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

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

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.

Anahtar Kelimeler:

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

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

Keywords:

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

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