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\section*{nonabelian cohomology}

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

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

[[!include cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{postnikov_decomposition_and_whitehead_principle}{Postnikov decomposition and Whitehead principle}\dotfill \pageref*{postnikov_decomposition_and_whitehead_principle} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{nonabelian_group_cohomology}{Nonabelian group cohomology}\dotfill \pageref*{nonabelian_group_cohomology} \linebreak
\noindent\hyperlink{NonabelianSheafCohomology}{Nonabelian sheaf cohomology with constant coefficients}\dotfill \pageref*{NonabelianSheafCohomology} \linebreak
\noindent\hyperlink{petit_sheaf_topos}{Petit $(\infty,1)$-sheaf $(\infty,1)$-topos}\dotfill \pageref*{petit_sheaf_topos} \linebreak
\noindent\hyperlink{gros_sheaf_topos}{Gros $(\infty,1)$-sheaf $(\infty,1)$-topos}\dotfill \pageref*{gros_sheaf_topos} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{objects_classified_by_nonabelian_cohomology}{Objects classified by nonabelian cohomology}\dotfill \pageref*{objects_classified_by_nonabelian_cohomology} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of [[cohomology]] finds its natural general formulation in terms of [[hom-space]]s in an [[(∞,1)-topos]], as described at [[cohomology]]. Much of the cohomologies which have been traditionally considered, such as [[abelian sheaf cohomology|sheaf cohomology]] turn out to be just a special case of the general situation, for objects which are sufficiently abelian in the sense of [[stable (infinity,1)-category|stable (∞,1)-categories]].

Therefore to amplify that one is looking at general [[cohomology]] without restricting to [[abelian sheaf cohomology|abelian cohomology]] one sometimes speaks of \textbf{nonabelian cohomology}.

\hypertarget{history}{}\subsection*{{History}}\label{history}

It was originally apparently John Roberts who understood (remarkably: while thinking about [[quantum field theory]] in the guise of [[AQFT]]) that general cohomology is about coloring simplices in $\infty$-categories.

\begin{itemize}%
\item John E. Roberts, \emph{Mathematical Aspects of Local Cohomology} talk at Colloqium on Operator Algebras and their Applications to Mathematical Physics, Marseille 20-24 June, 1977 .

\end{itemize}
This is recounted for instance by Ross Street in

\begin{itemize}%
\item [[Ross Street]], \emph{Categorical and combinatorial aspects of descent theory} (\href{http://arxiv.org/ftp/math/papers/0303/0303175.pdf}{pdf})

\end{itemize}
and

\begin{itemize}%
\item [[Ross Street]], \emph{An australian conspectus on higher categories} (\href{http://www.math.mq.edu.au/~street/Minneapolis.pdf}{pdf}).

\end{itemize}
Parallel to this development of the notion of [[descent|descent and codescent]] there was the development of [[homotopical cohomology theory]] as described in

\begin{itemize}%
\item Kenneth S. Brown, \emph{Abstract Homotopy Theory and Generalized Sheaf Cohomology}, Transactions of the American Mathematical Society, Vol. 186 (1973), 419-458 (\href{http://www.math.uni-hamburg.de/home/schreiber/Abstract%20homotopy%20theory%20and%20generalized%20sheaf%20cohomology.pdf}{pdf})

\end{itemize}
The two approaches are different, but closely related. Their relation is via the notion of [[descent|codescent]].

There is a chain of inclusions

\begin{displaymath}
AbelianGroups \hookrightarrow
  ChainComplexesOfAbelianGroups
  \hookrightarrow
  CrossedComplexes
  \hookrightarrow
  \omega Groupoids
  \hookrightarrow
  \omega Categories
\end{displaymath}
along which one can generalize the coefficient objects of ordinary cohomology. (See [[strict omega-groupoid]], [[strict omega-category]]). Since doing so in particular generalizes abelian groups to nonabelian groups (but goes much further!) this is generally addressed as leading to \emph{nonabelian cohomology}.

Depending on the [[model category|models]] chosen, there are different concrete realizations of nonabelian cohomology.

For instance nonabelian [[Cech cohomology]] played a special role in the motivation of the notion of [[gerbe]]s (see in particular [[gerbe (in nonabelian cohomology)]]), concretely thought of in terms of [[pseudofunctor]]s at least in the context of nonabelian [[group cohomology]], while more abstract (and less explicit) [[homotopy theory]] methods dominate the discussion of [[infinity-stack]]s.

Either way, one obtains a notion of \emph{cohomology on $\infty$-categories with coefficients in $\infty$-catgories}. This is, most generally, the setup of ``[[nonabelian cohomology]]''.

