group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The cohomology of a category $C$ is often defined to be the groupoid cohomology of the ∞-groupoid that is presented by the nerve of the category.
Hence for $\mathbf{A}$ some coefficient ∞-groupoid – typically, but not necessarily, an Eilenberg-MacLane object $\mathbf{A} = K(A,n) = \mathbf{B}^n A$ – regarded as a Kan complex, the cohomology of $C$ in this sense is
where
$N(C)$ is the nerve of $C$;
$F(N(C))$ is the Kan fibrant replacement of $N(C)$;
∞Grpd is the (∞,1)-category of ∞-groupoids.
Using the standard model structure on simplicial sets this is the same as the hom-set in the homotopy category of SSet
One can also use the Thomason model structure on Cat to say the same: due to the Quillen equivalence $Cat_{Thomason} \stackrel{Quillen}{\simeq} SSet_{Quillen}$ we have for $\alpha$ any category whose groupoidification is equivalent to $\mathbf{A}$, i.e. any cateory such that $F(N(\alpha)) \simeq \mathbf{A}$ in ∞Grpd, we have
Last revised on August 12, 2016 at 08:53:32. See the history of this page for a list of all contributions to it.