Closure hyperdoctrines, with paths
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Spatial logics are modal logics whose modalities are interpreted using topological concepts of neighbourhood and connectivity. Recently, these logics have been extended to (pre)closure spaces, a generalization of topological spaces covering also the notion of neighbourhood in discrete structures. In this paper we introduce an abstract theoretical framework for the systematic investigation of the logical aspects of closure spaces. To this end we define the categorical notion of closure (hyper)doctrine, which are doctrines endowed with inflationary operators (and subject to suitable conditions). The generality and effectiveness of this notion is demonstrated 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). In order to model also surroundedness, closure hyperdoctrines are then endowed with paths; this construction allows us to cover all the logical constructs of the Spatial Logic for Closure Spaces. By leveraging general categorical constructions, we provide a first axiomatisation and sound and complete semantics for propositional/regular/first order 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 various applications.
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