Epireflective subcategories and formal closure operators

Mathieu Duckerts-Antoine ; Marino Gran ; Zurab Janelidze

Theory and applications of categories, Tome 32 (2017), pp. 526-546

Résumé

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.

Détail

Publié le : 2017-04-19

Classification : 18A40, 18A20, 18A22, 18A32, 18D30, 08C15
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