Homotopical Dold-Kan Correspondences

TL;DR

This paper extends the classical Dold-Kan correspondence to a homotopical setting within test categories, establishing conditions under which homology functors induce equivalences of derived categories.

Contribution

It introduces a general framework for homotopical Dold-Kan correspondences in the context of test categories, including the construction of model structures and identification of Whitehead categories.

Findings

Homology functor induces equivalence between localized categories for Whitehead test categories.

Important examples like Joyal's category Θ are shown to be Whitehead categories.

Constructed model category structures on abelian presheaves over Whitehead test categories.

Abstract

This work originates from chapters V and VII of Grothendieck's manuscript Pursuing Stacks, which contains a series of questions, as well as a previously unexplored formalism, concerning the interactions between the notion of test categories and homology. The main objective of this thesis is to exhibit homotopical Dold-Kan correspondences in the context of test categories. More precisely, we introduce, following Grothendieck, a functor generalizing simplicial homology, from the category of abelian presheaves over any small category to the derived category of abelian groups in non-negative homological degree. We then look for conditions ensuring that this functor induces an equivalence of categories, after localization by the class of morphisms whose image in the derived category is an isomorphism. Generally, there exists a second class of weak equivalences, arising from the theory of test categories, on the category of abelian presheaves, and we call Whitehead Categories those small categories for which these two classes coincide, generalizing the case of $\Delta$. We show that important examples of test categories are Whitehead categories, notably Joyal's category $\Theta$. We construct, for any Whitehead local test category, a model category structure on its category of abelian presheaves with the weak equivalences mentioned above. We then prove that for any Whitehead test category, the homology functor does induce an equivalence between the localized categories. We obtain this way many examples of homotopical Dold-Kan correspondences, including, among others, the category $\Theta$.