Thomason cohomology and Quillen's Theorem A
Given a functor $φ: \mathcal{C} \to \mathcal{D}$ between two small categories, there is a homotopy equivalence $κ: hocolim _{\mathcal{D}} N(φ/-) \to N\mathcal{C}$ where $N(φ/-)$ is the functor which sends every object $d$ in $\mathcal{D}$ 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 $\int _{\mathcal{D}} F$ of a functor $F: \mathcal{D} \to Cat$ using the isomorphism in the main theorem.
math.AT↗