闭包hyperdoctrine，带路径
Closure hyperdoctrines, with paths
摘要：
（前）闭包空间是拓扑空间的推广，也涵盖了离散结构中的邻域概念，广泛用于分布式系统的空间方面的建模和推理。在本文中，我们介绍了一个抽象的理论框架，系统的调查的逻辑方面的封闭空间。为此，我们引入了闭包（超）理论的概念，即赋予暴胀算子（并服从适当条件）的理论。这个概念的一般性和有效性由许多自然产生的例子证明，这些例子来自拓扑空间、模糊集、代数结构、余代数，并且同时覆盖了已知的情况，例如Kripke框架和概率框架（即，马尔可夫链）。然后，我们展示了如何通过赋予超理论一个一般的路径概念来解释关于可达性和可扩展性的空间逻辑结构。通过利用一般的范畴结构，我们提供了公理化和声音和完整的语义的各种片段的逻辑闭包运算符。因此，闭包超理论对于精炼和改进现有的空间逻辑理论都是有用的，但特别是对于为新的应用定义新的空间逻辑。
英文摘要：
(Pre)closure spaces are a generalization of topological spaces covering also the notion of neighbourhood in discrete structures, widely used to model and reason about spatial aspects of distributed systems. In this paper we introduce an abstract theoretical framework for the systematic investigation of the logical aspects of closure spaces. To this end, we introduce the notion of closure (hyper)doctrines, i.e. doctrines endowed with inflationary operators (and subject to suitable conditions). The generality and effectiveness of this concept is witnessed by many examples arising naturally from topological spaces, fuzzy sets, algebraic structures, coalgebras, and covering at once also known cases such as Kripke frames and probabilistic frames (i.e., Markov chains). Then, we show how spatial logical constructs concerning surroundedness and reachability can be interpreted by endowing hyperdoctrines with a general notion of paths. By leveraging general categorical constructions, we provide axiomatisations and sound and complete semantics for various fragments of logics for closure operators. Therefore, closure hyperdoctrines are useful both for refining and improving the theory of existing spatial logics, but especially for the definition of new spatial logics for new applications.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
