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Categorical structures enriched in a quantaloid: categories, distributors and functors. (English) Zbl 1079.18005
The complete semilattices with complete semilattice homomorphisms form a symmetric monoidal closed category ${\cal S}up$. A quantoloid $\Cal Q$ is a ${\cal S}up$-enriched category. For a quantoloid $\Cal Q$, definitions of $\Cal Q$-enriched categories, of distributors and functors between $\Cal Q$-enriched categories are given. The theory of enriched categories is developed for $\Cal Q$-enriched categories. The notions and properties of adjoint functors, of Kan extensions, of weighted colimits and/or limits, of presheaves, of free cocompletion, of Cauchy completion, and of Morita equivalence are studied in $\Cal Q$-enriched categories for a quantoloid $\Cal Q$. Several examples illustrating obtained results are presented. The appendix is devoted to distributor calculus.

##### MSC:
 18D20 Enriched categories (over closed or monoidal categories) 06F07 Quantales 18A30 Limits; colimits 18A40 Adjoint functors 18B35 Preorders, orders and lattices (viewed as categories)
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