nLab cohomology group

Recall that cohomology in an (∞,1)-topos $\mathbf{H}$ on an object $X$ with coefficients in an object $A$ is the hom-set in the homotopy category of an (∞,1)-category

$$H(X,A) = \pi_0 \mathbf{H}(X,A) \,.$$

This is the cohomology set .

It is a pointed set if $A$ is a pointed object.

In the case that $A$ moreover carries the structure of a group object, the set $H(X,A)$ inherits naturally itself the structure of a group. In this case one speaks of the cohomology group of $X$ with coefficients in $A$.

Dually, is this why n-spheres are good for homotopy as they are cogroups? —David

Examples

generalized abelian cohomology

In all of what is called generalized cohomology – which is really generalized abelian cohomology, compare nonabelian cohomology – the coefficient object is taken to be not just a group object but a “maximally abelian” group object called a stable object in general and called a spectrum in the case that $\mathbf{H}$ = Top.

In that case all the deloopings $\mathbf{B}^n A$ of $A$ exists and are still stably abelian group objects.

So in that case not only is the cohomology set $H(X,A)$ naturally an abelian group, but there is an infinite sequence of such cohomology groups, one for each delooping $\mathbf{B}^n A$. This yields the traditional notation for graded cohomology groups by setting

$$H^n(X,A) := H(X, \mathbf{B}^n A) \,.$$

“ordinary” (integral/Eilenberg-Mac Lane-) cohomology

The standard example are the “ordinary” cohomology groups that come from taking $\mathbf{H} =$ Top or = ∞Grpd (see homotopy hypothesis) and choosing the coefficient object to be the Eilenberg-Mac Lane spectrum

$$A := \mathbf{B} \mathbb{Z} \,.$$

The for $X \in \mathbf{H}$ any object (a topological space or an ∞-groupoid) the “ordinary” cohimology of $X$ in degree $n$ is

$$H^n(X) := H^n(X,\mathbb{Z}) := H(X, \mathbf{B}^n \mathbb{Z}) = \pi_0 \mathbf{H}(X, \mathbf{B}\mathbb{Z}) =: [X, K(n, \mathbb{Z})] \,.$$

Here on the left we have the standard notation for the ordinary cohomology groups, and on the right their expression in terms of homotopy classes of maps into an Eilenberg-Mac Lane space.

Cohomology groups in nonabelian cohomology

The standard counter-example to keep in mind for a nonabelian cohomology set that does not carry a group structure is “nonabelian cohomology in degree 1” that classifies $G$-principal bundles, for $G$ some nonabelian group.

This cohomology set

$$H^1(X,G) := H(X, \mathbf{B}G) =: [X, \mathbf{B} G] \simeq G Bund(X)/_\sim$$

clearly has no natural group structure on it, unless $G$ is in fact abelian (in which case $\mathbf{B}G$ is indeed a group object, namely a 2-group).

But when we pass from group-principal bundles to groupoid-principal bundles, then there may be cohomology sets with group structure even in nonabelian cohomology.

Let for instance $G_{(2)}$ be a 2-group, i.e. a groupoid with group structure, such as the automrophism 2-group $G_{(2)} := AUT(H) := Aut_{Grpd}(\mathbf{B}H)$ of an ordinary group $H$, then there is the nonabelian cohomology set

$$H^1(X, G_{(2)}) := H(X, G_{(2)}) \simeq G_{(2)} GrpdBund(X)/_\sim \,.$$

and this does carry a nonabelian (in general) group structure.

This is to be distinguished from the cohomology set

$$H^2(X, G_{(2)}) := H(X, \mathbf{B} G_{(2)}) \simeq G_{(2)} Bund(X)/_\sim$$

that classifies $G_{(2)}$ principal 2-bundles as opposed to groupoid principal 1-bundles and which is not in general a group (unless $G_{(2)}$ in turn is sufficiently abelian).

For $G_{(2)} = AUT(H)$ both these cohomology sets play a role in the description of gerbes (see gerbe (as a stack) and gerbe (in nonabelian cohomology)).