Recall that cohomology in an (∞,1)-topos on an object with coefficients in an object is the hom-set in the homotopy category of an (∞,1)-category
This is the cohomology set .
It is a pointed set if is a pointed object.
In the case that moreover carries the structure of a group object, the set inherits naturally itself the structure of a group. In this case one speaks of the cohomology group of with coefficients in .
Dually, is this why n-spheres are good for homotopy as they are cogroups? —David
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 = Top.
In that case all the deloopings of exists and are still stably abelian group objects.
So in that case not only is the cohomology set naturally an abelian group, but there is an infinite sequence of such cohomology groups, one for each delooping . This yields the traditional notation for graded cohomology groups by setting
“ordinary” (integral/Eilenberg-Mac Lane-) cohomology
The standard example are the “ordinary” cohomology groups that come from taking Top or = ∞Grpd (see homotopy hypothesis) and choosing the coefficient object to be the Eilenberg-Mac Lane spectrum
The for any object (a topological space or an ∞-groupoid) the “ordinary” cohimology of in degree is
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 -principal bundles, for some nonabelian group.
This cohomology set
clearly has no natural group structure on it, unless is in fact abelian (in which case 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 be a 2-group, i.e. a groupoid with group structure, such as the automrophism 2-group of an ordinary group , then there is the nonabelian cohomology set
and this does carry a nonabelian (in general) group structure.
This is to be distinguished from the cohomology set
that classifies principal 2-bundles as opposed to groupoid principal 1-bundles and which is not in general a group (unless in turn is sufficiently abelian).
For both these cohomology sets play a role in the description of gerbes (see gerbe (as a stack) and gerbe (in nonabelian cohomology)).