Presheaves of groupoids as models for homotopy types
Léonard Guetta
TL;DR
This work extends Grothendieck's test-category framework to presheaves valued in groupoids. By introducing groupoidal versions of pseudo-, weak-, local-, and strict test categories and establishing precise adjunction-based criteria, the authors show that when a small category is groupoidally a test category, it is automatically a (set-valued) test category, and vice versa. This yields numerous new homotopy-type models, notably categories of groupoids internal to cubical, cellular, semisimplicial, and dendroidal sets, among others, and demonstrates a robust equivalence of homotopy theories between Set- and Grpd-valued presheaves in these contexts. A key technical device is the Grothendieck construction $I_{\mathsf{A}}$ and its adjoint, which translate Grpd-valued presheaves into ordinary categories, together with a nerve-based approach to weak equivalences that recovers classical diagonals of bisimplicial nerves. The paper also proves a bonus result: Grpd$(\mathsf{Cat})$ alone models homotopy types, implying that Grpd-valued presheaves over any (weak) test category provide canonical homotopy-type models. Overall, the results deepen the relationship between test-category theory and groupoid-valued homotopy theories, offering a versatile toolkit for modeling and comparing homotopy types across varied combinatorial settings.
Abstract
We introduce the notion of groupoidal (weak) test category, which is a small category A such that the groupoid-valued presheaves over A models homotopy types in a "canonical and nice" way. The definition does not require a priori that A is a (weak) test category, but we prove twon important comparison results: (1) every weak test category is a groupoidal weak test category, (2) a category is a test category if and only if it is a groupoidal test category. As an application, we obtain new models for homotopy types, such as the category of groupoids internal to cubical sets with or without connections, the category of groupoids internal to cellular sets, the category of groupoids internal to semi-simplicial sets, etc. We also prove, as a by-product result, that the category of groupoids internal to the category of small categories models homotopy types.