Categories enriched over a quantaloid: Isbell adjunctions and Kan adjunctions

Abstract

Each distributor between categories enriched over a small quantaloid Q gives rise to two adjunctions between the categories of contravariant and covariant presheaves, and hence to two monads. These two adjunctions are respectively generalizations of Isbell adjunctions and Kan extensions in category theory. It is proved that these two processes are functorial with infomorphisms playing as morphisms between distributors; and that the free cocompletion functor of Q-categories factors through both of these functors.

28 pages, final version

References in corpus (4)

• Categorical structures enriched in a quantaloid: categories, distributors and functors
• Categorical structures enriched in a quantaloid: tensored and cotensored categories
• Limits of small functors
• Higher Operads, Higher Categories

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• Monads on Q-Cat and their lax extensions to Q-Dist
• Enriched categories and tropical mathematics
• The powerset monad on quantale-valued sets
• Diagonals between $\mathcal{Q}$-distributors
• Density in categorical topology via quantaloid-enriched categories
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