Closure Hyperdoctrines

Authors Davide Castelnovo, Marino Miculan

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Davide Castelnovo
  • Department of Mathematics, Computer Science and Physics, University of Udine, Italy
Marino Miculan
  • Department of Mathematics, Computer Science and Physics, University of Udine, Italy

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Davide Castelnovo and Marino Miculan. Closure Hyperdoctrines. In 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 211, pp. 12:1-12:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


(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 present 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). By leveraging general categorical constructions, we provide axiomatisations and sound and complete semantics for various fragments of logics for closure operators. Hence, closure hyperdoctrines are useful both for refining and improving the theory of existing spatial logics, and for the definition of new spatial logics for new applications.

Subject Classification

ACM Subject Classification
  • Theory of computation → Modal and temporal logics
  • categorical logic
  • topological semantics
  • closure operators
  • spatial logic


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