#
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.

http://www.tac.mta.ca/tac/volumes/32/15/32-15.pdf

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