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Girard quantaloids. (English) Zbl 0761.18008
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.

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