TL;DR: It is shown that adequate semantics can be provided for imperative higher order concurrent languages simply using syntactical final coalgebras using hypersets and c.m.s. 's.
Abstract: We show that adequate semantics can be provided for imperative higher order concurrent languages simply using syntactical final coalgebras. In particular we investigate and compare various behavioural equivalences on higher order processes defined by finality using hypersets and c.m.s. 's. Correspondingly, we derive various coinduction and mixed induction-coinduction proof principles for establishing these equivalences.
TL;DR: For deterministic systems expressed as coalgebras over polynomial functors, every tree t (an element of the final coalgebra) turns out to represent a new coalgebra At, and every system is a filtered colimit of finitely presentable systems.