Presheaves of groupoids as models for homotopy types

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.