nLab nonabelian cohomology

Idea

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.

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.

• John E. Roberts, Mathematical Aspects of Local Cohomology talk at Colloqium on Operator Algebras and their Applications to Mathematical Physics, Marseille 20-24 June, 1977 .

This is recounted for instance by Ross Street in

• Ross Street, Categorical and combinatorial aspects of descent theory (pdf)

and

• Ross Street, An australian conspectus on higher categories (pdf).

Parallel to this development of the notion of descent and codescent there was the development of homotopical cohomology theory as described in

• Kenneth S. Brown, Abstract Homotopy Theory and Generalized Sheaf Cohomology, Transactions of the American Mathematical Society, Vol. 186 (1973), 419-458 (pdf)

The two approaches are different, but closely related. Their relation is via the notion of codescent.

There is a chain of inclusions

$$AbelianGroups \hookrightarrow ChainComplexesOfAbelianGroups \hookrightarrow CrossedComplexes \hookrightarrow \omega Groupoids \hookrightarrow \omega Categories$$

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 Toen 02

Properties

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 (Toën) the Whitehead principle of nonabelian cohomology.

Special cases

Nonabelian group 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

• John Baez, Mike Shulman, Lectures on n-Categories and Cohomology (arXiv).

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

• Ronnie Brown, P. Higgins, R. Sivera, Nonabelian algebraic topology.

For more see

• nonabelian group cohomology.

Nonabelian sheaf cohomology with constant coefficients

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:

$$H(X,A) := \pi_0 Top(X, |A|) \simeq \pi_0 \infty Func(Sing X, A) \,,$$

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 $X$).

Petit $(\infty,1)$-sheaf $(\infty,1)$-topos

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

$$\mathbf{H} := Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X))$$

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

$$(LConst \dashv \Gamma) : Sh_{(\infty,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd$$

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 $A$

$$H'(X,A) := \pi_0 Sh_{(\infty,1)}(X)(X, LConst A) \,.$$

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

$$\cdots \simeq \pi_0 \Gamma LConst A \,.$$

Theorem

If $X$ is a paracompact space, then these two definitins of nonabelian cohomology of $X$ with constant coefficients $A \in \infty Grpd$ agree:

$$H(X,A) := \pi_0 \infty Grpd(Sing X,A) \simeq Sh_{(\infty,1)}(X)(X,LConst A) \,.$$

This is HTT, theorem 7.1.0.1. See also (∞,1)-category of (∞,1)-sheaves for more.

Gros $(\infty,1)$-sheaf $(\infty,1)$-topos

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

$$(\Pi \dashv LConst \dashv \Gamma) : ETop\infty Grpd \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd \,,$$

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:

Theorem

For paracompact $X$ we have an equivalence of cocycle ∞-groupoids

$$ETop\infty Grpd(X, LConst A) \simeq Top(X, |A|)$$

and hence in particular an isomorphism on cohomology

$$H(X,A) \simeq \pi_0 ETop\infty Grpd(X, LConst A)$$

Proof

See ETop∞Grpd.

Examples

Examples of generalized nonabelian cohomology (with general classifying spaces):

• unstable topological K-theory

• Cohomotopy

Objects classified by nonabelian cohomology

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

• principal bundle

• principal 2-bundle, gerbe

• principal 3-bundle, 2-gerbe

• principal ∞-bundle, ∞-gerbe

• twisted forms .

Related concepts

• nonabelian Poincaré duality

• nonabelian de Rham cohomology

• differential non-abelian cohomology

• generalized cohomology

• Chern-Weil theory

References

Classical theory – bundles and groups

The classical notion of non-abelian (Čech-)cohomology in degree 1 and its relation to fiber bundles/principal bundles:

• Alexander Grothendieck, Chapter V of: A General Theory of Fibre Spaces With Structure Sheaf, University of Kansas, Report No. 4 (1955, 1958) [pdf, pdf]

• Jean Frenkel, Cohomologie à valeurs dans un faisceau non abélien, C. R. Acad. Se., t. 240 (1955) 2368-2370

• Jean Frenkel, Cohomologie non abélienne et espaces fibrés, Bulletin de la Société Mathématique de France, 85 (1957) 135-220 [numdam:BSMF_1957__85__135_0]

Review:

• Torsten Wedhorn, Torsors and Non-abelian Čech Cohomology, chapter 7 in: Manifolds, Sheaves, and Cohomology, Springer (2016) [doi:10.1007/978-3-658-10633-1]

Review in topological spaces (via classifying spaces):

• 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, pdf]

• Gerd Rudolph, Matthias Schmidt, Thm. 3.5.1 of: Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields, Springer (2017) [doi:10.1007/978-94-024-0959-8]

See also:

• 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]

The case of nonabelian group cohomology:

• James Milne, Section 27.a (of the pdf, not present in the published version) of: Algebraic Groups, Cambridge University Press 2017 (doi:10.1017/9781316711736, webpage, pdf)

Categorified theory – 2-bundles/gerbes

Early discussion, of higher non-abelian cohomology with coefficients in certain 2-groups (implicitly):

