Seyed Naser HOSSEINI, Mehdi NODEHI

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

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

In this article, given a category X , with Ω the subobject classifier in $set^ {chi op}$ Printed: 23.09.2013 , we set up a one-to-one correspondence between certain (i) classes of X -morphisms, (ii) Ω -subpresheaves, (iii) Ω -automorphisms, and (iv) universal operators. As a result we give necessary and sufficient conditions on a morphism class so that the associated (i) Ω -subpresheaf is a (weak) Grothendieck topology, (ii) Ω -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.

PDF

___

[1] Adamek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories. New York, John Wiley and Sons (1990).
[2] Dikranjan, D., Tholen, W.: Categorical Structure of Closure Operators. Dordrecht, Kluwer (1995).
[3] Hosseini, S.N., Mousavi, S.S.: A relation between closure operators on a small category and its category of presheaves. Appl. Categor. Struct. 14, 99ğ110 (2006).
[4] Johnstone, P.T.: Topos Theory. London, Academic Press (1977).
[5] MacLane, S., Moerdijk, I.: Sheaves in Geometry and Logic. A First Introduction to Topos Theory. Berlin, Springer (1992).