_1, g_2)}} &&& \downarrow \\ {}^{\mathllap{\psi(g_1)}}\uparrow &&{}^{\mathllap{\psi(g_1 g_2)}}\nearrow&& \downarrow^{\mathrlap{\psi(g_3)}} \\ \uparrow &&& {}^{\mathllap{\chi(g_1 g_2, g_2)}}\Downarrow & \downarrow \\ \bullet &&\stackrel{\psi(g_3)}{\to}&& \bullet } \;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\; \array{ \bullet &&\stackrel{\psi(g_2)}{\to}&& \bullet \\ \uparrow &&& {}^{\mathllap{\chi(g_2, g_3)}} \Downarrow & \downarrow \\ {}^{\mathllap{\psi(g_1)}}\uparrow &&\searrow^{\mathrlap{\psi(g_2 g_3)}}&& \downarrow^{\mathrlap{\psi(g_3)}} \\ \uparrow & \Downarrow{}^{\mathrlap{\chi(g_1 , g_2 g_3)}} &&& \downarrow \\ \bullet &&\stackrel{\psi(g_3)}{\to}&& \bullet }$$

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 $\mathbf{B}AUT(K)$ 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 guise of 2-groupoids where the thing classified by a cocycle is nothing but its homotopy fiber.

Proposition

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

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

Proof

In fact, we claim a bit more: we claim that the fibration sequence to the left which is induced by the cocycle $(\psi, \chi) \,\colon\, \mathbf{B}G \to \mathbf{B}AUT(K)$ is

$$\cdots \longrightarrow Aut(K) \longrightarrow Aut(K) \sslash K \longrightarrow Aut(K) \sslash \hat G \longrightarrow \mathbf{B}G \overset{(\psi,\xi)}{\to} \mathbf{B}AUT(K) \,,$$

where

$$\hat G \,\coloneqq\, K \times_{(\psi,\chi)} G$$

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

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

To see this, we compute the homotopy pullback

$$\array{ Aut(K)\sslash\hat G & \to & {*} \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{(\psi,\chi)}{\to}& \mathbf{B}AUT(K) }$$

as the ordinary pullback

$$\array{ Aut(K)\sslash\hat G & \to & \mathbf{E}AUT(K) \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{(\psi,\chi)}{\to}& \mathbf{B}AUT(K) }$$

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

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

$$\array{ && \bullet \\ & {}^{\mathllap{\alpha}}\swarrow &{}^\mathrlap{k}\swArrow& \searrow^{\mathrlap{\beta}} \\ \bullet &&\stackrel{\gamma}{\to}&& \bullet }$$

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

Accordingly, the pullback $\mathbf{B}G \times_{(\psi,\xi)} \mathbf{E}AUT(K)$ has

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

• morphisms $\alpha \to \beta$ are pairs

  (1)$$(k,g) \;\;\; \coloneqq \left( \array{ && \bullet \\ & {}^{\mathllap{\alpha}}\swarrow &{}^\mathrlap{k}\swArrow& \searrow^{\mathrlap{\beta}} &&&&& \in \mathbf{AUT}(K) \\ \bullet &&\stackrel{\psi(g)}{\to}&& \bullet \\ \\ \bullet &&\stackrel{g}{\to}&& \bullet &&&& \in \mathbf{B}G } \right)$$

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

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

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

Homotopies between 2-cocycles

Given two 2-cocycles

$$(\psi_1, \chi_1), (\psi_2, \chi_2) : \mathbf{B}G \to \mathbf{B}AUT(K)$$

a homotopy (coboundary) between them is a transformation

$$\lambda : (\psi_1, \chi_1) \Rightarrow (\psi_2, \chi_2) \,.$$

Its components

$$\lambda \,\colon\, (\bullet \stackrel{g}{\to} \bullet) \;\; \mapsto \;\; \;\; \left( \;\;\;\; \array{ \bullet &\overset{\psi_1(g)}{\longrightarrow}& \bullet \\ {}^{\mathllap{\lambda(\bullet)}} \big\downarrow &\;{}^{\mathllap{\lambda(g)}}\!\swArrow& \big\downarrow^{\mathrlap{\lambda(\bullet)}} \\ \bullet &\overset{\psi_2(g)}{\longrightarrow}& \bullet } \;\;\;\; \right)$$

are given in terms of group elements by

• $\lambda(\bullet) \in Aut(K)$

• $\{\lambda(g) \in K | g \in G\}$

such that

$$\lambda(\bullet) \psi_1(g) = Ad(\lambda(g)) \psi_2(g) \lambda(\bullet) \,.$$

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

$$\array{ && \bullet \\ & {}^{\mathllap{\psi_1(g_1)}}\nearrow &\Downarrow^{\chi_1(g_1,g_2)}& \searrow^{\mathrlap{\psi_1(g_2)}} \\ \bullet &&\stackrel{\psi_1(g_2 g_1)}{\to}&& \bullet \\ {}^{\mathllap{\lambda(\bullet)}}\downarrow && {}^{\mathllap{\lambda(g_1, g_2)}}\swArrow && \downarrow^{\mathrlap{\lambda(\bullet)}} \\ \bullet &&\stackrel{\psi_2(g_2 g_1)}{\to}&& \bullet } \;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\; \array{ & {}^{\mathllap{\psi_1(g_1)}}\nearrow &\downarrow& \searrow^{\mathrlap{\psi_1(g_1)}} \\ \downarrow &{}^{\lambda(g_2)}\swArrow & \downarrow^{\mathrlap{\lambda(\bullet)}} &{}^{\lambda(g_2)}\swArrow& \downarrow \\ {}^{\mathllap{\lambda(\bullet)}}\downarrow & {}^{\mathllap{\psi_2(g_1)}} \nearrow & \Downarrow^{\chi_2(g_1,g_2)} & \searrow^{\mathrlap{\psi_2(g_2)}} & \downarrow^{\mathrlap{\lambda(\bullet)}} \\ \bullet &&\stackrel{\psi_2(g2 g_1)}{\to}&& \bullet }$$

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

(2)$$\lambda(g_2 g_1) \; \rho(\lambda(\bullet))(\chi_1(g_1,g_2)) = \chi_2(g_1, g_2) \; \rho(\psi_1(g_2))(\lambda(g_2)) \; \lambda(g_2) \,.$$

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.

Related concepts

• non-abelian cohomology

• group cohomology

  • nonabelian group cohomology, groupoid cohomology

• group extension

• Lie group cohomology

  • ∞-Lie groupoid cohomology

References

Non-abelian group cohomology in degree 1, as the set of crossed-conjugation-equivalence classes of crossed homomorphisms:

• Philippe Gille, Tamás Szamuely, Def. 2.3.2 in: Central Simple Algebras and Galois Cohomology, Cambridge University Press 2006 (doi:10.1017/CBO9780511607219, pdf)

• James Milne, Sec. 3.k (27.a of the pdf) in: Algebraic Groups, Cambridge University Press 2017 (doi:10.1017/9781316711736, webpage, pdf)

• Groupprops, First cohomology set with coefficients in a non-abelian group