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

Abstract

In this article, given a category X, with \Omega the subobject classifier in Set^{X^{op}, we set up a one-to-one correspondence between certain (i) classes of X-morphisms, (ii) \Omega-subpresheaves, (iii) \Omega-automorphisms, and (iv) universal operators. As a result we give necessary and sufficient conditions on a morphism class so that the associated (i) \Omega-subpresheaf is a (weak) Grothendieck topology, (ii) \Omega-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.