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

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