nLab nonabelian cohomology

Redirected from "nonabelian cohomology theory".
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Cohomology

cohomology

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Special notions

Variants

Extra structure

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Theorems

Contents

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 speaks of nonabelian cohomology.


To say this in a bit more detail:

Cohomology via classifying spaces

It is a classical and yet possibly undervalued fact that reasonable cohomology theories have classifying spaces (and more generally classifying stacks). To quickly recall (more details and pointers in FSS23, §2):

Ordinary cohomology. This begins with the observation that (reduced) ordinary singular cohomology, with coefficients in a discrete abelian group AA, is classified in degree nn by Eilenberg-MacLane spaces K(A,n)K(A,n) – in that on well-behaved topological spaces XX, notably on smooth manifolds, there are natural isomorphisms between the ordinary cohomology groups and the connected components of the respective (pointed) mapping spaces:

(1)H n(X;A)π 0Maps(X,K(A,n)),H˜ n(X;A)π 0Maps */(X,K(A,n)). H^n(X;\, A) \;\simeq\; \pi_0 \, \mathrm{Maps} \big( X,\, K(A,n) \big) \,, \;\;\;\;\;\;\;\;\;\;\; \widetilde H^n(X;\, A) \;\simeq\; \pi_0 \, \mathrm{Maps}^{\ast/\!} \big( X,\, K(A,n) \big) \,.

This equivalence makes manifest the characteristic properties of cohomology: homotopy invariance, exactness and wedge property, since these are now immediately implied by general abstract properties of mapping spaces.

Moreover, these EM-spaces are in fact loop spaces of each other, via weak homotopy equivalences

(2)σ n:K(A,n)ΩK(A,n+1) \sigma_n \;\colon\; K(A,n) \xrightarrow{\;\sim\;} \Omega K(A,n+1)

that thereby represent the suspension isomorphisms between ordinary cohomology groups, as follows:

H˜ n(X;A)Maps */(X,K(A,n))(σ n) *Maps */(X,ΩK(A,n+1))Maps */(ΣX,K(A,n+1))H˜ n+1(ΣX;A). \widetilde H^{n}(X;A) \;\simeq\; \mathrm{Maps}^{\ast/\!}\big( X ,\, K(A,n) \big) \xrightarrow{ (\sigma_n)_\ast } \mathrm{Maps}^{\ast/\!}\big( X ,\, \Omega K(A,n+1) \big) \;\simeq\; \mathrm{Maps}^{\ast/\!}\big( \Sigma X ,\, K(A,n+1) \big) \;\simeq\; \widetilde H^{n+1}\big( \Sigma X ;\, A \big) \,.


Ordinary non-abelian cohomology. Note here that it is the loop space property (2), and hence the corresponding suspension isomorphism, which reflect the fact that the coefficient AA has been assumed to be an abelian group: For a non-abelian group GG, an Eilenberg-MacLane space K(G,1)BGK(G, 1) \,\simeq\, B G still exists (cf. classifying space), but is not a loop space.

While the suspension isomorphism is thus lost for non-abelian coefficients, the assignment

(3)XH 1(X;G)π 0Maps(X,BG)Set */ X \;\; \mapsto \;\; H^1\big( X ;\, G \big) \;\; \coloneqq \;\; \pi_0 \, \mathrm{Maps}\big( X ,\, B G \big) \;\;\; \in\; \mathrm{Set}^{\ast/}

still satisfies homotopy invariance, exactness and wedge property, just as before by the general properties of mapping spaces, and hence has all the characteristic properties of ordinary cohomology – except for its abelian-ness. Accordingly, (3) is known as non-abelian cohomology, famous from early applications in Chern-Weil theory.


Whitehead-generalized cohomology. If or as long as we do insist on abelian cohomology groups related by suspension isomorphisms, we may still immediately generalize ordinary cohomology in the form (1), simply by using any other sequence of classifying spaces (E n) n=0 (E_n)_{n=0}^\infty, being successive loop spaces of each other as in (2),

σ n:E nΩE n+1, \sigma_n \,:\, E_n \xrightarrow{\; \sim \;} \Omega E_{n+1} \,,

as such called a sequential Ω \Omega -spectrum of spaces, or just a spectrum, for short. The Brown representability theorem says that the resulting assignments

XE n(X)π 0Maps(X,E n) X \;\; \mapsto \;\; E^n(X) \,\coloneqq\, \pi_0 \, \mathrm{Maps}\big( X ,\, E_n \big)

are equivalently the “generalized cohomology theories” as introduced by Whitehead, including examples such as K-theory, elliptic cohomology and cobordism cohomology.


