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Rosenthal, Kimmo I.

Girard quantaloids. (English) Zbl 0761.18008

Math. Struct. Comput. Sci. 2, No. 1, 93-108 (1992).
This paper synthesizes the author’s previous work on Girard quantales [Cah. Topologie Géom. Différ. Catégoriques 31, 3-11 (1990; Zbl 0713.06005)] and on quantaloids [J. Pure Appl. Algebra 72, 67-82 (1991; Zbl 0729.18007)]. “Quantaloids” is the name which the author insists on using for categories enriched in the closed monoidal category of sup- lattices. Here he shows how the notion of cyclic dualizing element, which in a quantale (a one-object quantaloid) is used to model Girard’s linear negation, may be extended to quantales with several objects. A number of examples of, and constructions on, Girard quantaloids are given, but no specific applications are suggested.
Reviewer: P.T.Johnstone (Cambridge)

Cited in 12 Documents

MSC:

18D20 Enriched categories (over closed or monoidal categories)
03G30 Categorical logic, topoi
06F05 Ordered semigroups and monoids

Keywords:

Girard quantales; quantaloids; categories enriched in the closed monoidal category of sup-lattices; cyclic dualizing element; Girard’s linear negation

Citations:

Zbl 0713.06005; Zbl 0729.18007
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\textit{K. I. Rosenthal}, Math. Struct. Comput. Sci. 2, No. 1, 93--108 (1992; Zbl 0761.18008)
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References:

[1] DOI: 10.1016/0022-4049(87)90065-X · Zbl 0615.18006
[2] Freyd, Categories, Allegories (1990)
[3] Carboni, Pac. J. Math 124 pp 275– (1986) · Zbl 0565.18001
[4] DOI: 10.1016/0022-4049(83)90100-7 · Zbl 0571.18004
[5] Barr, *-Autonomous Categories, Springer Lecture Notes in Mathematics 752 (1979)
[6] DOI: 10.1112/plms/s3-57.3.433 · Zbl 0619.18005
[7] Pin, Varieties of Formal Languages (1986)
[8] DOI: 10.1111/j.1746-8361.1969.tb01194.x · Zbl 0341.18002
[9] Lambek, Introduction to Higher Order Categorical Logic (1986) · Zbl 0596.03002
[10] Kasangian, Cah. de Top. et Géom. Diff. Cat XXVII pp 137– (1986)
[11] Kelly, Basic Concepts of Enriched Category Theory (1982) · Zbl 0478.18005
[12] Johnstone, Stone Spaces (1982)
[13] DOI: 10.2307/2274953 · Zbl 0701.03026
[14] Vickers, Topology via Logic. Camb. Tracts Theor. Comp. Sci (1989)
[15] Seely, Categories in Computer Science and Logic. Cont. Math pp 371– (1989) · Zbl 0674.03007
[16] DOI: 10.1016/0022-4049(91)90130-T · Zbl 0729.18007
[17] Rosenthal, Cah. de Top. et Géom. Diff. Cat XXXI pp 3– (1990)
[18] Rosenthal, Quantales and their Applications. Pitman Research Notes in Math 234 (1990) · Zbl 0703.06007
[19] DOI: 10.1017/S0305004100065403 · Zbl 0658.06007
[20] Girard, Categories in Comp. Sci. and Logic, Cont. Math pp 69– (1989)
[21] DOI: 10.1016/0304-3975(87)90045-4 · Zbl 0625.03037
[22] DOI: 10.1016/0022-4049(87)90065-X · Zbl 0615.18006
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