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

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.