Thomason Cohomology and Quillen’s Theorem A

Abstract

Given a functor \(\varphi : \mathcal {C}\rightarrow \mathcal {D}\) between small categories, Quillen showed that there is a homotopy equivalence \(\kappa :\mathop {\textrm{hocolim}}\limits _{\mathcal {D}} N(\varphi /-) \rightarrow N\mathcal {C}\), where \(N(\varphi /-)\) is the functor that sends each object d of \(\mathcal {D}\) to the nerve of the comma category \(\varphi /d\). We show that the homotopy equivalence \(\kappa \) induces an isomorphism on the Gabriel-Zisman cohomology of the associated simplicial sets. As a consequence, we obtain a version of Quillen’s Theorem A for the Thomason cohomology of categories. We also construct a spectral sequence converging to the Thomason cohomology of the Grothendieck construction \(\int _{\mathcal {D}} F\) of a functor \(F:\mathcal {D}\rightarrow \textbf{Cat}\) with arbitrary coefficients.