• Paul Dedecker, Cohomologie à coefficients non abéliens et espaces fibrés, Bulletins de l’Académie Royale de Belgique 41 (1955) 1132-1146 [persee:barb_0001-4141_1955_num_41_1_69497]

• Paul Dedecker, Cohomologie de dimension 2 à coefficients non abéliens, C. R. Acad. Sci. Paris 247 (1958) 1160-1163 [BnF]

• Paul Dedecker, Sur La Cohomologie Non Abelienne I (Dimension Deux), Canadian Journal of Mathematics 12 (1960) 231-251 [doi:10.4153/CJM-1960-019-7]

and with coefficients in certain 3-groups presented by crossed squares:

• 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)

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

Of algebraic structures:

• René Lavendhomme, J. R. Roisin, Cohomologie non abélienne de structures algèbriques, Journal of Algebra 67 2 (1980) 385-414 [doi:10.1016/0021-8693(80)90168-4]

Discussion in terms of gerbes:

• 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)

• 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)

• John Duskin: Non-abelian cohomology in a topos, reprinted as: Reprints in Theory and Applications of Categories 23 (2013) 1-165 [tac:tr23]

• Ieke Moerdijk: Lie Groupoids, Gerbes, and Non-Abelian Cohomology, K-Theory 28 3 (2003) 207-258 [journal, doi:10.1023/A:1026251115381]

• Amnon Yekutieli, Combinatorial descent data for gerbes, Journal of Noncommutative Geometry 8 4 (2014) 1083–1099 [arXiv:1109.1919, webpage]

• Alexander Campbell, A higher categorical approach to Giraud’s non-abelian cohomology, PhD thesis, Macquarie University (2016) [hdl:1959.14/1261186]

Existence of classifying spaces for principal 2-bundles/nonabelian gerbes:

• John Baez, Danny Stevenson, The Classifying Space of a Topological 2-Group, In: Baas N., Friedlander E., Jahren B., Østvær P. (eds.) Algebraic Topology, Abel Symposia, vol 4. Springer 2009 (arXiv:0801.3843, doi:10.1007/978-3-642-01200-6_1)

General theory – $\infty$-bundles/$\infty$-gerbes

Discussion of the general theory via principal ∞-bundles and/or ∞-gerbes and/or ∞-stacks:

• Paul Glenn, Realization of cohomology classes in arbitrary exact categories, J. Pure Appl. Algebra 25, 1982, no. 1, 33- 105 (doi:10.1016/0022-4049(82)90094-9)

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, Def. 6.0.6 in: 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: Nils Baas, Eric Friedlander, B. Jahren , Arne Østvær (eds.) Algebraic Topology, Abel Symposia, vol 4. Springer 2009 (arXiv:math/0605198, doi:10.1007/978-3-642-01200-6_8)

• Jacob Lurie, Thm. 7.1.0.1 in: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press 2009 (pup:8957, pdf)

• Matthias Wendt, Classifying spaces and fibrations of simplicial sheaves, Journal of Homotopy and Related Structures 6 1 (2011) 1-38 [arXiv:1009.2930, JHRS:6-1]

• David Roberts, Danny Stevenson, Simplicial principal bundles in parametrized spaces, New York Journal of Mathematics Volume 22 (2016) 405-440 (arXiv:1203.2460, nyjm:22-19)

• 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 10 4 (2015) 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 10 3 (2015) 565-622 [doi:10.1007/s40062-014-0077-4, arXiv:1207.0249]

• John F. Jardine: Non-abelian Cohomology, chapter 9 in: Local Homotopy Theory, Springer Monographs in Mathematics (2015) [doi:10.1007/978-1-4939-2300-7_9]

On cohomology operations on components of Whitehead-generalized cohomology theories seen in non-abelian cohomology:

• John Michael Boardman, David Copeland Johnson, W. Stephen Wilson, Unstable Operations in Generalized Cohomology (pdf), in: Ioan Mackenzie James (ed.) Handbook of Algebraic Topology Oxford 1995 (doi:10.1016/B978-0-444-81779-2.X5000-7)

The suggestion that every (pointed) space may be regarded as being the classifying space of a non-abelian cohomology theory, and that this is what defines non-abelian cohomology in general:

• Urs Schreiber: Background fields in twisted differential nonabelian cohomology, talk at Oberwolfach Workshop, June 2009 – Strings, Fields, Topology, Oberwolfach Report 28 (2009) [pdf]

  (in twisted and differential generality)

• Jacob Lurie, Def. 6 in: Nonabelian Poincaré Duality, Lecture 8 in Tamagawa Numbers via Nonabelian Poincare Duality (282y) (2014) [pdf, pdf]

  (in view of nonabelian Poincaré duality)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, The Character Map in Nonabelian Cohomology — Twisted, Differential, Generalized, World Scientific (2023) [arXiv:2009.11909, doi:10.1142/13422]

  (generalizing the Chern-Dold character to nonabelian cohomology in this sense)