The stable category of preorders in a pretopos I: general theory

TL;DR

This paper develops a general categorical framework for the stable category of internal preorders within coherent categories, extending previous work and identifying conditions under which certain sequences are preserved, with implications for universal properties.

Contribution

It introduces an alternative construction of the stable category for internal preorders in any coherent category, highlighting its categorical nature and preservation properties in pretoposes.

Findings

The quotient functor preserves finite coproducts in pretoposes.

Identifies classes of pretoposes where short $ ext{Z}$-exact sequences map to short exact sequences.

Establishes foundational properties for the universal characterization of the stable category.

Abstract

In a recent article Facchini and Finocchiaro considered a natural pretorsion theory in the category of preordered sets inducing a corresponding stable category. In the present work we propose an alternative construction of the stable category of the category $\mathsf{PreOrd} (\mathbb C)$ of internal preorders in any coherent category $\mathbb C$, that enlightens the categorical nature of this notion. When $\mathbb C$ is a pretopos we prove that the quotient functor from the category of internal preorders to the associated stable category preserves finite coproducts. Furthermore, we identify a wide class of pretoposes, including all $\sigma$-pretoposes and all elementary toposes, with the property that this functor sends any short $\mathcal Z$-exact sequences in $\mathsf{PreOrd} (\mathbb C)$ (where $\mathcal Z$ is a suitable ideal of trivial morphisms) to a short exact sequence in the stable category. These properties will play a fundamental role in proving the universal property of the stable category, that will be the subject of a second article on this topic.