Presheaves of groupoids as models for homotopy types

Guetta, Léonard

doi:10.2140/agt.2025.25.3833

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Presheaves of groupoids as models for homotopy types

Léonard Guetta

Algebraic & Geometric Topology 25 (2025) 3833–3874

DOI: 10.2140/agt.2025.25.3833

Abstract

We introduce the notion of groupoidal (weak) test category, which is a small category $A$ such that the $G r p d$ -valued presheaves over $A$ model 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 two 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 semisimplicial 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.

Keywords

test categories, homotopy types, groupoids, presheaves

Mathematical Subject Classification

Primary: 18N40, 20L05

Secondary: 55P99, 55U35, 55U40

References

Publication

Received: 29 September 2022

Revised: 6 July 2024

Accepted: 21 August 2024

Published: 29 October 2025

Authors

Léonard Guetta
Mathematical Institute Utrecht University Utrecht Netherlands
https://leoguetta.github.io/

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Volume 25, issue 7 (2025)