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

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Summary

Introduction

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.

We prove here the following
Preliminaries
Cohomology of Small Categories
Baues–Wirsching Cohomology of Small Categories
Cohomology of Simplicial Sets
The Nerve of a Small Category
The Involved Constructions
A Free Resolution of the Natural System Z over
Findings
A Projective Resolution of the
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