A homotopical Dold-Kan correspondence for Joyal's category $Θ$ and other test categories
Léo Hubert
TL;DR
The paper develops a homotopical Dold–Kan framework for abelian presheaves on test categories, defining a homology functor $H_A$ as the left derived colimit and identifying when it yields a DK-type equivalence via a strong Whitehead condition. It introduces integrators (notably the Bousfield–Kan integrator) to compute homology concretely and proves that aspherical functors preserve homology, enabling Quillen equivalences between abelian presheaves and chain complexes. Under local test and strong Whitehead hypotheses, the Grothendieck–Cisinski model structure transfers to abelian presheaves, giving Quillen equivalences that realize a homotopical DK correspondence for categories such as $\Delta^n$ and $\Theta$. The results culminate with Joyal's $\Theta$ shown to be a homologically pseudo-test category, thereby extending the Dold–Kan paradigm to a broad class of test categories and yielding a robust homotopical interpretation of abelian presheaves in non-simplicial settings.
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 $Θ$.