[Paper Review] Adjunctions in Quantaloid-enriched Categories

Lili Shen|arXiv (Cornell University)|Aug 1, 2014

Rough Sets and Fuzzy Logic64 references17 citations

TL;DR

This paper develops a unified framework for adjunctions in categories enriched over a quantaloid, introducing Isbell and Kan adjunctions between Q-categories via distributors. It establishes functoriality of these constructions through infomorphisms, leading to new factorizations of the free cocompletion functor and extensions of formal concept analysis and rough set theory to fuzzy settings.

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.

Motivation & Objective

• To generalize Isbell and Kan adjunctions to categories enriched over a quantaloid Q.
• To establish functoriality of these adjunctions using infomorphisms between distributors.
• To characterize fixed points of monads arising from adjunctions as skeletal complete Q-categories.
• To extend formal concept analysis and rough set theory to fuzzy relations between fuzzy sets.
• To provide new factorizations of the free cocompletion functor in Q-categories.

Proposed method

• Introduces infomorphisms between Q-distributors to form a category of distributors.
• Defines Isbell adjunction φ↓ ⊣ φ↑ and Kan adjunction φ∗ ⊣ φ∗ for each distributor φ: A −◦−/ B.
• Constructs monads φ↓◦φ↑ on PA and φ∗◦φ∗ on PB, where PA, PB are Q-categories of presheaves.
• Introduces Q-closure spaces as Q-categories equipped with a monad on their presheaf category.
• Proves that the assignments φ ↦ (A, φ↓◦φ↑) and φ ↦ (B, φ∗◦φ∗) are covariant and contravariant functors to the category of Q-closure spaces.
• Shows that fixed points of these monads form skeletal complete Q-categories, with the assignments being functorial.

Experimental results

Research questions

• RQ1 How can Isbell and Kan adjunctions be generalized to Q-categories enriched over a quantaloid?
• RQ2 What is the functorial behavior of the Isbell and Kan adjunctions under infomorphisms between distributors?
• RQ3 How do the fixed points of the monads φ↓◦φ↑ and φ∗◦φ∗ relate to completeness in Q-categories?
• RQ4 Can the free cocompletion functor for Q-categories be factored using these adjunctions?
• RQ5 How can formal concept analysis and rough set theory be extended to fuzzy relations between fuzzy sets?

Key findings

• The correspondence φ ↦ (A, φ↓◦φ↑) is a covariant functor from the category of Q-distributors and infomorphisms to the category of Q-closure spaces.
• The correspondence φ ↦ (B, φ∗◦φ∗) is a contravariant functor to the category of Q-closure spaces.
• The fixed objects of the monad φ↓◦φ↑ on PA form a complete Q-category, and the assignment is covariant functorial.
• The fixed objects of φ∗◦φ∗ on PB also form a complete Q-category, with a contravariant functorial assignment.
• Three new factorizations of the free cocompletion functor for Q-categories are derived from the functoriality of the constructions.
• Formal concept analysis and rough set theory are generalized to fuzzy settings using Q-categories and Q-distributors.