Dergiler / Turkish Journal of Mathematics / 2013 / Cilt: 37 - Sayı: 5
Morphism classes producing (weak) Grothendieck topologies, (weak) Lawvere--Tierney topologies, and universal closure operations
- Sayfa
- 818–829
- DOI
- —
Özet
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.
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.