nLab
nonabelian cohomology

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

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. 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.

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

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)

Both 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

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) := \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

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

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

A readable survey on nonabelian cohomology is

A useful motivation is

  • Nicolas Addington, Fiber bundles and nonabelian cohomology (pdf)

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 GG-gerbes by cohomology with coefficients in the automorphism 2-group AUT(G)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)

  • 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.

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.

Revised on March 26, 2014 19:26:50 by Jon Beardsley (68.49.92.24)