Non-abelian generalized cohomology. But as we just saw, suspension isomorphisms are to be regarded as extra structure on cohomology. Not necessarily requiring them leads to consider any pointed space 𝒜\mathscr{A} (which we may as well assume to be connected) as the classifying space of a non-abelian generalized cohomology theory, defined in evident generalization of (3) simply by

(4)H 1(X;Ω𝒜)π 0Maps(X,𝒜). H^1\big( X ;\, \Omega\mathscr{A} \big) \;\; \coloneqq \;\; \pi_0 \, \mathrm{Maps}\big( X ,\, \mathscr{A} \big) \,.

Here the notation on the left is suggestive of the fact that any loop space Ω𝒜\Omega \mathscr{A} canonically carries the structure of a higher homotopy-coherent group – a groupal A A_\infty-space or \infty -group, for short – whose de-looping is equivalent to the connected component of the original space:

(5)𝒜BΩ𝒜. \mathscr{A} \;\; \simeq \;\; B \, \Omega \mathscr{A} \,.

For example, in the archetypical case where 𝒜S n\mathscr{A} \,\equiv\, S^n is the n n -sphere, then the non-abelian generalized cohomology theory that it classifies is known as (unstable) Cohomotopy π n\pi^n

(6)H˜ 1(X;ΩS n)π 0Maps */(X,S n)π n(X), \widetilde H^1\big( X ;\, \Omega S^n \big) \;\; \equiv \;\; \pi_0 \, \mathrm{Maps}^{\ast/\!}\big( X ,\, S^n \big) \;\; \equiv \;\; \pi^n(X) \,,

in dual reference to the familar homotopy groups

π n(X)π 0Maps */(S n,X). \pi_n(X) \;\; \simeq \;\; \pi_0 \, \mathrm{Maps}^{\ast/\!}\big( S^n ,\, X \big) \,.

Another example of non-abelian generalized cohomology is unstable topological K-theory, whose classifying spaces are taken to be finite stages U(n)\mathrm{U}(n) of the sequential colimits which construct the classifying spaces of topological K-theory.

Developing non-abelian cohomology

Fundamental, elementary and compelling as the notion of non-abelian generalized cohomology in (4) is, it has long remained underappreciated. For example, none of the original authors on Cohomotopy (6) address their subject as a cohomology theory, instead the early development revolves around partial fixes for the perceived defect of co-homotopy sets to not in general carry group structure. The situation does not improve with the early development of “non-abelian gerbes”, whose original description appears unwieldy.

Explicit acknowledgement of (stacky) non-abelian generalized cohomology in the transparent guise (4) appears only in a lecture of Toën 2002, (possibly following Simpson 2002 where non-abelian generalization of de Rham cohomology is considered).

Two independent developments around 2009 put non-abelian generalized cohomology into practical context:


With non-abelian generalized cohomology thus recognized as a worthwhile subject, one is led to generalize familiar constructions in abelian cohomology, as far as possible, and to explore the consequences.

First, one may straightforwardly equip non-abelian cohomology with further attributes: Considering the right hand side of (4) not just for plain spaces but for sheaves of spaces (higher stacks) leads to non-abelian generalized sheaf cohomology, including, in particular, non-abelian generalized versions of twisted cohomology, of equivariant cohomology and nonabelian differential cohomology.


Equivariant non-abelian cohomology. Via the above identification of cohomology sets with homotopy-classes of maps to a classifying space, every flavor of homotopy theory comes with its corresponding flavor of cohomology theories.

In equivariant homotopy theory one considers topological spaces 𝒜\mathscr{A} equipped with the action G𝒜G \curvearrowright \mathscr{A} of a (finite, for our purposes) group GG and with GG-equivariant maps between them – and the corresponding flavor of cohomology is equivariant cohomology:

(7)H G 1(X;Ω𝒜)=π 0Maps(GX,G𝒜) G. H^1_G\big( X;\, \Omega\mathscr{A} \big) \;\; = \;\; \pi_0 \, \mathrm{Maps}\big( G \curvearrowright X ,\, G \curvearrowright \mathscr{A} \big)^G \,.

