Limits in categories of Vietoris coalgebras

Abstract

Motivated by the need to reason about hybrid systems, we study limits in categories of coalgebras whose underlying functor is a Vietoris polynomial one -intuitively, the topological analogue of a Kripke polynomial functor. Among other results, we prove that every Vietoris polynomial functor admits a final coalgebra if it respects certain conditions concerning separation axioms and compactness.When the functor is restricted to some of the categories induced by these conditions the resulting categories of coalgebras are even complete.As a practical application, we use these developments in the specification and analysis of nondeterministic hybrid systems, in particular to obtain suitable notions of stability, and behaviour.

Coalgebras for Vietoris and monadicity

“…A first hint of the characterisation of codirected limits in CH is in Bourbaki [16]. The characterisation was proved in [27] (see also [29,Theorem 3.29]), and it is called the Bourbaki criterion. Here, we formulate it in the setting of compact ordered spaces.…”

“…Indeed, the Vietoris functor admits very natural extensions to several categories of compact spaces, and thus the study of Stone duality above dimension 0 [41] has attracted a lot of attention. In [29,28] the authors show quasivariety results for some of these categories.…”

Saturated Kripke Structures as Vietoris Coalgebras

“…Hence, when extending the Vietoris functor to act on arbitrary topological spaces X = (X, τ ), one has the choice to take as base set for V(X ) all closed subsets or all compact subsets of X. In [13] the authors show that both choices lead to endofunctors on the category T op of topological spaces, the "lower" Vietoris functor, and the compact Vietoris functor. Here we shall only need to work with the latter, which for us then is "the" Vietoris functor:…”

“…Named the lower Vietoris functor, resp. the compact Vietoris functor, these endofunctors on the category T op of topological spaces with continuous functions were explored in recent work by Hofmann, Neves and Nora [13].…”

Languages and models for hybrid automata: A coalgebraic perspective

“…Such is indeed not the case for the powerset functor and the distribution functor and it suggests a shift from the category Set to other categories where this kind of behaviour can be better handled. For example the category of topological spaces Top [32] and the category of Polish spaces Pol [33].…”