Article Title

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

Authors

SEYED NASER HOSSEINI
MEHDI NODEHI

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.

Keywords

(Preordered) morphism class, \Omega-subpresheaf, (weak) Grothendieck topology, \Omega-automorphism, (weak) Lawvere--Tierney topology, universal operation, (idempotent) universal closure operation

First Page

818

Last Page

829

Recommended Citation

HOSSEINI, SEYED NASER and NODEHI, MEHDI (2013) "Morphism classes producing (weak) Grothendieck topologies, (weak) Lawvere--Tierney topologies, and universal closure operations," Turkish Journal of Mathematics: Vol. 37: No. 5, Article 9. https://doi.org/10.3906/mat-1206-16
Available at: https://journals.tubitak.gov.tr/math/vol37/iss5/9