Here the notion of GG-homotopy equivalence of maps is straightforward but, at face value, technically cumbersome to reason about. However, Elmendorf's theorem reveals that GG-homotopy equivalences (between GG-cell complexes) are nothing but systems of ordinary weak homotopy equivalences between the HH-fixed spaces 𝒜 H\mathscr{A}^H for all subgroups HGH \subset G. These systems of fixed spaces are conveniently re-packaged as presheaves on a small category called the orbit category Orb(G)\mathrm{Orb}(G) of GG, whence GG-equivariant homotopy theory is equivalently the homotopy theory of presheaves of spaces on Orb(G)\mathrm{Orb}(G).


Twisted non-abelian cohomology. Somewhat similarly, given any space \mathscr{B} in any homotopy theory, the \mathscr{B} -slice is the homotopy theory whose objects are spaces fibered over \mathscr{B} with maps between them respecting the fibration up to specified homotopy. If we assume, without essential restriction, that the base space is connected, then we may identify it as B𝒢\mathscr{B} \,\simeq\, B \mathscr{G}, as in (5), which exhibits any fibration over it as the Borel construction 𝒜𝒢\mathscr{A} \sslash \mathscr{G} of the homotopy-quotient of a homotopy-coherent action 𝒢acts𝒜\mathscr{G} \acts \mathscr{A}.

If we now think of a domain object XτB𝒢X \xrightarrow{\tau} B\mathscr{G} in this B𝒢B \mathscr{G}-slice as a twist and of a codomain object 𝒜𝒢pB𝒢\mathscr{A} \sslash \mathscr{G} \xrightarrow{p} B \mathscr{G} as a local coefficient bundle, then the corresponding non-abelian cohomology is just the homotopy classes of sections of the τ\tau-associated 𝒜\mathscr{A}-fiber bundle, and as such is τ\tau-twisted 𝒜\mathscr{A}-cohomology:

This works generally: If all spaces here are in addition equipped with GG-actions as in (7), hence if we are looking at a slice of equivariant homotopy, then the above is automatically twisted equivariant non-abelian cohomology.


The non-abelian character. One famous construction on abelian cohomology is the Chern-Dold character map to de Rham cohomology, which in the case of K-cohomology becomes the familar Chern character (and which on ordinary cohomology is essentially just the de Rham theorem). One may think of the Chern-Dold character as universally extracting the non-torsion data of cohomology. Its generalization to non-abelian cohomology was developed in FSS23:

Observing that the Chern-Dold character is essentially just the cohomology operation induced by rationalization of the classifying space,

𝒜η L 𝒜 \mathscr{A} \xrightarrow{ \phantom{--} \eta^{\mathbb{Q}} \phantom{--} } L^{\!\mathbb{Q}}\mathscr{A}

as such it makes sense in the generality of non-abelian classifying spaces (immediately so under mild technical assumptions, such as nilpotency, but with more work also more generally). In view of this, the fundamental theorem of dg-algebraic rational homotopy theory may be re-cast as a non-abelian de Rham theorem which identifies, over smooth manifolds XX, the resulting non-abelian rational cohomology with the concordance classes of flat differential forms having coefficients in the real Whitehead-bracket L L_\infty -algebra 𝔩𝒜\mathfrak{l}\mathscr{A} of the classifying space:

Since generalized cohomology theories are typically hard to analyze, in particular non-abelian ones, this character map may be regarded as extracting the first non-trivial stage of more tractable invariants. For instance, the character of a non-abelian class is the first obstruction to the trivialization of that class.


Nonabelian cohomology and gauge fields. In the mentioned application to physics, the flux densities of a higher gauge field are sourced by charges which appear as classes in non-abelian de Rham cohomology on the right, and the completion of the higher gauge theory by a flux-quantization law means to lift these charges through the character map to classes in a chosen non-abelian cohomology theory on the left.

CohomologyGauge fields
-theoryflux-quantization law
cocyclefield configuration
coboundarygauge transformation
characterflux densities
ordinary-electromagnetic
differential-gauge potentials
twisted-background fields
equivariant-on orbifolds
Real-on orientifolds
nonabelian-non-linear Gauss law

Cohomology and gauge fields. While cohomology has of course many and diverse applications, in physics no less than in other fields, the role of cohomology specifically in the global description of (higher) gauge fields (“force fields”) is profound: In generalization of the seminal historical observation (“Dirac charge quantization”) that electromagnetic field configurations are globally to be identified with 2-cocycles in ordinary differential cohomology of spacetime, higher gauge field species are similarly to be identified with generalized cohomology theories whose further properties and attributes closely reflect the field’s physical nature, as indicated in the above table.


