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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 𝒮up. A quantoloid 𝒬 is a 𝒮up-enriched category. For a quantoloid 𝒬, definitions of 𝒬-enriched categories, of distributors and functors between 𝒬-enriched categories are given. The theory of enriched categories is developed for 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 𝒬-enriched categories for a quantoloid 𝒬. Several examples illustrating obtained results are presented. The appendix is devoted to distributor calculus.

MSC:
18D20Enriched categories (over closed or monoidal categories)
06F07Quantales
18A30Limits; colimits
18A40Adjoint functors
18B35Preorders, orders and lattices (viewed as categories)