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 : 13 | Download : 4

Publication Date : 0000-00-00

Article Type : Research Paper

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 insert ignore into journalissuearticles values(i); classes of X-morphisms, insert ignore into journalissuearticles values(ii); W-subpresheaves, insert ignore into journalissuearticles values(iii); W-automorphisms, and insert ignore into journalissuearticles values(iv); universal operators. As a result we give necessary and sufficient conditions on a morphism class so that the associated insert ignore into journalissuearticles values(i); W-subpresheaf is a insert ignore into journalissuearticles values(weak); Grothendieck topology, insert ignore into journalissuearticles values(ii); W-automorphism is a insert ignore into journalissuearticles values(weak); Lawvere--Tierney topology, and insert ignore into journalissuearticles values(iii); universal operation is an insert ignore into journalissuearticles values(idempotent); universal closure operation. We also finally give several examples of morphism classes yielding insert ignore into journalissuearticles values(weak); Grothendieck topologies, insert ignore into journalissuearticles values(weak); Lawvere--Tierney topologies, and insert ignore into journalissuearticles values(idempotent); universal closure operations.
Keywords : Preordered, morphism class, W subpresheaf, weak, Grothendieck topology, W automorphism, weak, Lawvere Tierney