More 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, in: Algèbres d’opérateurs et leurs applications en physique mathématique, Colloques Internationaux du Centre National de la Recherche Scientifique (C.N.R.S) 274, Paris (1979) 321–332 [ISBN:2-222-02441-2, pdf, pdf]

This is recounted in

  • Ross Street, pages 9-10 of: An Australian conspectus of higher category theory, talk at Institute for Mathematics and its Applications Summer Program: nn-Categories: Foundations and Applications at the University of Minnesota (Minneapolis, 7–18 June 2004), in: Towards Higher Categories, The IMA Volumes in Mathematics and its Applications 152, Springer (2010) 237-264 [pdf, pdf, doi:10.1007/978-1-4419-1524-5]

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

AbelianGroupsChainComplexesOfAbelianGroupsCrossedComplexesωGroupoidsωCategories 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

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

Nonabelian sheaf cohomology with constant coefficients

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

H(X,A)π 0Top(X,|A|)π 0Func(SingX,A), H(X,A) \;\coloneqq\; \pi_0 Top(X, |A|) \simeq \pi_0 \infty Func(Sing X, A) \,,

where |A||A| is the geometric realization of AA and SingXSing X the fundamental ∞-groupoid of XX.

But both XX and AA here naturally can be regarded, in several ways, as objects of (∞,1)-sheaf (∞,1)-toposes H=Sh (,1)(C)\mathbf{H} = Sh_{(\infty,1)}(C) over nontrivial (∞,1)-sites CC. The intrinsic cohomology of such H\mathbf{H} is a nonabelian sheaf cohomology. The following discusses two such choices for H\mathbf{H} such that the corresponding nonabelian sheaf cohomology coincides with H(X,A)H(X,A) (for paracompact XX).

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

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

HSh (,1)(X)Sh (,1)(Op(X)) \mathbf{H} \;\coloneqq\; Sh_{(\infty,1)}(X) \;\coloneqq\; Sh_{(\infty,1)}(Op(X))

be the (∞,1)-category of (∞,1)-sheaves on XX. The space XX itself is naturally identified with the terminal object X=*Sh (,1)(X)X = * \in Sh_{(\infty,1)}(X). This is the petit topos incarnation of XX.

Write

(LConstΓ):Sh (,1)(X)ΓLConstGrpd (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 LConstLConst an an ∞-groupoid AGrpdA \in \infty Grpd is regarded as an object LConstASh (,1)(X)LConst A \in Sh_{(\infty,1)}(X).

There is therefore the intrinsic cohomology of the (,1)(\infty,1)-topos Sh (,1)(X)Sh_{(\infty,1)}(X) with coefficients in the constant (∞,1)-sheaf on AA

H(X,A)π 0Sh (,1)(X)(X,LConstA). H'(X,A) \;\coloneqq\; \pi_0 Sh_{(\infty,1)}(X)(X, LConst A) \,.

This is cohomology with constant coefficients.

Notice that since XX is in fact the terminal object of Sh (,1)(X)Sh_{(\infty,1)}(X) and that Sh (,1)(X)(X,)Sh_{(\infty,1)}(X)(X,-) is in fact that global sections functor, this is equivalently

π 0ΓLConstA. \cdots \simeq \pi_0 \Gamma LConst A \,.
Theorem

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

H(X,A)π 0Grpd(SingX,A)Sh (,1)(X)(X,LConstA). H(X,A) \;\coloneqq\; \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 (,1)(\infty,1)-sheaf (,1)(\infty,1)-topos

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

(ΠLConstΓ):ETopGrpdΓLConstΠGrpd, (\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 Π\Pi to LConstLConst 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 XX we have an equivalence of cocycle ∞-groupoids

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

and hence in particular an isomorphism on cohomology

H(X,A)π 0ETopGrpd(X,LConstA) 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):

Objects classified by nonabelian cohomology

For g:XAg : X \to A a cocycle in nonabelian cohomology, we say the homotopy fibers of gg is the object classified by gg.

For examples and discussion of this see

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:

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:

Categorified theory – 2-bundles/gerbes

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

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:

Discussion in terms of gerbes:

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

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

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

In a context of nonabelian Hodge theory:

Generally:

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

The suggestion that every (pointed) space/ \infty -stack 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:

Last revised on November 18, 2024 at 11:02:20. See the history of this page for a list of all contributions to it.