group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
This entry largely discusses Schreier theory of nonabelian group extensions – but from the nPOV.
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 $k \in \mathbb{N}$ of a group $G$ with coefficients in an abelian group $K$ is the set of equivalence classes of morphisms
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 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 $\mathbf{B}G$ with coefficients in such approximations.
More precisely, 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
But for every group $K$ there is also its automorphism 2-group $AUT(K)$. Its delooping corresponds to the crossed complex
where the boundary map $\delta$ is the one that sends an element $k \in K$ to the automorphism $Ad(k) : k' \mapsto k k' k^{-1}$.
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
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 $\mathbf{B}G \to \mathbf{B}^n K$ we may associate its fibration sequence
(where both squares are homotopy pullback squares). In particular, for $n = 2$ we get ordinary central extensions
which may be looped to yield exact sequences of morphisms of groups
In Schreier theory one notices that similarly nonabelian group cohomology in degree 2 classifies nonabelian group extensions, i.e. sequences
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
where the double slashes denote action groupoids (and ${*}//G = \mathbf{B}G$).
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 $\mathbf{B}AUT(AUT(K))$.
And so on.
We work out in detail what nonabelian group cocycles, such as morphisms
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 of nonabelian group cohomology on $G$ with coefficients in $K$ are given by the following data:
a map $\psi : G \to Aut(K)$;
a map $\chi : G \times G \to K$
subject to the constraint that for all $g_1, g_2 \in G$ we have
and subject to the cocycle condition that for all $g_1, g_2, g_3 \in G$ we have
Use the identification of $\mathbf{B}AUT(K)$ with its crossed module $(A \stackrel{Ad}{\to} Aut(K))$ 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
The cocycle condition is the fact that this assignment has to make all tetrahedras commute (since there are only trivial k-morphisms with $k \geq 3$ in $\mathbf{B}AUT(K)$):
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.
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.
Cohomology classes of nonabelian 2-cocycles $(\psi, \chi) : \mathbf{B}G \to \mathbf{B}AUT(K)$ are in bijection with equivalence classes of extensions
In fact, we claim a bit more: we claim that the fibration sequence to the left defined by the cocycle $(\psi, \chi) : \mathbf{B}G \to \mathbf{B}AUT(K)$ is
where
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
To see this, we compute the homotopy pullback
as the ordinary pullback
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
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 are pairs
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
Translating these diagrams into forumas using the convention LB as described at strict 2-group – in terms of crossed modules yields the given formulas.
Given two 2-cocycles
a homotopy (coboundary) between them is a transformation
Its components
are given in terms of group elements by
$\lambda(\bullet) \in Aut(K)$
$\{\lambda(g) \in K | g \in G\}$
such that
The naturality condition on this datat is that for all $g_1, g_2 \in G$ we have
In terms of the conventionl LB at strict 2-group – in terms of crossed modules, this is equivalent to the equation
Compare this to the discussion of 2-coboundaries of extensions at group extension.
When the groups in question are Lie groups, there is an infinitesimal version of nonabelian group cohomology:
See there for details.