On quantaloids and quantal categories
In ``Quantal sets and sheaves over quantales'' [Lith. Math. J. 34, No. 1, 8-29 (1994) and Liet. Mat. Rink. 34, No. 1, 9-31 (1994; Zbl 0837.18003)], the author presented an approach to quantal sets and sheaves on quantales somewhat more general than those that had been considered earlier. The restriction of idempotence on the quantale was replaced by rather technical, but less restrictive axioms. This approach to sheaves on quantales was a modification of some ideas of Höhle on \(M\)-valued sets and integral commutative \(cl\)-monoids. The article under review extends this earlier work of the author on quantales to the setting of quantaloids. Quantaloids are a natural and useful categorical generalization of quantales (which are the one object quantaloids). The recent book by the reviewer [``The theory of quantaloids'', Pitman Res. Notes Math. 348 (1996; Zbl 0845.18003)] contains a detailed look at quantaloids and some of their applications. The author looks at a certain class of quantaloids, (not necessarily unital) and proceeds to define a quantal category for a quantaloid \(Q\) as a set \(X\) ``enriched'' in \(Q\) with an existence predicate. This notion differs from that of \(Q\)-enriched category considered by the reviewer based on the standard notion of category enriched in a base bicategory. The author does provide a useful comparison between the two concepts. He then proceeds to define a category of quantal categories on which is defined a monad assigning to every \(Q\)-category its ``singletons''. The algebras for this monad then lead to the notion of sheaf on \(Q\). No concrete examples are considered and the next step would be to apply these ideas in a particular setting.
scientific article; zbMATH DE number 939528
scientific article