group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The notion of cohomology finds its natural general formulation in terms of hom-spaces in an (∞,1)-topos, as described at cohomology. Much of the cohomologies which have been traditionally considered, such as 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 (∞,1)-categories.
Therefore to amplify that one is looking at general cohomology without restricting to abelian cohomology one sometimes speaks of nonabelian cohomology.
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.
This is recounted for instance by Ross Street in
and
Parallel to this development of the notion of descent and codescent there was the development of homotopical cohomology theory as described in
The two approaches are different, but closely related. Their relation is via the notion of codescent.
There is a chain of inclusions
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 nonabelian cohomology.
Depending on the models chosen, there are different concrete realizations of nonabelian cohomology.
For instance nonabelian Čech cohomology played a special role in the motivation of the notion of gerbes (see in particular gerbe (in nonabelian cohomology)), concretely thought of in terms of pseudofunctors at least in the context of nonabelian group cohomology, while more abstract (and less explicit) homotopy theory methods dominate the discussion of infinity-stacks.
Either way, one obtains a notion of 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, using (infinity,1)-categories of (infinity,1)-sheaves.
This perspective on nonabelian cohomology is discussed for instance in
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 (Toën) the Whitehead principle of nonabelian cohomology.
Sometimes the term 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 groups, which are groupoids with a single object.
This kind of nonabelian cohomology is discussed for instance in
That and how ordinary group cohomology is reproduced from the homotopical cohomology theory of strict omega-groupoids is discussed in detail in chapter 12 of
For more see
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:
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)-sheaf (∞,1)-toposes $\mathbf{H} = Sh_{(\infty,1)}(C)$ over nontrivial (∞,1)-sites $C$. The intrinsic cohomology of such $\mathbf{H}$ is a 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 $X$).
For $X$ a topological space and $Op(X)$ its category of open subsets equipped with the canonical structure of an (∞,1)-site, let
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
be the global sections 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 intrinsic cohomology of the $(\infty,1)$-topos $Sh_{(\infty,1)}(X)$ with coefficients in the constant (∞,1)-sheaf on $A$
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 sections functor, this is equivalently
If $X$ is a paracompact space, then these two definitins of nonabelian cohomology of $X$ with constant coefficients $A \in \infty Grpd$ agree:
This is HTT, theorem 7.1.0.1. See also (∞,1)-category of (∞,1)-sheaves for more.
Another alternative is to regard the space $X$ as an object in the cohesive (∞,1)-topos ETop∞Grpd.
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 ∞-groupoids are equivalent:
For paracompact $X$ we have an equivalence of cocycle ∞-groupoids
and hence in particular an isomorphism on cohomology
See ETop∞Grpd.
For $g : X \to A$ a cocycle in nonabelian cohomology, we say the homotopy fibers of $g$ is the object classified by $g$.
For examples and discussion of this see
A readable survey on nonabelian cohomology is
A useful motivation is
Early original references include
Paul Dedecker, Cohomologie de dimension 2 à coefficients non abéliens, C. R. Acad. Sci. Paris, 247 (1958), 1160–1163;
(with coefficients in certain 2-group)
John Duskin, Non-abelian cohomology in a topos, reprinted as: Reprints in Theory and Applications of Categories, No. 23 (2013) pp. 1-165 (TAC)
Paul Dedecker, A. Frei, Les relations d’équivalence des morphismes de la suite exacte de cohomologie non abêlienne, C. R. Acad. Sci. Paris, 262(1966), 1298-1301
Paul Dedecker, Three dimensional non-abelian cohomology for groups, Category theory, homology theory and their applications, II (Battelle Institute Conf.) 1969 (MathSciNet)
(with coefficients in certain 3-groups presented by crossed squares)
The standard classical monograph focusing on low-dimensional cases is
J. Giraud, Cohomologie non abélienne , Springer (1971)
(aspects of classification of $G$-gerbes by cohomology with coefficients in the automorphism 2-group $AUT(G)$, but imposes extra constraints)
The correct definition using crossed modules of sheaves then appeared in
Raymond Debremaeker, Cohomologie met waarden in een gekruiste groepenschoof op een situs, PhD thesis, 1976 (Katholieke Universiteit te Leuven). English translation: Cohomology with values in a sheaf of crossed groups over a site, arXiv:1702.02128
Larry Breen, Bitorseurs et cohomologie non-Abélienne , The Grothendieck Festschrift: a collection of articles written in honour of the 60th birthday of Alexander Grothendieck, Vol. I, edited P.Cartier, et al., Birkhäuser, Boston, Basel, Berlin, 401-476, (1990)
Ieke Moerdijk, Lie Groupoids, Gerbes, and Non-Abelian Cohomology (journal)
The classification of ∞-gerbes is secretly in
see the discussion at ∞-gerbe for more on this.
Carlos Simpson has studied nonabelian Hodge theory.
Carlos Simpson, The Hodge filtration on nonabelian cohomology (arXiv)
Carlos Simpson, Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology (arXiv)
Carlos Simpson, Algebraic aspects of higher nonabelian Hodge theory (arXiv)
Carlos Simpson, Tony Pantev, Ludmil Katzarkov, Nonabelian mixed Hodge structures (arXiv)
Some links and references can be found at Alsani’s descent and category theory page.
In as far as nonabelian cohomology is nothing but the study of hom-spaces between ∞-stacks, see also the references at ∞-stack.