Adjunctions in Quantaloid-enriched Categories

Abstract

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 Kan extensions in category theory, so, they will be called the Isbell adjunction and Kan adjunction, respectively. The functoriality of these constructions is the central topic of this dissertation. In order to achieve this, infomorphisms between distributors are introduced to organize distributors (as objects) into a category. Then we proceed as follows: first, the Isbell adjunction and Kan adjunction associated with each distributor between Q-categories give rise to two monads, which are respectively (covariant) functorial and contravariant functorial from the category of distributors and infomorphisms to the category of Q-closure spaces; second, it is shown that the assignments of a distributor to the fixed points of the two monads are respectively (covariant) functorial and contravariant functorial from the category of distributors and infomorphisms to that of skeletal complete Q-categories and left adjoint functors. As consequences of the functoriality of the above processes, three factorizations of the free cocompletion functor of Q-categories are presented. Finally, as applications, the theory of formal concept analysis and that of rough sets are extended to theories based on fuzzy relations between fuzzy sets.

167 pages, an unofficial English version of the author's PhD thesis. Some typos are corrected in v2

References in corpus (1)

• Categorical structures enriched in a quantaloid: tensored and cotensored categories

Cited by in corpus (14)

• Topological categories, quantaloids and Isbell adjunctions
• Fuzzy Galois connections on fuzzy sets
• $\mathcal{Q}$-closure spaces
• The groupoid structure of groupoid morphisms
• Lax Distributive Laws for Topology, I
• Chu connections and back diagonals between $\mathcal{Q}$-distributors
• Regularity vs. constructive complete (co)distributivity
• Multi-adjoint concept lattices via quantaloid-enriched categories
• Completeness and injectivity
• The powerset monad on quantale-valued sets
• Enriched categories and tropical mathematics
• Isbell adjunctions and Kan adjunctions via quantale-enriched two-variable adjunctions
• Density in categorical topology via quantaloid-enriched categories
• Diagonals between $\mathcal{Q}$-distributors