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 \(Sets^{{\mathcal {X}}^{op}}\).