A new Galois structure in the category of internal preorders

Abstract

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, Hébert 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 Paré) 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 $\mathsf{PreOrd}(\mathsf{Set})$.

24 pages, minor corrections

Cited by in corpus (3)

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