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Turkish Journal of Mathematics

DOI

10.3906/mat-1206-16

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.

First Page

818

Last Page

829

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Mathematics Commons

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