Thomason cohomology and Quillen's Theorem A

Abstract

Given a functor $\phi :\mathcal{C}\to \mathcal{D}$ between two small categories, there is a homotopy equivalence $\kappa :hocoli{m}_{\mathcal{D}}N(\phi /-)\to N\mathcal{C}$ where $N(\phi /-)$ is the functor which sends every object $d$ in $\mathcal{D}$ to the nerve of the comma category $\phi /d$. We prove that the homotopy equivalence $\kappa$ induces an isomorphism on cohomology with coefficients in any coefficient system. As a consequence, we obtain a version of Quillen's Theorem A for the Thomason cohomology of categories. We also construct a spectral sequence for the Thomason cohomology of the Grothendieck construction ${\int }_{\mathcal{D}}F$ of a functor $F:\mathcal{D}\to Cat$ using the isomorphism in the main theorem.

15 pages. Accepted version. Revised and shortened after the referee report