Epireflective subcategories and formal closure operators

Abstract

On a category $\mathscr{C}$ with a designated (well-behaved) class $\mathcal{M}$ of monomorphisms, a closure operator in the sense of D. Dikranjan and E. Giuli is a pointed endofunctor of $\mathcal{M}$, seen as a full subcategory of the arrow-category $\mathscr{C}^\mathbf{2}$ whose objects are morphisms from the class $\mathcal{M}$, which "commutes" with the codomain functor $\mathsf{cod}\colon \mathcal{M}\to \mathscr{C}$. In other words, a closure operator consists of a functor $C\colon \mathcal{M}\to\mathcal{M}$ and a natural transformation $c\colon 1_\mathcal{M}\to C$ such that $\mathsf{cod} \cdot C=C$ and $\mathsf{cod}\cdot c=1_\mathsf{cod}$. In this paper we adapt this notion to the domain functor $\mathsf{dom}\colon \mathcal{E}\to\mathscr{C}$, where $\mathcal{E}$ is a class of epimorphisms in $\mathscr{C}$, and show that such closure operators can be used to classify $\mathcal{E}$-epireflective subcategories of $\mathscr{C}$, provided $\mathcal{E}$ is closed under composition and contains isomorphisms. Specializing to the case when $\mathcal{E}$ is the class of regular epimorphisms in a regular category, we obtain known characterizations of regular-epireflective subcategories of general and various special types of regular categories, appearing in the works of the second author and his coauthors. These results show the interest in investigating further the notion of a closure operator relative to a general functor. They also point out new links between epireflective subcategories arising in algebra, the theory of fibrations, and the theory of categorical closure operators.