This is conceptually best understood today in terms of [[Higher Topos Theory|higher topos theory]], using [[(infinity,1)-category of (infinity,1)-sheaves|(infinity,1)-categories of (infinity,1)-sheaves]].

This perspective on nonabelian cohomology is discussed for instance in \hyperlink{Toen02}{Toen 02}

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{postnikov_decomposition_and_whitehead_principle}{}\subsubsection*{{Postnikov decomposition and Whitehead principle}}\label{postnikov_decomposition_and_whitehead_principle}

In an [[(∞,1)-topos]] every object has a [[Postnikov tower in an (∞,1)-category]]. This means that in some sense general nonabelian cohomology can be decomposed into nonabelian cohomology in degree 1 and abelian cohomology in higher degrees, twisted by this nonabelian cohomology. This has been called (\hyperlink{Toen02}{To\"e{}n}) the [[Whitehead principle of nonabelian cohomology]].

\hypertarget{special_cases}{}\subsection*{{Special cases}}\label{special_cases}

\hypertarget{nonabelian_group_cohomology}{}\subsubsection*{{Nonabelian group cohomology}}\label{nonabelian_group_cohomology}

Sometimes the term \emph{nonabelian cohomology} is used in a more restrictive sense. Often people mean [[nonabelian group cohomology]] when they say nonabelian cohomology, hence restricting to the domains to [[group|groups]], which are [[groupoid|groupoids]] with a single object.

This kind of nonabelian cohomology is discussed for instance in

\begin{itemize}%
\item [[John Baez]], [[Mike Shulman]], [[Lectures on n-Categories and Cohomology]] (\href{http://arxiv.org/abs/math.CT/0608420}{arXiv}).

\end{itemize}
That and how ordinary [[group cohomology]] is reproduced from the [[homotopical cohomology theory]] of [[strict omega-groupoid|strict omega-groupoids]] is discussed in detail in chapter 12 of

\begin{itemize}%
\item [[Ronnie Brown]], P. Higgins, R. Sivera, [[nonabelian algebraic topology|Nonabelian algebraic topology]].

\end{itemize}
For more see

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

\end{itemize}
\hypertarget{NonabelianSheafCohomology}{}\subsubsection*{{Nonabelian sheaf cohomology with constant coefficients}}\label{NonabelianSheafCohomology}

For $X$ a [[topological space]] and $A$ an [[∞-groupoid]], the standard way to define the [[nonabelian cohomology]] of $X$ with coefficients in $A$ is to define it as the intrinsic cohomology as seen in [[∞Grpd]] $\simeq$ [[Top]]:

\begin{displaymath}
H(X,A) := \pi_0 Top(X, |A|) \simeq \pi_0 \infty Func(Sing X, A)
  \,,
\end{displaymath}
where $|A|$ is the [[geometric realization]] of $A$ and $Sing X$ the [[fundamental ∞-groupoid]] of $X$.

But both $X$ and $A$ here naturally can be regarded, in several ways, as objects of [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaf (∞,1)-topos]]es $\mathbf{H} = Sh_{(\infty,1)}(C)$ over nontrivial [[(∞,1)-site]]s $C$. The intrinsic cohomology of such $\mathbf{H}$ is a [[nonabelian cohomology|nonabelian sheaf cohomology]]. The following discusses two such choices for $\mathbf{H}$ such that the corresponding nonabelian sheaf cohomology coincides with $H(X,A)$ (for [[paracompact space|paracompact]] $X$).

\hypertarget{petit_sheaf_topos}{}\paragraph*{{Petit $(\infty,1)$-sheaf $(\infty,1)$-topos}}\label{petit_sheaf_topos}

For $X$ a [[topological space]] and $Op(X)$ its [[category of open subsets]] equipped with the canonical structure of an [[(∞,1)-site]], let

\begin{displaymath}
\mathbf{H}
  :=
  Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X))
\end{displaymath}
be the [[(∞,1)-category of (∞,1)-sheaves]] on $X$. The space $X$ itself is naturally identified with the [[terminal object]] $X = * \in Sh_{(\infty,1)}(X)$. This is the [[petit topos]] incarnation of $X$.

Write

\begin{displaymath}
(LConst \dashv \Gamma) :  Sh_{(\infty,1)}(X)
  \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}
  \infty Grpd
\end{displaymath}
be the [[global section]]s terminal [[geometric morphism]].

Under the [[constant (∞,1)-sheaf]] functor $LConst$ an an [[∞-groupoid]] $A \in \infty Grpd$ is regarded as an object $LConst A \in Sh_{(\infty,1)}(X)$.

