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. Many 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 Cech 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 Toen 02
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
The classical theory via principal bundles and Lie groups/algebraic groups:
Nicolas Addington, Fiber bundles and nonabelian cohomology, 2007 (pdf)
Stephen Mitchell, around Theorem 7.4 in: Notes on principal bundles and classifying spaces, Lecture Notes. University of Washington, 2011 (pdf)
Jinpeng An. Zhengdong Wang, Nonabelian cohomology with coefficients in Lie groups,Trans. Amer. Math. Soc. 360 (2008), 3019-3040 (doi:10.1090/S0002-9947-08-04278-5)
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 is
Jean Giraud, Cohomologie non abélienne , Springer (1971) (doi:10.1007/978-3-662-62103-5)
(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
Discussion in terms of gerbes:
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) (doi:10.1007/978-0-8176-4574-8_10)
Ieke Moerdijk, Lie Groupoids, Gerbes, and Non-Abelian Cohomology (journal)
Amnon Yekutieli, Combinatorial descent data for gerbes, Journal of Noncommutative Geometry Volume 8, Issue 4, 2014, pp. 1083–1099, arXiv:1109.1919 (webpage)
Alexander Campbell, A higher categorical approach to Giraud’s non-abelian cohomology, PhD thesis, Macquarie University 2016 http://hdl.handle.net/1959.14/1261186
Existence of classifying spaces for principal 2-bundles/nonabelian gerbes:
Discussion of the general theory via principal ∞-bundles and/or ∞-gerbes and/or ∞-stacks:
In a context of nonabelian Hodge theory:
Carlos Simpson, The Hodge filtration on nonabelian cohomology, in: János Kollár, Robert Lazarsfeld, David Morrison (eds.) Algebraic Geometry Santa Cruz 1995, Part 2, Proceedings of Symposia in Pure Mathematics Volume 62.2, AMS 1997 (arXiv:alg-geom/9604005, doi:10.1090/pspum/062.2)
Carlos Simpson, Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology (arXiv:alg-geom/9712020)
Carlos Simpson, Algebraic aspects of higher nonabelian Hodge theory, in: Fedor Bogomolov, Ludmil Katzarkov (eds.), Motives, polylogarithms and Hodge theory, Part II (Irvine, CA, 1998), Int. Press Lect. Ser., 3, II, Int. Press, 2002, 2016, 417-604. (arXiv:math/9902067, ISBN:9781571462909)
Carlos Simpson, Tony Pantev, Ludmil Katzarkov, Nonabelian mixed Hodge structures (arXiv:math/0006213)
Generally:
Bertrand Toën, Stacks and Non-abelian cohomology, lecture at Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory, MSRI 2002 (slides, ps, pdf, pdf)
J. F. Jardine, Z. Luo, Higher principal bundles, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 140, Issue 2 March 2006 , pp. 221-243 (pdf, doi:10.1017/S0305004105008911)
J. F. Jardine, Cocycle categories, In: Baas N., Friedlander E., Jahren B., Østvær P. (eds) Algebraic Topology, Abel Symposia, vol 4. Springer 2009 (arXiv:math/0605198, doi:10.1007/978-3-642-01200-6_8)
Matthias Wendt, Classifying spaces and fibrations of simplicial sheaves , Journal of Homotopy and Related Structures 6(1), 2011, pp. 1–38. (arXiv:1009.2930) (published version)
David Roberts, Danny Stevenson, Simplicial principal bundles in parametrized spaces, New York Journal of Mathematics Volume 22 (2016) 405-440 (arXiv:1203.2460)
Danny Stevenson, Classifying theory for simplicial parametrized groups (arXiv:1203.2461)
Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Principal ∞-bundles – General theory, Journal of Homotopy and Related Structures, Volume 10, Issue 4 (2015), pages 749-801 (doi:10.1007/s40062-014-0083-6, arXiv:1207.0248)
Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Principal ∞-bundles – Presentations, Journal of Homotopy and Related Structures, Volume 10, Issue 3 (2015), pages 565-622 (doi:10.1007/s40062-014-0077-4, arXiv:1207.0249)
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