The stable category of preorders in a pretopos II: the universal property

TL;DR

This paper establishes a universal property of the stable category derived from internal preorders in a pretopos, linking pretorsion and torsion theories categorically, extending previous special case results.

Contribution

It proves that the stable category satisfies a universal property, connecting pretorsion theories to classical torsion theories in a broad categorical context.

Findings

The stable category satisfies a universal property.

The canonical functor transforms pretorsion theories into torsion theories.

Provides categorical insight into the construction of the stable category.

Abstract

We prove that the stable category associated with the category $\mathsf{PreOrd}(\mathbb C)$ of internal preorders in a pretopos $\mathbb C$ satisfies a universal property. The canonical functor from $\mathsf{PreOrd}(\mathbb C)$ to the stable category $\mathsf{Stab}(\mathbb C)$ universally transforms a pretorsion theory in $\mathsf{PreOrd}(\mathbb C)$ into a classical torsion theory in the pointed category $\mathsf{Stab}(\mathbb C)$. This also gives a categorical insight into the construction of the stable category first considered by Facchini and Finocchiaro in the special case when $\mathbb C$ is the category of sets.