Thomason’s homotopy colimit theorem and cohomology of categories

Abstract

In 1978, R.W. Thomason [1] proves that there is a homotopy equivalence η:hocolim_C N(F) → N(∫_C F) for any functor F: C → Cat. Here hocolim_C N(F) is the diagonalization of the bisimplicial set ⊔_{σ∈N(C)} N(F(σ(0))) and ∫_C F is the Grothendieck construction whose objects are given by the pairs (c, x) with c ∈ Ob(C) and x ∈ Ob(F(c)), and whose morphisms are given by the pairs (α, γ):(c, x) → (c′, x′) with α ∈ Hom_C(c, c′) and γ ∈ Hom_{F(c′)}(F (α)(x), x′). We prove that Grothendieck construction is a precofibred category over the canonical functor π : ∫_C F → C. We also prove that for any functor φ : D → C there is a homotopy equivalence λ : hocolim_C N(φ/c) → N(D). We show that these two together prove Thomason’s homotopy colimit theorem from a conceptual point of view. We further investigate how our conceptual approximation for the proof of Thomason’s homotopy colimit theorem becomes useful for the cohomology version of Thomason’s theorem which was proven by A. M. Cegarra [2] in 2020.