Let $\mathsf{ PreOrd}(\mathbb C)$ be the category of internal preorders in an exact category $\mathbb C$. We show that the pair $(\mathsf{ Eq}(\mathbb C), \mathsf{ ParOrd}(\mathbb C))$ is a pretorsion theory in $\mathsf{ PreOrd}(\mathbb C)$, where $\mathsf{ Eq}(\mathbb C)$ and $\mathsf{ ParOrd}(\mathbb C)$) are the full subcategories of internal equivalence relations and of internal partial orders in $\mathbb C$, respectively. We observe that $\mathsf{ ParOrd}(\mathbb C)$ is a reflective subcategory of $\mathsf{ PreOrd}(\mathbb C)$ such that each component of the unit of the adjunction is a pullback-stable regular epimorphism. The reflector $F:\mathsf{ PreOrd}(\mathbb C)\to \mathsf{ ParOrd}(\mathbb C)$ turns out to have stable units in the sense of Cassidy, H\'ebert and Kelly, thus inducing an admissible categorical Galois structure. In particular, when $\mathbb C$ is the category $\mathsf{ Set}$ of sets, we show that this reflection induces a monotone-light factorization system (in the sense of Carboni, Janelidze, Kelly and Par\'e) in $\mathsf{ PreOrd}(\mathsf{ Set})$. A topological interpretation of our results in the category of Alexandroff-discrete spaces is also given, via the well-known isomorphism between this latter category and $\mathsf{ PreOrd}(\mathsf{ Set})$.
Categories: math.CT