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 PreOrd(C) of internal preorders in any coherent category C, that enlightens the categorical nature of this notion. When 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 σ-pretoposes and all elementary toposes, with the property that this functor sends any short Z-exact sequences in PreOrd(C) (where 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. Introduction In a recent article [9] Facchini and Finocchiaro observed that in the category PreOrd of preordered sets there is a natural pretorsion theory (T ,F) = (Eq,ParOrd), where Eq is the “torsion subcategory” of equivalence relations and ParOrd the “torsion-free” subcategory of partial orders. Let us write Z = Eq ∩ ParOrd for the full subcategory of PreOrd whose objects are discrete equivalence relations, and call Z-trivial a morphism in PreOrd that factors through an object in Z. Then the fact that (Eq,ParOrd) is a pretorsion theory can be expressed as follows: (1) any morphism f : (X, τ) → (Y, σ) from an equivalence relation (X, τ) to a partial order (Y, σ) is Z-trivial; (2) for any preorder (A, ρ) there is a canonical short Z-exact sequence (A,∼ρ) IdA // (A, ρ) π // (A/∼ρ, π(ρ)) where ∼ρ= ρ ∩ ρ is the equivalence relation given by the intersection of ρ with its opposite relation ρ, π : A → A/∼ρ is the canonical quotient and π(ρ) is the partial order induced on A/∼ρ by ρ. 2020 Mathematics Subject Classification. Primary 06A75, 18B25, 18B50, 18B35, 18E08, 18E40.