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\begin{document}

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\section*{nonabelian group cohomology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

This entry largely discusses [[Schreier theory]] of nonabelian [[group extension]]s -- but from the [[nPOV]].

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea_and_definition}{Idea and Definition}\dotfill \pageref*{idea_and_definition} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{degree_2_cocycles}{Degree 2 cocycles}\dotfill \pageref*{degree_2_cocycles} \linebreak
\noindent\hyperlink{extensions_classified_by_degree_2cocycles}{Extensions classified by degree 2-cocycles}\dotfill \pageref*{extensions_classified_by_degree_2cocycles} \linebreak
\noindent\hyperlink{Homotopy2Cocycle}{Homotopies between 2-cocycles}\dotfill \pageref*{Homotopy2Cocycle} \linebreak
\noindent\hyperlink{nonabelian_lie_algebra_cohomology}{Nonabelian Lie algebra cohomology}\dotfill \pageref*{nonabelian_lie_algebra_cohomology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea_and_definition}{}\subsection*{{Idea and Definition}}\label{idea_and_definition}

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

By the [[category theory|general abstract]] definition of [[cohomology]], the \emph{[[abelian group|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]]

\begin{displaymath}
H^n(G,K) = \{ \mathbf{B}G  \to \mathbf{B}^n K \}_\sim
\end{displaymath}
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 \emph{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

\begin{displaymath}
[\mathbf{B}^2 K]
  = 
  \left(
    K \to {*} \stackrel{\to}{\to} {*}
  \right)
  \,.
\end{displaymath}
But for every [[group]] $K$ there is also its [[2-group|automorphism 2-group]] $AUT(K)$. Its delooping corresponds to the [[crossed complex]]

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

\begin{displaymath}
H^2_{nonab}(G,K) := \{ \mathbf{B}G \to \mathbf{B}AUT(K) \}_\sim
  \,.
\end{displaymath}
Notice that when $K$ has nontrivial [[automorphism]]s, 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 extension]]s. 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]]

\begin{displaymath}
\itexarray{
    \mathbf{B}^{n-1} K& \to&\mathbf{B}\hat G &\to& {*}
    \\
    \downarrow &&\downarrow && \downarrow 
    \\
    {*}& \to& \mathbf{B}G &\to& \mathbf{B}^n K
  }
\end{displaymath}
(where both squares are [[homotopy pullback]] squares). In particular, for $n = 2$ we get ordinary central extensions

\begin{displaymath}
\mathbf{B}K \to \mathbf{B}\hat G \to \mathbf{B}B
  \,.
\end{displaymath}
which may be looped to yield exact sequences of morphisms of groups

\begin{displaymath}
K \to \hat G \to B
  \,.
\end{displaymath}
In [[Schreier theory]] one notices that similarly nonabelian group cohomology in degree 2 classifies nonabelian [[group extension]]s, i.e. sequences

\begin{displaymath}
K \to \hat G \to G
  \,.
\end{displaymath}
As we shall discuss below, by following the [[category theory|abstract nonsense]] as described above, nonabelian degree 2 cocycles really classify something slightly richer, namely exact sequences of groupoids

\begin{displaymath}
Aut(K)//K \to Aut(K)//\hat G \to {*}//G
  \,,
\end{displaymath}
where the double slashes denote [[action groupoid]]s (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 $\mathbf{B}AUT(AUT(K))$.

\begin{displaymath}
H^3_{nonab}(G,K) = \{\mathbf{B}G \to \mathbf{B}AUT(AUT(K))\}_\sim
  \,.
\end{displaymath}
And so on.

\hypertarget{details}{}\subsection*{{Details}}\label{details}

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

\begin{displaymath}
\mathbf{B}G \to \mathbf{B}AUT(K)
\end{displaymath}
correspond to in terms of claassical group data, using the relation between [[strict 2-group]]s and [[crossed module]]s that is spelled out in detail at \href{http://ncatlab.org/nlab/show/strict+2-group#InTermsOfCrossedModules}{strict 2-group -- in terms of crossed modules}.

For making the translation we follow the \textbf{convention LB} there.

\hypertarget{degree_2_cocycles}{}\subsubsection*{{Degree 2 cocycles}}\label{degree_2_cocycles}

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

\begin{itemize}%
\item a map $\psi : G \to Aut(K)$;


\item a map $\chi : G \times G \to K$


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

\begin{displaymath}
\psi(g_1 g_2)
  = 
  Ad(\chi(g_1, g_2)) \psi(g_2) \psi(g_1)  
  \,.
\end{displaymath}

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

\begin{displaymath}
\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)
\end{displaymath}


\end{itemize}
\end{uprop}
\begin{proof}
Use the identification of $\mathbf{B}AUT(K)$ with its [[crossed module]] $(A \stackrel{Ad}{\to} Aut(K))$ in the \emph{convention L B} as described at \href{http://ncatlab.org/nlab/show/strict+2-group#InTermsOfCrossedModules}{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

\begin{displaymath}
(\psi, \chi) : 
  \left(
    \itexarray{
       && \bullet
       \\
       & {}^{\mathllap{g_1}}\nearrow &
       \Downarrow^{\mathrlap{=}}&
       \searrow^{\mathrlap{g_2}}
       \\
       \bullet &&\stackrel{g_1 g_2}{\to} && \bullet
    }
  \right)
  \;\;\;
  \mapsto
  \;\;\;
  \left(
    \itexarray{
       && \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)
  \,.
\end{displaymath}
The cocycle condition is the fact that this assignment has to make all tetrahedras commute (since there are only trivial [[k-morphism]]s with $k \geq 3$ in $\mathbf{B}AUT(K)$):

\begin{displaymath}
\itexarray{
    \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
  }
  \;\;\;\;\;\;\;\;
  =
  \;\;\;\;\;\;\;\;
  \itexarray{
    \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
  }
\end{displaymath}
\end{proof}
\begin{uremark}
Precisely the same kind of ``twisted'' cocycles appear as the cocycles of nonabelian [[gerbe]]s and [[principal 2-bundle]]s: 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.

