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

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 $Θ$.

💡 Research Summary

The paper revisits Grothendieck’s theory of test categories, originally introduced in the “Pursuing Stacks” manuscript, and establishes a homotopical version of the classical Dold‑Kan correspondence that works for a broad class of test categories, including Joyal’s Θ.

The author begins by recalling the basic definitions: for a small category A, the presheaf category ̂A = Set^{A^{op}} and its category of elements i_A(X)=A/X give rise to a class of weak equivalences W_A (those morphisms whose image under i_A becomes a weak equivalence of categories). A is called a pseudo‑test category if the induced functor Hot_A → Hot (the homotopy category of spaces) is an equivalence; further refinements lead to weak, local, and strict test categories.

The central construction is a homology functor H_A : ̂A_ab → Hot_ab, where ̂A_ab = Ab^{A^{op}} denotes abelian presheaves. H_A is defined as the left derived functor of the colimit functor; concretely, it sends a presheaf X to the homology of its category of elements i_A(X). This yields a class of weak equivalences W_ab A consisting of those morphisms that become isomorphisms under H_A. In the simplicial case (A = Δ), H_Δ coincides with the usual normalized chain complex functor, and W_ab Δ matches the usual quasi‑isomorphisms.

A key hypothesis introduced is the “strong Whitehead condition”: the class W_ab A coincides with the inverse image U^{-1}W_A of the underlying weak equivalences of presheaves (U being the forgetful functor to sets). When this condition holds, H_A sends every morphism in U^{-1}W_A to an isomorphism in Hot_ab, and the localization of ̂A_ab at W_ab A is equivalent to the non‑negative derived category of abelian groups, denoted Hot_ab.

The paper then studies a broader class W_ab^∞ of morphisms that induce isomorphisms in homology (homology equivalences). For a functor u : A → B that is W_ab^∞‑aspherical (i.e., each slice u/b lies in W_ab^∞), the restriction functor u^* : ̂B_ab → ̂A_ab commutes with H and therefore preserves weak equivalences.

Using the Grothendieck‑Cisinski model structure on presheaf categories over a local test category, the author equips ̂A_ab with a cofibrantly generated model structure whose weak equivalences are precisely W_ab A and whose cofibrations are monomorphisms. Theorem 5.9 asserts the existence of this model structure under the strong Whitehead condition. Theorem 5.11 shows that if u : A → B is an aspherical functor between local test categories, then u^* is a left Quillen equivalence for the above model structures. Consequently, the equivalence of localized categories in Theorem 5.14 can be upgraded to a Quillen equivalence between ̂A_ab and the standard projective model structure on non‑negative chain complexes of abelian groups.

The final part applies these results to Joyal’s Θ. Building on work of Ara and Maltsiniotis, the author observes that the cellular nerve functor Δ → ̂Θ and Street’s oriental functor are aspherical. By the stability results for aspherical functors, Θ inherits the strong Whitehead condition. Therefore, Θ satisfies the homotopical Dold‑Kan correspondence: the homology functor H_Θ induces an equivalence between the homotopy category of abelian presheaves on Θ and Hot_ab.

The paper also notes that the strong Whitehead condition is not necessary in all cases; for example, the test category Δ/N(Gr) (the simplex category over the reflexive globular category) fails the condition but still fulfills a version of the homotopical Dold‑Kan correspondence, as shown in the author’s PhD thesis.

In summary, the work provides a conceptual bridge between test category theory, homological algebra of presheaves, and model category theory. It generalizes the classical Dold‑Kan equivalence from simplicial abelian groups to abelian presheaves on any test category satisfying the strong Whitehead condition, and demonstrates that important higher‑categorical structures such as Θ fall within this framework. This opens the door to systematic homological calculations in higher‑dimensional category theory and offers a robust Quillen‑theoretic foundation for future developments.