A Relation Between Closure Operators on a Small Category and Its Category of Presheaves

Abstract

In this paper we show that each factorization structure ${\mathcal {M}}$ on a small category ${\mathcal {X}}$, satisfying certain conditions, yields a presheaf ${{\boldsymbol{M}}}$ on ${\mathcal {X}}$ and a morphism of presheaves ${\mathbf{m}}:\Omega \xrightarrow{.}{\mathbf{M}}$. We then give connections, and set up one to one correspondences, between subclasses of the following classes: (a) closure operators on ${\mathcal {X}}$ (b) subobjects of ${\boldsymbol{M}}$ (c) morphisms from ${\boldsymbol{M}}$ to ${\boldsymbol{\Omega}}$ (d) weak Lawvere–Tierney topologies (e) weak Grothendieck topologies (f) closure operators on $Sets^{{\mathcal {X}}^{op}}$.