The Change-Base Issue for Ω-Categories

Abstract

Let G : Ω → Ω′ be a closed unital map between commutative, unital quantales. G induces a functor Ḡ from the category of Ω-categories to that of Ω′-categories. This paper is concerned with some basic properties of Ḡ. The main results are: (1) when Ω, Ω′ are integral, G : Ω → Ω′ and F : Ω′ → Ω are closed unital maps, $\bar F$ is a left adjoint of Ḡ if and only if F is a left adjoint of G; (2) Ḡ is an equivalence of categories if and only if G is an isomorphism in the category of commutative unital quantales and closed unital maps; and (3) a sufficient condition is obtained for Ḡ to preserve completeness in the sense that ḠA is a complete Ω′-category whenever A is a complete Ω-category.

Keywords

Commutative unital quantale / Closed unital map / Enriched category / Change-base

References

[1] Adámek J., Herrlich H., Strecker G. E.. Abstract and Concrete Categories. The Joy of Cats, 1990, New York: John Wiley & Sons

[2] Bonsangue M., Breugel F., Rutten J. J. M. M.. Generalized metric spaces: completion, topology, and powerdomains via the Yoneda embedding. Theoret. Comput. Sci., 1998, 193: 1-51

[3] Borceux F.. Handbook of Categorical Algebra. Categories and Structures, 1994, Cambridge: Cambridge University Press

[4] Eilenberg S., Kelly G. M.. Closed categories, Proceedings of the Conference on Categorical Algebra, La Jolla 1965, 1966, Berlin: Springer 421-562

[5] Gierz G., Hofmann K. H.. Continuous Lattices and Domains, 2003, Cambridge: Cambridge University Press

[6] Höhle U.. Höhle U., Klement E. P.. Commutative, residuated l-monoids. Non-classical Logics and Their Applications to Fuzzy Subsets: A Handbook on the Mathematical Foundations of Fuzzy Set Theory, 1995, Dordrecht: Kluwer

[7] Kelly G. M.. Basic Concepts of Enriched Category Theory. London Mathematical Soceity Lecture Notes Series 64, 1982, Cambridge: Cambridge University Press

[8] Klement E. P., Mesiar R., Pap E.. Triangular Norms, Trends in Logic, 2000, Dordrecht: Kluwer Academic Publishers

[9] Lawvere F. W.. Metric spaces, generalized logic, and closed categories. Rend. Sem. Mat. Fis. Milano, 1973, 43: 135-166

[10] Lai H., Zhang D.. Complete and directed complete Ω-categories. Theoret. Comput. Sci., 2007, 388: 1-25

[11] Mac Lane S.. Categories for the Working Mathematician, 1998 2 Berlin: Springer

[12] Rosenthal K. I.. Quantales and Their Applications, 1990, London: Longman Scientific & Technical

[13] Schweizer B., Sklar A.. Probabilistic Metric Spaces, 1983, New York: North-Holland

[14] Stubbe I.. Categorical structures enriched in a quantaloid: categories, distributors and functors. Theory Appl. Categ., 2005, 14: 1-45

[15] Stubbe I.. Categorical structures enriched in a quantaloid: tensored and cotensored categories. Theory Appl. Categ., 2006, 16: 283-306

[16] Wagner K. R.. Liminf convergence in Ω-categories. Theoret. Comput. Sci., 1997, 184: 61-104

[17] Zhao H., Zhang D.. Many valued lattices and their representations. Fuzzy Sets and Syst., 2008, 159: 81-94