There is therefore the \emph{intrinsic} [[cohomology]] of the $(\infty,1)$-topos $Sh_{(\infty,1)}(X)$ with coefficients in the [[constant ∞-stack|constant (∞,1)-sheaf]] on $A$

\begin{displaymath}
H'(X,A) := \pi_0 Sh_{(\infty,1)}(X)(X, LConst A)
  \,.
\end{displaymath}
This is [[cohomology with constant coefficients]].

Notice that since $X$ is in fact the [[terminal object]] of $Sh_{(\infty,1)}(X)$ and that $Sh_{(\infty,1)}(X)(X,-)$ is in fact that [[global section]]s functor, this is equivalently

\begin{displaymath}
\cdots \simeq \pi_0 \Gamma LConst A
  \,.
\end{displaymath}
\begin{utheorem}
If $X$ is a [[paracompact space]], then these two definitins of [[nonabelian cohomology]] of $X$ with [[constant ∞-stack|constant coefficients]] $A \in \infty Grpd$ agree:

\begin{displaymath}
H(X,A) := \pi_0 \infty Grpd(Sing X,A)  \simeq Sh_{(\infty,1)}(X)(X,LConst A)
  \,.
\end{displaymath}
\end{utheorem}
This is [[Higher Topos Theory|HTT, theorem 7.1.0.1]]. See also [[(∞,1)-category of (∞,1)-sheaves]] for more.

\hypertarget{gros_sheaf_topos}{}\paragraph*{{Gros $(\infty,1)$-sheaf $(\infty,1)$-topos}}\label{gros_sheaf_topos}

Another alternative is to regard the space $X$ as an object in the [[cohesive (∞,1)-topos]] [[ETop∞Grpd]].

\begin{displaymath}
(\Pi \dashv LConst \dashv \Gamma)
  :
  ETop\infty Grpd
  \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}}
  \infty Grpd
  \,,
\end{displaymath}
with the further [[left adjoint]] $\Pi$ to $LConst$ being the intrinsic [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] functor. The intrinsic [[nonabelian cohomology]] in there also coincides with nonabelian cohomology in [[Top]]; even the full [[cocycle]] [[∞-groupoid]]s are equivalent:

\begin{utheorem}
For [[paracompact space|paracompact]] $X$ we have an equivalence of [[cocycle]] [[∞-groupoid]]s

\begin{displaymath}
ETop\infty Grpd(X, LConst A)
  \simeq
  Top(X, |A|)
\end{displaymath}
and hence in particular an isomorphism on cohomology

\begin{displaymath}
H(X,A) \simeq \pi_0   ETop\infty Grpd(X, LConst A)
\end{displaymath}
\end{utheorem}
\begin{proof}
See [[ETop∞Grpd]].

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[Cohomotopy]]

\end{itemize}
\hypertarget{objects_classified_by_nonabelian_cohomology}{}\subsection*{{Objects classified by nonabelian cohomology}}\label{objects_classified_by_nonabelian_cohomology}

For $g : X \to A$ a [[cocycle]] in nonabelian cohomology, we say the [[fibration sequence|homotopy fibers]] of $g$ is the object \emph{classified} by $g$.

For examples and discussion of this see

\begin{itemize}%
\item [[principal bundle]]


\item [[principal 2-bundle]], [[gerbe]]


\item [[principal 3-bundle]], [[2-gerbe]]


\item [[principal ∞-bundle]], [[∞-gerbe]]


\item [[twisted form|twisted forms]] .



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

A readable survey on nonabelian cohomology is

\begin{itemize}%
\item [[Bertrand Toën]], \emph{Stacks and Non-abelian cohomology}, lecture at \emph{\href{https://www.msri.org/realvideo/index04.html}{Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory}}, MSRI 2002 (\href{http://www.msri.org/publications/ln/msri/2002/introstacks/toen/1/index.html}{slides}, \href{http://www.msri.org/publications/ln/msri/2002/introstacks/toen/1/meta/aux/toen.ps}{ps}, \href{https://perso.math.univ-toulouse.fr/btoen/files/2015/02/msri2002.pdf}{pdf})

\end{itemize}
A useful motivation is

\begin{itemize}%
\item Nicolas Addington, \emph{Fiber bundles and nonabelian cohomology} (\href{http://pages.uoregon.edu/adding/notes/gstc2007.pdf}{pdf})

\end{itemize}
Early original references include

\begin{itemize}%
\item [[Paul Dedecker]], \emph{Cohomologie de dimension 2 \`a{} coefficients non ab\'e{}liens}, C. R. Acad. Sci. Paris, 247 (1958), 1160--1163;

(with coefficients in certain [[2-group]])


\item [[John Duskin]], \emph{Non-abelian cohomology in a topos}, reprinted as: Reprints in Theory and Applications of Categories, No. 23 (2013) pp. 1-165 (\href{http://www.tac.mta.ca/tac/reprints/articles/23/tr23abs.html}{TAC})


