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\begin{document}

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\section*{cohomology group}

Recall that [[cohomology]] in an [[(∞,1)-topos]] $\mathbf{H}$ on an object $X$ with coefficients in an object $A$ is the [[hom-set]] in the [[homotopy category of an (∞,1)-category]]

\begin{displaymath}
H(X,A) = \pi_0 \mathbf{H}(X,A)
  \,.
\end{displaymath}
This is the \emph{cohomology set} .

It is a [[pointed set]] if $A$ is a [[pointed object]].

In the case that $A$ moreover carries the structure of a [[group object]], the set $H(X,A)$ inherits naturally itself the structure of a [[group]]. In this case one speaks of the \emph{cohomology group} of $X$ with coefficients in $A$.

Dually, is this why n-spheres are good for homotopy as they are cogroups? ---David

\hypertarget{examples}{}\section*{{Examples}}\label{examples}

\hypertarget{generalized_abelian_cohomology}{}\section*{{generalized abelian cohomology}}\label{generalized_abelian_cohomology}

In all of what is called [[generalized cohomology]] -- which is really generalized \emph{abelian} cohomology, compare [[nonabelian cohomology]] -- the coefficient object is taken to be not just a [[group object]] but a ``maximally abelian'' group object called a [[stable (infinity,1)-category|stable object]] in general and called a [[spectrum]] in the case that $\mathbf{H}$ = [[Top]].

In that case all the [[delooping hypothesis|deloopings]] $\mathbf{B}^n A$ of $A$ exists and are still stably abelian group objects.

So in that case not only is the cohomology set $H(X,A)$ naturally an abelian group, but there is an infinite sequence of such cohomology groups, one for each delooping $\mathbf{B}^n A$. This yields the traditional notation for graded cohomology groups by setting

\begin{displaymath}
H^n(X,A) := H(X, \mathbf{B}^n A)
  \,.
\end{displaymath}
\hypertarget{ordinary_integraleilenbergmac_lane_cohomology}{}\subsection*{{``ordinary'' (integral/Eilenberg-Mac Lane-) cohomology}}\label{ordinary_integraleilenbergmac_lane_cohomology}

The standard example are the \textbf{``ordinary'' cohomology groups} that come from taking $\mathbf{H} =$ [[Top]] or = [[∞Grpd]] (see [[homotopy hypothesis]]) and choosing the coefficient object to be the [[Eilenberg-Mac Lane spectrum]]

\begin{displaymath}
A := \mathbf{B} \mathbb{Z}
  \,.
\end{displaymath}
The for $X \in \mathbf{H}$ any object (a [[topological space]] or an [[∞-groupoid]]) the ``ordinary'' cohimology of $X$ in degree $n$ is

\begin{displaymath}
H^n(X) := H^n(X,\mathbb{Z})
  :=
  H(X, \mathbf{B}^n \mathbb{Z})
  =
  \pi_0 \mathbf{H}(X, \mathbf{B}\mathbb{Z})
  =: [X, K(n, \mathbb{Z})]
  \,.
\end{displaymath}
Here on the left we have the standard notation for the ordinary cohomology groups, and on the right their expression in terms of homotopy classes of maps into an [[Eilenberg-Mac Lane space]].

\hypertarget{cohomology_groups_in_nonabelian_cohomology}{}\subsection*{{Cohomology groups in nonabelian cohomology}}\label{cohomology_groups_in_nonabelian_cohomology}

The standard \textbf{counter-example} to keep in mind for a [[nonabelian cohomology]] set that does \emph{not} carry a group structure is ``nonabelian cohomology in degree 1'' that classifies $G$-[[principal bundles]], for $G$ some nonabelian group.

This cohomology set

\begin{displaymath}
H^1(X,G)
  := H(X, \mathbf{B}G)
  =:
  [X, \mathbf{B} G]
  \simeq
  G Bund(X)/_\sim
\end{displaymath}
clearly has no natural group structure on it, unless $G$ is in fact abelian (in which case $\mathbf{B}G$ is indeed a [[group object]], namely a [[2-group]]).

But when we pass from group-principal bundles to groupoid-principal bundles, then there may be cohomology sets with group structure even in nonabelian cohomology.

Let for instance $G_{(2)}$ be a [[2-group]], i.e. a [[groupoid]] with group structure, such as the automrophism 2-group $G_{(2)} := AUT(H) := Aut_{Grpd}(\mathbf{B}H)$ of an ordinary group $H$, then there is the nonabelian cohomology set

\begin{displaymath}
H^1(X, G_{(2)}) := H(X, G_{(2)})
  \simeq G_{(2)} GrpdBund(X)/_\sim
  \,.
\end{displaymath}
and this does carry a \emph{nonabelian} (in general) group structure.

This is to be distinguished from the cohomology set

\begin{displaymath}
H^2(X, G_{(2)}) := H(X, \mathbf{B} G_{(2)})
  \simeq G_{(2)} Bund(X)/_\sim
\end{displaymath}
that classifies $G_{(2)}$ [[principal 2-bundles]] as opposed to groupoid principal 1-bundles and which is not in general a group (unless $G_{(2)}$ in turn is sufficiently abelian).

For $G_{(2)} = AUT(H)$ both these cohomology sets play a role in the description of [[gerbes]] (see [[gerbe (as a stack)]] and [[gerbe (in nonabelian cohomology)]]).

[[!redirects cohomology groups]]



\end{document}