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

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 $\mathsf{P}\mathsf{r}\mathsf{e}\mathsf{O}\mathsf{r}\mathsf{d}(\mathbb{C})$ 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 $\mathsf{P}\mathsf{r}\mathsf{e}\mathsf{O}\mathsf{r}\mathsf{d}(\mathbb{C})$ (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.