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The topological interpretation of modal logics provides descriptive languagesand proof systems for reasoning about points of topological spaces. Recent workhas been devoted to model checking of spatial logics on discrete spatialstructures, such as finite graphs and digital images, with applications invarious case studies including medical image analysis. These recentdevelopments required a generalisation step, from topological spaces to closurespaces. In this work we initiate the study of bisimilarity and minimisationalgorithms that are consistent with the closure spaces semantics. For thispurpose we employ coalgebraic models. We present a coalgebraic definition ofbisimilarity for quasi-discrete models, which is adequate with respect to aspatial logic with reachability operators, complemented by a free andopen-source minimisation tool for finite models. We also discuss thenon-quasi-discrete case, by providing a generalisation of the well-knownset-theoretical notion of topo-bisimilarity, and a categorical definition, inthe same spirit as the coalgebraic rendition of neighbourhood frames, butemploying the covariant power set functor, instead of the contravariant one. Weprove its adequacy with respect to infinitary modal logic.

Towards Spatial Bisimilarity for Closure Models: Logical and Coalgebraic Characterisations