Thomason cohomology and Quillen's Theorem A
Mehmet Kirtisoglu, Ergun Yalcin
TL;DR
The article proves that the Thomason cohomology of a small category is invariant under the Quillen A-type homotopy equivalence κ: hocolim_{𝒟}N(φ/−)→N𝒞, yielding a Thomason-versions of Quillen's Theorem A for arbitrary coefficient systems and enabling a spectral sequence for the Thomason cohomology of Grothendieck constructions. It develops the necessary framework connecting Quillen, Baues–Wirsching, and Thomason cohomologies via coefficient systems on nerves and simplicial replacements, and provides a detailed double-complex construction to establish the main isomorphism. The work also extends cohomology theories to bisimplicial contexts, derives Dold–Puppe type equivalences, and presents a spectral-sequence toolkit for computing Thomason cohomology of ∫_{𝒟}F, thereby unifying several cohomological approaches for small categories. The results have implications for understanding how functorial and categorical constructions preserve cohomological information, and they generalize Serre-type spectral sequences to the setting of Thomason cohomology.
Abstract
Given a functor $\varphi : \mathcal{C} \to \mathcal{D}$ between two small categories, there is a homotopy equivalence $κ: hocolim _{\mathcal{D}} N(\varphi /-) \to N\mathcal{C}$ 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 $κ$ 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.