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Zbl 1079.18005
Stubbe, Isar
Categorical structures enriched in a quantaloid: categories, distributors and functors.
(English)
[J] Theory Appl. Categ. 14, 1-45, electronic only (2005). ISSN 1201-561X/e

The complete semilattices with complete semilattice homomorphisms form a symmetric monoi\-dal 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.
[Václav Koubek (Praha)]
MSC 2000:
*18D20 Enriched categories
06F07 Quantales
18A30 Limits
18B35 Generalizations of lattices viewed as categories

Keywords: quantaloid; enriched category; limit; colimit; adjoint functor; Kan extension; continuous functor; presheaves; free cocompletion; Cauchy completion; Morita equivalence; distributor; quantale

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