[论文解读] Adjunctions in Quantaloid-enriched Categories

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

Rough Sets and Fuzzy Logic参考文献 64被引用 17

一句话总结

本文提出了一套在关于量值半群（quantaloid）的范畴上进行富集的范畴中，对伴随函子的统一框架，通过分配器（distributors）在Q-范畴之间引入Isbell和Kan伴随函子。通过信息同态（infomorphisms）建立这些构造的函子性，从而导出自由共完备化函子的新分解方式，并将形式概念分析和粗糙集理论推广至模糊设定。

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.

研究动机与目标

• 将Isbell和Kan伴随函子推广至关于量值半群Q的富集范畴。
• 通过分配器之间的信息同态，建立这些伴随函子的函子性。
• 将伴随函子所生成的单子（monads）的不动点表征为骨架完备的Q-范畴。
• 将形式概念分析与粗糙集理论推广至模糊集合之间的模糊关系。
• 为Q-范畴中的自由共完备化函子提供新的分解方式。

提出的方法

• 引入Q-分配器之间的信息同态，构成分配器的范畴。
• 对每个分配器φ: A −◦−/ B，定义Isbell伴随函子φ↓ ⊣ φ↑与Kan伴随函子φ∗ ⊣ φ∗。
• 在PA与PB上构造单子φ↓◦φ↑与φ∗◦φ∗，其中PA、PB为预层（presheaves）的Q-范畴。
• 引入Q-闭包空间（Q-closure spaces）作为在其预层范畴上配备单子的Q-范畴。
• 证明映射φ ↦ (A, φ↓◦φ↑)与φ ↦ (B, φ∗◦φ∗)是到Q-闭包空间范畴的协变与反变函子。
• 证明这些单子的不动点构成骨架完备的Q-范畴，且映射具有函子性。

实验结果

研究问题

• RQ1 如何将Isbell与Kan伴随函子推广至关于量值半群的Q-范畴？
• RQ2 在分配器之间的信息同态下，Isbell与Kan伴随函子的函子行为如何？
• RQ3 单子φ↓◦φ↑与φ∗◦φ∗的不动点如何与Q-范畴中的完备性相关？
• RQ4 能否利用这些伴随函子对Q-范畴的自由共完备化函子进行分解？
• RQ5 如何将形式概念分析与粗糙集理论推广至模糊集合之间的模糊关系？

主要发现

• 映射φ ↦ (A, φ↓◦φ↑)是从Q-分配器与信息同态的范畴到Q-闭包空间范畴的协变函子。
• 映射φ ↦ (B, φ∗◦φ∗)是从Q-分配器与信息同态的范畴到Q-闭包空间范畴的反变函子。
• 单子φ↓◦φ↑在PA上的不动点构成一个完备的Q-范畴，且该映射是协变函子性。
• 单子φ∗◦φ∗在PB上的不动点也构成一个完备的Q-范畴，且映射具有反变函子性。
• 从构造的函子性中导出了Q-范畴的自由共完备化函子的三种新分解方式。
• 利用Q-范畴与Q-分配器，将形式概念分析与粗糙集理论推广至模糊设定。