Limits and colimits of quantaloid-enriched categories and their distributors

Categorical structures enriched in a quantaloid: categories, distributors and functors

We thoroughly treat several familiar and less familiar definitions and results concerning categories, functors and distributors enriched in a base quantaloid Q. In analogy with V-category theory we discuss such things as adjoint functors, (pointwise) left Kan extensions, weighted (co)limits, presheaves and free (co)completion, Cauchy completion and Morita equivalence. With an appendix on the universality of the quantaloid Dist(Q) of Q-enriched categories and distributors.

Adjunctions in Quantaloid-enriched Categories

This dissertation is devoted to a study of adjunctions concerning categories enriched over a quantaloid Q (or Q-categories for short), with the following types of adjunctions involved: (1) adjoint functors between Q-categories; (2) adjoint distributors between Q-categories; (3) adjoint functors between categories consisting of Q-categories. For a small quantaloid Q and a distributor between Q-categories, two adjunctions between the Q-categories of contravariant and covariant presheaves are presented. These adjunctions respectively extend the fundamental construction of Isbell adjunctions and K…

Categories enriched over a quantaloid: Isbell adjunctions and Kan adjunctions

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.

Diagonals between $\mathcal{Q}$-distributors

For a small quantaloid $\mathcal{Q}$, it is shown that the category of $\mathcal{Q}$-distributors and diagonals is equivalent to a quotient category of the category of $\mathcal{Q}$-interior spaces and continuous $\mathcal{Q}$-distributors. Kan adjunctions induced by $\mathcal{Q}$-distributors play a crucial role in establishing this equivalence.

Categorical structures enriched in a quantaloid: tensored and cotensored categories

Our subject is that of categories, functors and distributors enriched in a base quantaloid Q. We show how cocomplete Q-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored, cotensored and order-cocomplete. Bearing this in mind, we analyze how Sup-valued homomorphisms on Q are related to Q-categories. With an appendix on action, representation and variation.

Quantaloid-enriched范畴：分解、弱分类子与对称性

Quantaloid-enriched categories: Factorization, weak classifiers, and symmetry

本文研究了quantaloid-enriched范畴的基础性质，证明了其左伴随分配子范畴存在(epi, extremal mono)-分解系统，并在稳定性条件下满足弱子对象分类子公理，进而将结构结果推广至quantale值集。