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.
Keywords: Quantaloid, Q-distributor, complete Q-category, Q-closure space, Isbell adjunction, Kan adjunction
2010 MSC: 18A40, 18D20
Theory and Applications of Categories, Vol. 28, 2013, No. 20, pp 577-615.