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Free quantaloids. (English) Zbl 0729.18007
A quantaloid is the author’s (rather inelegant) name for a category enriched in the closed category of complete semilattices. Such categories have been studied before [e.g. by A. M. Pitts, Proc. Lond. Math. Soc., III. Ser. 57, 433-480 (1988; Zbl 0619.18005)]. However, the author’s concern here is to view them as a generalization of the quantales introduced by C. J. Mulvey [Rend. Circ. Mat. Palermo, II. Ser., Suppl. 12, 99-104 (1986; Zbl 0633.46005)], which are simply quantaloids with one object. The free functor from ordinary (locally small) categories to quantaloids is simply constructed, by applying the power-set functor to each hom-set; the author observes that the resulting adjunction is monadic, and also studies an appropriate notion of nuclei for quantaloids. Finally he studies categories enriched in a free quantaloid, and advances the claim that they provide a suitable notion of “nondeterministic functor” which may have applications in computer science.

MSC:
18D20 Enriched categories (over closed or monoidal categories)
06B23 Complete lattices, completions
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