# Epireflective subcategories and formal closure operators

## Mathieu Duckerts-Antoine, Marino Gran, and Zurab Janelidze

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.

Keywords: category of morphisms, category of epimorphisms, category of monomorphisms, cartesian lifting, closure operator, codomain functor, cohereditary operator, domain functor, epimorphism, epireflective subcategory, form, minimal operator, monomorphism, normal category, pointed endofunctor, reflection, reflective subcategory, regular category, subobject, quotient

2010 MSC: 18A40, 18A20, 18A22, 18A32, 18D30, 08C15

Theory and Applications of Categories, Vol. 32, 2017, No. 15, pp 526-546.

Published 2017-04-19.

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