Morphism classes producing (weak) Grothendieck topologies, (weak) Lawvere–Tierney topologies, and universal closure operations

Abstract

In this article, given a category X , with Ω the subobject classifier in $set^ {chi op}$ Printed: 23.09.2013 , we set up a one-to-one correspondence between certain (i) classes of X -morphisms, (ii) Ω -subpresheaves, (iii) Ω -automorphisms, and (iv) universal operators. As a result we give necessary and sufficient conditions on a morphism class so that the associated (i) Ω -subpresheaf is a (weak) Grothendieck topology, (ii) Ω -automorphism is a (weak) Lawvere–Tierney topology, and (iii) universal operation is an (idempotent) universal closure operation. We also finally give several examples of morphism classes yielding (weak) Grothendieck topologies, (weak) Lawvere–Tierney topologies, and (idempotent) universal closure operations.