nLab cohomology of a category

Idea

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

$$H^n(C,A) := \pi_0 \infty Grpd( F(N(C)), \mathbf{A} ) \,,$$

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

$$\cdots = Ho_{SSet}(N(C), \mathbf{A}) \,.$$

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

$$\cdots = Ho_{Cat_{Thomason}}(C,\alpha) \,.$$