Limits in Categories of Vietoris Coalgebras

TL;DR

This paper investigates limits in categories of Vietoris coalgebras, establishing conditions for the existence of final coalgebras and completeness, with applications to hybrid systems analysis.

Contribution

It introduces conditions under which Vietoris polynomial functors admit final coalgebras and are complete, advancing the theoretical framework for hybrid systems modeling.

Findings

Every Vietoris polynomial functor admits a final coalgebra under certain conditions.

Categories of coalgebras can be complete when restricted appropriately.

Applications include stability and behavior analysis of non-deterministic hybrid systems.

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 non-deterministic hybrid systems, in particular to obtain suitable notions of stability, and behaviour.