## 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 $${\mathcal S}up$$. A quantoloid $$\mathcal Q$$ is a $${\mathcal S}up$$-enriched category. For a quantoloid $$\mathcal Q$$, definitions of $$\mathcal Q$$-enriched categories, of distributors and functors between $$\mathcal Q$$-enriched categories are given. The theory of enriched categories is developed for $$\mathcal Q$$-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 $$\mathcal Q$$-enriched categories for a quantoloid $$\mathcal Q$$. 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 and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) 18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) 18B35 Preorders, orders, domains and lattices (viewed as categories)
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