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. 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 $\mathrm{\infty }$-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)

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

There is a chain of inclusions

$$\mathrm{AbelianGroups}\hookrightarrow \mathrm{ChainComplexesOfAbelianGroups}\hookrightarrow \mathrm{CrossedComplexes}\hookrightarrow \omega \mathrm{Groupoids}\hookrightarrow \omega \mathrm{Categories}$$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 $\mathrm{\infty }$-categories with coefficients in $\mathrm{\infty }$-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

• Bertrand Toen, Non-abelian cohomology (slides ps)

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:

$$H(X,A):={\pi }_0\mathrm{Top}(X,\mid A\mid )\simeq {\pi }_0\mathrm{\infty }\mathrm{Func}(\mathrm{Sing}X,A),$$H(X,A) := \pi_0 Top(X, |A|) \simeq \pi_0 \infty Func(Sing X, A) \,,

where $\mid A\mid$ is the geometric realization of $A$ and $\mathrm{Sing}X$ 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 $H={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ over nontrivial (∞,1)-sites $C$. The intrinsic cohomology of such $H$ is a nonabelian sheaf cohomology. The following discusses two such choices for $H$ such that the corresponding nonabelian sheaf cohomology coincides with $H(X,A)$ (for paracompact $X$).

Petit $(\mathrm{\infty }\mathrm{,1})$-sheaf $(\mathrm{\infty }\mathrm{,1})$-topos

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

$$H:={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(X):={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{Op}(X))$$\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 $X=*\in {\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(X)$. This is the petit topos incarnation of $X$.

Write

$$(\mathrm{LConst}⊣\Gamma ):{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(X)\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Gamma }{\to }}\mathrm{\infty }\mathrm{Grpd}$$(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 $\mathrm{LConst}$ an an ∞-groupoid $A\in \mathrm{\infty }\mathrm{Grpd}$ is regarded as an object $\mathrm{LConst}A\in {\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(X)$.

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

$$H\prime (X,A):={\pi }_0{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(X)(X,\mathrm{LConst}A).$$H'(X,A) := \pi_0 Sh_{(\infty,1)}(X)(X, LConst A) \,.

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

$$\cdots \simeq {\pi }_0\Gamma \mathrm{LConst}A.$$\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 $A\in \mathrm{\infty }\mathrm{Grpd}$ agree:

$$H(X,A):={\pi }_0\mathrm{\infty }\mathrm{Grpd}(\mathrm{Sing}X,A)\simeq {\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(X)(X,\mathrm{LConst}A).$$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 $(\mathrm{\infty }\mathrm{,1})$-sheaf $(\mathrm{\infty }\mathrm{,1})$-topos

Another alternative is to regard the space $X$ as an object in the gros (∞,1)-sheaf topos ${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})$ over the site CartSp, as described at ∞-Lie groupoid. This has the special property that it is a locally ∞-connected (∞,1)-topos, which means that the terminal geometric morphism is an essential geometric morphism

$$(\Pi ⊣\mathrm{LConst}⊣\Gamma ):{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Gamma }{\to }}}\mathrm{\infty }\mathrm{Grpd},$$(\Pi \dashv LConst \dashv \Gamma) : Sh_{(\infty,1)}(CartSp) \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd \,,

with the further left adjoint $\Pi$ to $\mathrm{LConst}$ being the intrinsic path ∞-groupoid 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

$${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})(X,\mathrm{LConst}A)\simeq \mathrm{Top}(X,\mid A\mid )$$Sh_{(\infty,1)}(CartSp)(X, LConst A) \simeq Top(X, |A|)

and hence in particular an isomorphism on cohomology

$$H(X,A)\simeq {\pi }_0{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{CartSp})(X,\mathrm{LConst}A)$$H(X,A) \simeq \pi_0 Sh_{(\infty,1)}(CartSp)(X, LConst A)

Proof

The key point is that for paracompact $X$, the nerve theorem asserts that $\Pi (X)$ is weak homotopy equivalent to $\mathrm{Sing}X$, the standard fundamental ∞-groupoid of $X$. This is discussed in detail in the section geometric realization at path ∞-groupoid.

Using this, the statement follows by the (∞,1)-adjunction $(\Pi ⊣\mathrm{LConst})$, that is discussed in detail at Unstructured homotopy ∞-groupoid.

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

• gerbe

• principal 2-bundle

• principal ∞-bundle .

References

In as far as nonabelian cohomology is nothing but the study of hom-spaces between ∞-stacks, see also the references at ∞-stack.

A readable survey on nonabelian cohomology is

• Bertrand Toen, Non-abelian cohomology (ps)

A useful motivation is

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

Early references include

• P. Dedecker, Cohomologie de dimension 2 à coefficients non abéliens, C. R. Acad. Sci. Paris, 247 (1958), 1160–1163;

• P. 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

• P. Dedecker, Three dimensional non-abelian cohomology for groups, Category theory, homology theory and their applications, II (Battelle Institute Conf.) 1969

The standard classical monograph focusing on low-dimensional cases is

• J. Giraud, Cohomologie non abélienne , Springer (1971)

See also nonabelian gerbe.

• 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)

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.