An homotopical description of small presheaves

TL;DR

This paper explores a homotopical approach to understanding small presheaves by linking cocompletion of categories with finite limits to homotopy categories of equivalence 2-groupoids, unifying various infinitary pretopos definitions.

Contribution

It introduces a novel homotopical framework connecting cocompletion and 2-groupoids, providing a new perspective on infinitary pretopos concepts.

Findings

Cocompletion of categories with finite limits can be described via homotopy categories of equivalence 2-groupoids.

Establishes a link between different definitions of infinitary pretopos.

Provides a simplified conceptual framework for understanding small presheaves.

Abstract

This article describes the cocompletion of a category $C$ with finite limits as the homotopy category of some equivalence 2-groupoids in coproducts of elements of $C$. This yields a simple link between several definitions of an infinitary pretopos.