A homotopical Dold-Kan correspondence for Joyal's category $\Theta$ and other test categories

TL;DR

This paper generalizes the Dold-Kan correspondence to presheaves of abelian groups over test categories, establishing a homotopical equivalence of categories via a Quillen model structure, with applications to Joyal's category Θ.

Contribution

It introduces a homotopical generalization of the Dold-Kan correspondence for presheaves over test categories, including Joyal's Θ, through a new model structure and Quillen equivalence.

Findings

Homotopy category of abelian presheaves is equivalent to derived category of abelian groups.

Established a Quillen equivalence for the model structure on abelian presheaves.

Applied the result specifically to Joyal's category Θ.

Abstract

We prove that for any test category $A$, in the sense of Grothendieck, satisfying a compatibility condition between homology equivalences and weak equivalences of presheaves, the homotopy category of abelian presheaves on $A$ is equivalent to the non-negative derived category of abelian groups. This provides a homotopical generalization of the Dold-Kan correspondence for presheaves of abelian groups over a wide range of test categories. This equivalence of homotopy categories comes from a Quillen equivalence for a model structure on abelian presheaves that we introduce under these conditions. We then show that this result applies to Joyal's category $\Theta$.