- Turkish Journal of Mathematics
- Vol: 37 Issue: 5
- Morphism classes producing (weak) Grothendieck topologies, (weak) Lawvere–Tierney topologies, and un...
Morphism classes producing (weak) Grothendieck topologies, (weak) Lawvere–Tierney topologies, and universal closure operations
Authors : Seyed Naser Hosseini, Mehdi Nodehi
Pages : 818-829
Doi:10.3906/mat-1206-16
View : 12 | Download : 4
Publication Date : 9999-12-31
Article Type : Makaleler
Abstract :In this article, given a category X, with W the subobject classifier in Set^{X^{op}, we set up a one-to-one correspondence between certain (i) classes of X-morphisms, (ii) W-subpresheaves, (iii) W-automorphisms, and (iv) universal operators. As a result we give necessary and sufficient conditions on a morphism class so that the associated (i) W-subpresheaf is a (weak) Grothendieck topology, (ii) W-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, W-subpresheaf, (weak) Grothendieck topology, W-automorphism, (weak) Lawvere--Tierney topology, universal operation, (idempotent) universal closure operation