Abstract
It is known since the late 1960's that the dual of the category of compact Hausdoroff spaces and continuous maps is a variety - not finitary, but bounded by aleph(1). In this note we show that the dual of the category of partially ordered compact spaces and monotone continuous maps is an aleph(1)-ary quasivariety, and describe partially its algebraic theory. Based on this description, we extend these results to categories of Vietoris coalgebras and homomorphisms on ordered compact spaces. We also characterise the aleph(1)-copresentable partially ordered compact spaces.

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.

Abstract
The recently introduced notions of guarded traced (monoidal) category and guarded (pre-)iterative monad aim at unifying di erent instances of partial iteration whilst keeping in touch with the established theory of total iteration and preserving its merits. In this paper we use these notions and the corresponding stock of results to examine di erent types of iteration for hybrid computations. As a starting point we use an available notion of hybrid monad restricted to the category of sets, and modify it in order to obtain a suitable notion of guarded iteration with guardedness interpreted as progressiveness in time – we motivate this modification by our intention to capture Zeno behaviour in an arguably general and feasible way. We illustrate our results with a simple programming language for hybrid computations and interpret it over the developed semantic foundations. © Sergey Goncharov, Julian Jakob, and Renato Neves.

Abstract
Hybrid programs combine digital control with differential equations, and naturally appear in a wide range of application domains, from biology and control theory to real-time software engineering. The entanglement of discrete and continuous behaviour inherent to such programs goes beyond the established computer science foundations, producing challenges related to e.g. infinite iteration and combination of hybrid behaviour with other effects. A systematic treatment of hybridness as a dedicated computational effect has emerged recently. In particular, a generic idealized functional language HYBCORE with a sound and adequate operational semantics has been proposed. The latter semantics however did not provide hints to implementing HYBCORE as a runnable language, suitable for hybrid system simulation (e.g. the semantics features rules with uncountably many premises). We introduce an imperative counterpart of HYBCORE, whose semantics is simpler and runnable, and yet intimately related with the semantics of HYBCORE at the level of hybrid monads. We then establish a corresponding soundness and adequacy theorem. To attest that the resulting semantics can serve as a firm basis for the implementation of typical tools of programming oriented to the hybrid domain, we present a web-based prototype implementation to evaluate and inspect hybrid programs, in the spirit of GHCI for HASKELL and UTOP for OCAML. The major asset of our implementation is that it formally follows the operational semantic rules.

Abstract
Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale V, which covers the cases of (in)equations and (ultra)metric equations among others. Our main result is the introduction of a V-equational deductive system for linear ?-calculus together with a proof that it is sound and complete (in fact, an internal language) for a class of enriched autonomous categories. In the case of inequations, we get an internal language for autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an internal language for autonomous categories enriched over (ultra)metric spaces. We use our results to obtain examples of inequational and metric equational systems for higher-order programs that contain real-time and probabilistic behaviour.

Abstract