Generalized Vietoris Bisimulations

We introduce and study bisimulations for coalgebras on Stone spaces [14]. Our notion of bisimulation is sound and complete for behavioural equivalence, and generalizes Vietoris bisimulations [4]. The main result of our paper is that bisimulation for a $\mathbf{Stone}$ coalgebra is the topological closure of bisimulation for the underlying $\mathbf{Set}$ coalgebra

Coalgebraic Geometric Logic: Basic Theory

Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor on some full subcategory of the category of topological spaces and continuous functions. We investigate derivation systems, soundness and completeness for such geometric modal logics, and we we specify a method of lifting an endofunctor on Set, accompanied by a collection of predicate liftings, to an endofunctor on the category of topological spaces, again accompanied by a collection of (open) predicate liftings. Furthermore, we compare the notions of modal equivalence, behavioural equivalence and bisimulation on the resulting class of models, and we provide a final object for the corresponding category

Aczel-Mendler Bisimulations in a Regular Category

Aczel-Mendler bisimulations are a coalgebraic extension of a variety of computational relations between systems. It is usual to assume that the underlying category satisfies some form of axiom of choice, so that the theory enjoys desirable properties, such as closure under composition. In this paper, we accommodate the definition in a general regular category - which does not necessarily satisfy any form of axiom of choice. We show that this general definition 1) is closed under composition without using the axiom of choice, 2) coincides with other types of coalgebraic formulations under milder conditions, 3) coincides with the usual definition when the category has the regular axiom of choice. We then develop the particular case of toposes, where the formulation becomes nicer thanks to the power-object monad, and extend the formalism to simulations. Finally, we describe several examples in Stone spaces, toposes for name-passing, and modules over a ring

Coalgebraic Geometric Logic

Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor T on some full subcategory of the category Top of topological spaces and continuous functions. We compare the notions of modal equivalence, behavioural equivalence and bisimulation on the resulting class of models, and we provide a final object for the corresponding category. Furthermore, we specify a method of lifting an endofunctor on Set, accompanied by a collection of predicate liftings, to an endofunctor on the category of topological spaces

Coalgebras of topological types

In This work, we focus on developing the basic theory of coalgebras over the category Top (the category of topological spaces with continuous maps). We argue that, besides Set, the category Top is an interesting base category for coalgebras. We study some endofunctors on Top, in particular, Vietoris functor and P-Vietoris Functor (where P is a set of propositional letters) that due to Hofmann et. al. [42] can be considered as the topological versions of the powerset functor and P-Kripke functor, respectively. We define the notion of compact Kripke structures and we prove that Kripke homomorphisms preserve compactness. Our definition of "compact Kripke structure" coincides with the notion of "modally saturated structures" introduced in Fine [27]. We prove that the class of compact Kripke structures has Hennessy-Milner property. As a consequence we show that in this class of Kripke structures, bihavioral equivalence, modal equivalence and Kripke bisimilarity all coincide.Furthermore, we generalize the notion of descriptive structures defined in Venema et. al. [11] by introducing a notion Vietoris models. We identify Vietoris models as coalgebras for the P-Vietoris functor on the category Top. One can see that each compact Kripke model can be modified to a Vietoris model. This yields an adjunction between the category of Vietoris structures (VS) and the category of compact Kripke structurs (CKS). Moreover, we will prove that the category of Vietoris models has a terminal object. We study the concept of a Vietoris bisimulation between Vietoris models, and we will prove that the closure of a Kripke bisimulation between underlying Kripke models of two Vietoris models is a Vietoris bisimulation. In the end, it will be shown that in the class of Vietoris models, Vietoris bisimilarity, bihavioral equivalence, modal equivalence, all coincide

Games for topological fixpoint logic

Topological fixpoint logics are a family of logics that admits topological models and where the fixpoint operators are defined with respect to the topological interpretations. Here we consider a topological fixpoint logic for relational structures based on Stone spaces, where the fixpoint operators are interpreted via clopen sets. We develop a game-theoretic semantics for this logic. First we introduce games characterising clopen fixpoints of monotone operators on Stone spaces. These fixpoint games allow us to characterise the semantics for our topological fixpoint logic using a two-player graph game. Adequacy of this game is the main result of our paper. Finally, we define bisimulations for the topological structures under consideration and use our game semantics to prove that the truth of a formula of our topological fixpoint logic is bisimulation-invariant