\item [[Paul Dedecker]], A. Frei, \emph{Les relations d'\'e{}quivalence des morphismes de la suite exacte de cohomologie non ab\^e{}lienne}, C. R. Acad. Sci. Paris, 262(1966), 1298-1301


\item [[Paul Dedecker]], \emph{Three dimensional non-abelian cohomology for groups}, Category theory, homology theory and their applications, II (Battelle Institute Conf.) 1969 (\href{http://www.ams.org/mathscinet/pdf/263894.pdf?pg1=IID&s1=55880&vfpref=html&r=15}{MathSciNet})

(with coefficients in certain [[3-group]]s presented by [[crossed square]]s)



\end{itemize}
The standard classical monograph focusing on low-dimensional cases is

\begin{itemize}%
\item [[J. Giraud]], \emph{Cohomologie non ab\'e{}lienne} , Springer (1971)

(aspects of classification of $G$-[[gerbe]]s by cohomology with coefficients in the [[automorphism 2-group]] $AUT(G)$, but imposes extra constraints)



\end{itemize}
The correct definition using crossed modules of sheaves then appeared in

\begin{itemize}%
\item Raymond Debremaeker, \emph{Cohomologie met waarden in een gekruiste groepenschoof op een situs}, PhD thesis, 1976 (Katholieke Universiteit te Leuven). English translation: \emph{Cohomology with values in a sheaf of crossed groups over a site}, arXiv:\href{https://arxiv.org/abs/1702.02128}{1702.02128}


\item [[Larry Breen]], \emph{Bitorseurs et cohomologie non-Ab\'e{}lienne} , The Grothendieck Festschrift: a collection of articles written in honour of the 60th birthday of Alexander Grothendieck, Vol. I, edited P.Cartier, et al., Birkh\"a{}user, Boston, Basel, Berlin, 401-476, (1990)


\item [[Ieke Moerdijk]], \emph{Lie Groupoids, Gerbes, and Non-Abelian Cohomology} (\href{http://www.springerlink.com/content/ul554x3077444545/}{journal})


\item [[Amnon Yekutieli]], Combinatorial descent data for gerbes, Journal of Noncommutative Geometry Volume 8, Issue 4, 2014, pp. 1083–1099, arXiv:1109.1919 (\href{https://www.math.bgu.ac.il/~amyekut/publications/comb-descent/comb-descent.html}{webpage})



\end{itemize}
The classification of [[∞-gerbe]]s is secretly in

\begin{itemize}%
\item [[Matthias Wendt]], \emph{Classifying spaces and fibrations of simplicial sheaves} , Journal of Homotopy and Related Structures 6(1), 2011, pp. 1--38. (\href{http://arxiv.org/abs/1009.2930}{arXiv}) (\href{http://tcms.org.ge/Journals/JHRS/volumes/2011/volume6-1.htm}{published version})

\end{itemize}
see the discussion at [[∞-gerbe]] for more on this.

Carlos Simpson has studied [[nonabelian Hodge theory]].

\begin{itemize}%
\item [[Carlos Simpson]], \emph{The Hodge filtration on nonabelian cohomology} (\href{http://arxiv.org/abs/alg-geom/9604005}{arXiv})


\item [[Carlos Simpson]], \emph{Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology} (\href{http://arxiv.org/abs/alg-geom/9712020}{arXiv})


\item [[Carlos Simpson]], \emph{Algebraic aspects of higher nonabelian Hodge theory} (\href{http://arxiv.org/abs/math/9902067}{arXiv})


\item [[Carlos Simpson]], [[Tony Pantev]], [[Ludmil Katzarkov]], \emph{Nonabelian mixed Hodge structures} (\href{http://arxiv.org/abs/math/0006213}{arXiv})



\end{itemize}
Some links and references can be found at Alsani's descent and category theory \href{http://north.ecc.edu/alsani/descent.html}{page}.

In as far as nonabelian cohomology is nothing but the study of [[(infinity,1)-categorical hom-space|hom-spaces]] between [[∞-stack]]s, see also the references at [[∞-stack]].

The following thesis continues Street's treatment of nonabelian cohomology

\begin{itemize}%
\item [[Alexander Campbell]], \emph{A higher categorical approach to Giraud's non-abelian cohomology}, PhD thesis, Macquarie University 2016 \href{http://hdl.handle.net/1959.14/1261186}{http\char58\char47\char47hdl\char46handle\char46net\char47\char49\char57\char53\char57\char46\char49\char52\char47\char49\char50\char54\char49\char49\char56\char54}

\end{itemize}
[[!redirects non-abelian cohomology]]



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