[论文解读] Fibrations of predicates and bicategories of relations

Finn Lawler|arXiv (Cornell University)|Feb 27, 2015

Homotopy and Cohomology in Algebraic Topology参考文献 38被引用 4

一句话总结

本文建立了正则纤维化2-范畴与正则装备2-范畴之间的等价性——即满足正则逻辑模型公理的笛卡尔装备。该等价性通过双函子的三范畴构建，将装备定义为该三范畴中的伪-单子，并表明纤维化中的 comprehension 与装备中的 tabulation 对应，且余单子的 Eilenberg–Moore 对象刻画了 tabulation。

ABSTRACT

We reconcile the two different category-theoretic semantics of regular theories in predicate logic. A 2-category of `regular fibrations' is constructed, as well as a 2-category of `regular proarrow equipments', and it is shown that the two are equivalent. A regular equipment is a `cartesian equipment' satisfying certain axioms, and a cartesian equipment is a slight generalization of a cartesian bicategory. This is done by defining a tricategory Biprof whose objects are bicategories and whose morphisms are category-valued profunctors, and then defining an equipment to be a pseudo-monad in this tricategory. The resulting notion of equipment is compared to several existing ones. Most importantly, this involves showing that every pseudo-monad in Biprof has a Kleisli object. A strict 2-category of equipments, over locally discrete base bicategories, is identified, and cartesian equipments are defined to be the cartesian objects in this 2-category. Thus cartesian equipments themselves form a 2-category, and this is shown to admit a 2-fully-faithful functor from the 2-category of regular fibrations. The cartesian equipments in the image of this functor are characterized as those satisfying certain axioms, and hence a 2-category of `regular equipments' is identified that is equivalent to that of regular fibrations. It is then shown that a regular fibration admits comprehension for predicates if and only if its corresponding regular equipment admits tabulation for morphisms, and further that the presence of tabulations for morphisms is equivalent to the existence of Eilenberg--Moore objects for co-monads. We conclude with a brief examination of the two different constructions of the effective topos, via triposes and via assemblies, in the light of the foregoing.

研究动机与目标

• 统一正则逻辑的两种范畴论语义：纤维化与关系双范畴。
• 构建正则纤维化的2-范畴与正则装备的2-范畴，并证明二者等价。
• 刻画正则纤维化通过 comprehension 与 tabulation 生成正则装备的条件。
• 证明装备中的 tabulation 与余单子的 Eilenberg–Moore 对象等价。
• 在纤维化-装备等价性的视角下，审视通过 tripos 与装配构造的有效拓扑。

提出的方法

• 以双范畴与双函子的三范畴 2-Prof 作为基础框架。
• 将装备定义为 2-Prof 中的伪-单子，推广笛卡尔双范畴的概念。
• 构建基于局部离散基双范畴的装备严格2-范畴。
• 将笛卡尔装备识别为该2-范畴中的笛卡尔对象。
• 证明 2-Prof 中每个伪-单子均存在 Kleisli 对象，从而为装备构造 Kleisli 2-范畴。
• 建立从正则纤维化到笛卡尔装备的2-全忠实函子，并通过公理刻画其像。

实验结果

研究问题

• RQ1 如何通过纤维化与关系双范畴统一建模正则逻辑？
• RQ2 在正则逻辑背景下，统一纤维化与关系双范畴的精确范畴结构为何？
• RQ3 在何种条件下，正则纤维化可通过 comprehension 与 tabulation 对应于正则装备？
• RQ4 余单子的 Eilenberg–Moore 对象如何与笛卡尔装备中的 tabulation 相关联？
• RQ5 通过纤维化-装备等价性，能否比较通过 tripos 与装配构造有效拓扑的两种方式？

主要发现

• 正则纤维化的2-范畴与正则装备的2-范畴等价。
• 正则纤维化中谓词的 comprehension 恰好对应于相应正则装备中态射的 tabulation。
• 笛卡尔装备中态射的 tabulation 等价于余单子的 Eilenberg–Moore 对象的存在性。
• 2-Prof 中每个伪-单子均存在 Kleisli 对象，确保任意装备均存在结构良好的 Kleisli 2-范畴。
• 纤维化与 S(C)-范畴化范畴之间的等价性暗示了在范畴化设定中解释正则结构的一般框架。
• 本研究为通过纤维化-装备对偶性比较可实现性构造（如 tripos 与装配中的构造）提供了结构性基础。