Special and general types
Groupoid cohomology is the cohomology specifically of ordinary groupoids, more generally that of internal groupoids.
Groupoid cohomology generalizes group cohomology, which is the cohomology of delooping groupoids of groups. Analogously to abelian and nonabelian group cohomology there is abelian and nonabelian groupoid cohomology.
Under the homotopy hypothesis theorem, plain (non-internal) groupoid cohomology is the same as the cohomology of homotopy 1-types.
A common special case of groupoid cohomology is the cohomology of action groupoids: this is (Borel)-equivariant cohomology.
General groupoid cohomology may be regarded as a generalization of equivariant cohomology exactly analogous to the passge from global action groupoids to orbifolds.
Let ∞Grpd be the (∞,1)-topos of ∞-groupoids. Let be an ordinary 1-groupoid. Let be an arbitrary -groupoid. Then the cohomology of with coefficients in is
Concrete formulas for this work exactly as described in detail at group cohomology, only wherever there we have the unique object , now there may be arbitrary objects.