Thomason cohomology and Quillen's Theorem A

Given a functor φ : C → D between two small categories, there is a homotopy equivalence κ: hocolim D N(φ /-) → NC where N(φ/-) is the functor which sends every object d in D to the nerve of the comma category φ/d. We prove that the homotopy equivalence κ 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 ∫ D F of a functor F: D → Cat using the isomorphism in the main theorem.