Contents

cohomology

# 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 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)

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 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

## 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 $X$ 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 $A$

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

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

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

See ETop∞Grpd.

## 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

## References

### Classical theory – bundles and groups

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)

### Categorified theory – 2-bundles/gerbes

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

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

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:

On the non-abelian Chern-Dold character:

Last revised on January 19, 2021 at 04:38:28. See the history of this page for a list of all contributions to it.