\end{uremark}
\hypertarget{extensions_classified_by_degree_2cocycles}{}\subsubsection*{{Extensions classified by degree 2-cocycles}}\label{extensions_classified_by_degree_2cocycles}

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 fiber]]s.

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

\begin{displaymath}
K \to \hat G \to G
\end{displaymath}
\end{uprop}
\begin{proof}
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

\begin{displaymath}
\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)
  \,,
\end{displaymath}
where

\begin{displaymath}
\hat G := K \times_{(\psi,\chi)} G
\end{displaymath}
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

\begin{displaymath}
(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)
  \,.
\end{displaymath}
To see this, we compute the [[homotopy pullback]]

\begin{displaymath}
\itexarray{
    Aut(K)//\hat G & \to & {*}
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{(\psi,\chi)}{\to}&
    \mathbf{B}AUT(K)
  }
\end{displaymath}
as the ordinary [[pullback]]

\begin{displaymath}
\itexarray{
    Aut(K)//\hat G & \to & \mathbf{E}AUT(K)
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{(\psi,\chi)}{\to}&
    \mathbf{B}AUT(K)
  }
\end{displaymath}
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

\begin{displaymath}
\itexarray{
     && \bullet
     \\
     & {}^{\mathllap{\alpha}}\swarrow
     &{}^\mathrlap{k}\swArrow& 
     \searrow^{\mathrlap{\beta}}
     \\
     \bullet 
     &&\stackrel{\gamma}{\to}&&
     \bullet
  }
\end{displaymath}
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

\begin{itemize}%
\item 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 [[groupoid]]s);


\item morphisms are pairs

\begin{displaymath}
(k,g)
  \;\;\;
  :=
  \left(
  \itexarray{
    && \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)
\end{displaymath}

\item 2-morphisms (thought of as 2-[[simplex]]es) 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

\begin{displaymath}
\itexarray{
    && \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
  }
  \,.
\end{displaymath}


\end{itemize}
Translating these diagrams into forumas using the \emph{convention LB} as described at \href{http://ncatlab.org/nlab/show/strict+2-group#InTermsOfCrossedModules}{strict 2-group -- in terms of crossed modules} yields the given formulas.

\end{proof}
\hypertarget{Homotopy2Cocycle}{}\subsubsection*{{Homotopies between 2-cocycles}}\label{Homotopy2Cocycle}

Given two 2-cocycles

\begin{displaymath}
(\psi_1, \chi_1), (\psi_2, \chi_2)
  : 
  \mathbf{B}G
  \to 
  \mathbf{B}AUT(K)
\end{displaymath}
a homotopy (coboundary) between them is a [[transfor|transformation]]

\begin{displaymath}
\lambda : (\psi_1, \chi_1) \Rightarrow (\psi_2, \chi_2)
  \,.
\end{displaymath}
Its components

\begin{displaymath}
\lambda
  : 
  (\bullet \stackrel{g}{\to} \bullet)
  \;\;
  \mapsto
  \;\;
  \left(
   \itexarray{
     \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)
\end{displaymath}
are given in terms of group elements by

\begin{itemize}%
\item $\lambda(\bullet) \in Aut(K)$


\item $\{\lambda(g) \in K | g \in G\}$



\end{itemize}
such that

\begin{displaymath}
\lambda(\bullet) \psi_1(g)
  =
  Ad(\lambda(g)) \psi_2(g) \lambda(\bullet)
  \,.
\end{displaymath}
The naturality condition on this datat is that for all $g_1, g_2 \in G$ we have

\begin{displaymath}
\itexarray{
    && \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
  }
  \;\;\;\;\;\;\;\;
  =
  \;\;\;\;\;\;\;\;
  \itexarray{
    & {}^{\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
  }
\end{displaymath}
In terms of the \emph{conventionl LB} at \href{http://ncatlab.org/nlab/show/strict+2-group#InTermsOfCrossedModules}{strict 2-group -- in terms of crossed modules}, this is equivalent to the equation

\begin{equation}
\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)
  \,.
\label{homotopy}\end{equation}
Compare this to the discussion of \href{http://ncatlab.org/nlab/show/group+extension#2Coboundaries}{2-coboundaries of extensions} at [[group extension]].

\hypertarget{nonabelian_lie_algebra_cohomology}{}\subsection*{{Nonabelian Lie algebra cohomology}}\label{nonabelian_lie_algebra_cohomology}

When the groups in question are [[Lie group]]s, there is an [[infinitesimal object|infinitesimal]] version of nonabelian group cohomology:

\begin{itemize}%
\item [[nonabelian Lie algebra cohomology]]

\end{itemize}
See there for details.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[group cohomology]]

\begin{itemize}%
\item \textbf{nonabelian group cohomology}, [[groupoid cohomology]]

\end{itemize}

\item [[group extension]]


\item [[Lie group cohomology]]

\begin{itemize}%
\item [[∞-Lie groupoid cohomology]]

\end{itemize}


\end{itemize}


\end{document}