Thomason cohomology and Quillen's Theorem A

Abstract

Given a functor $\varphi : \mathcal{C} \to \mathcal{D}$ between two small categories, there is a homotopy equivalence $\kappa: hocolim _{\mathcal{D}} N(\varphi /-) \to N\mathcal{C}$ where $N(\varphi/-)$ is the functor which sends every object $d$ in $\mathcal{D}$ to the nerve of the comma category $\varphi/d$. 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.