# nLab cohomology of a category

cohomology

## 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 $A$ some coeffiecient ∞-groupoid – typically, but not necessarily, an Eilenberg-MacLane object $A=K\left(A,n\right)={B}^{n}A$ – regarded as a Kan complex, the cohomology of $C$ in this sense is

${H}^{n}\left(C,A\right):={\pi }_{0}\infty \mathrm{Grpd}\left(F\left(N\left(C\right)\right),A\right)\phantom{\rule{thinmathspace}{0ex}},$H^n(C,A) := \pi_0 \infty Grpd( F(N(C)), \mathbf{A} ) \,,

where

• $N\left(C\right)$ is the nerve of $C$;

• $F\left(N\left(C\right)\right)$ is the Kan fibrant replacement of $N\left(C\right)$;

• ∞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

$\cdots ={\mathrm{Ho}}_{\mathrm{SSet}}\left(N\left(C\right),A\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 ${\mathrm{Cat}}_{\mathrm{Thomason}}\stackrel{\mathrm{Quillen}}{\simeq }{\mathrm{SSet}}_{\mathrm{Quillen}}$ we have for $\alpha$ any category whose groupoidification is equivalent to $A$, i.e. any cateory such that $F\left(N\left(\alpha \right)\right)\simeq A$ in ∞Grpd, we have

$\cdots ={\mathrm{Ho}}_{{\mathrm{Cat}}_{\mathrm{Thomason}}}\left(C,\alpha \right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots = Ho_{Cat_{Thomason}}(C,\alpha) \,.
Created on January 11, 2010 20:56:35 by Urs Schreiber (87.212.203.135)