Presheaves of groupoids as models for homotopy types

TL;DR

This paper introduces groupoidal test categories as a new framework for modeling homotopy types using presheaves of groupoids, establishing their equivalence with classical test categories and providing new models for homotopy types.

Contribution

It defines groupoidal weak test categories and proves their equivalence with test categories, offering novel models for homotopy types via presheaves of groupoids.

Findings

Every weak test category is a groupoidal weak test category.

A category is a test category iff it is a groupoidal test category.

New models for homotopy types include categories of groupoids internal to various set theories.

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.