Abstract
This paper deals with well-known weak homotopy equivalences that relate homotopy colimits of small categories and simplicial sets. We show that these weak homotopy equivalences have stronger cohomology-preserving properties than for local coefficients.
Highlights
Since the 1980 paper by Thomason [1], we know that the categories SSet, of simplicial sets, and Cat, of small categories, are equivalent from a homotopical point of view
There are interesting algebraic constructions, both on simplicial sets and on small categories, that are not invariants of their homotopy type. This is the case for Gabriel–Zisman cohomology groups H n ( X, A) ([2] Appendix II), of simplicial sets X with arbitrary coefficient systems on them, that is, with coefficients in abelian group valued functors A : ∆( X ) → Ab
The aim of this paper is to prove that two relevant and well-known weak homotopy equivalences have similar strong cohomology-preserving properties
Summary
Since the 1980 paper by Thomason [1], we know that the categories SSet, of simplicial sets, and Cat, of small categories, are equivalent from a homotopical point of view. Baues–Wirsching cohomology groups H n (C, A) [4], of a small category C with coefficients in natural systems A on it, that is, with coefficients in abelian group valued functors on its category of factorizations A : F(C) → Ab, are homotopy invariants of C only for local coefficients A : Π(C) → Ab. There are, some particular weak homotopy equivalences that have a stronger conservation property of cohomology than for local coefficients. The aim of this paper is to prove that two relevant and well-known weak homotopy equivalences have similar strong cohomology-preserving properties These come respectively associated to diagrams of small categories and simplicial sets. The first of them arises from the seminal Homotopy Colimit Theorem by Thomason ([5] Theorem 1.2) This theorem states that, for any indexing small category C and any functor F : Cop → Cat, there is a natural weak homotopy equivalence of simplicial sets.
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