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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 ..."
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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 Qcategories factors through both of these functors. 1.
COVARIANT PRESHEAVES AND SUBALGEBRAS
"... Abstract. For small involutive and integral quantaloids Q it is shown that covariant presheaves on symmetric Qcategories are equivalent to certain subalgebras of a speci c monad on the category of symmetric Qcategories. This construction is related to a weakening of the subobject classi er axiom w ..."
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Abstract. For small involutive and integral quantaloids Q it is shown that covariant presheaves on symmetric Qcategories are equivalent to certain subalgebras of a speci c monad on the category of symmetric Qcategories. This construction is related to a weakening of the subobject classi er axiom which does not require the classi cation of all subalgebras, but only guarantees that classi able subalgebras are uniquely classi able. As an application the identi cation of closed left ideals of noncommutative C∗algebras with certain open subalgebras of freely generated algebras is given.