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Categorical structures enriched in a quantaloid: tensored and cotensored categories. (English) Zbl 1119.18005
Categories enriched in a locally cocomplete bicategory were studied by the reviewer in a 1983 paper reprinted as [{\it R. H. Street}, “Enriched categories and cohomology”, Repr. Theory Appl. Categ. 14, 1--18 (2005; Zbl 1085.18010)]. Motivation for that work was {\it R. F. C. Walters}’ example of sheaves as enriched categories in which case the base bicategory is locally ordered. A quantaloid is a locally ordered, locally cocomplete bicategory. Categories enriched in a quantaloid exhibit special characteristics, in particular, as the author shows, when it comes to the universal operations of tensoring and cotensoring. Follow up papers will use this work in connection with causal duality and dynamic domains.

##### MSC:
 18D05 Double categories, $2$-categories, bicategories and generalizations 06F07 Quantales 18D20 Enriched categories (over closed or monoidal categories)
##### Keywords:
quantale; bicategory; enriched category; weighted (co)limit; module
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