A final coalgebra theorem

Initial algebra and final coalgebra semantics for concurrency

“…First the functor P 1 (A x ·) is treated, next arbitrary functors. Much of what follows in this subsection is an expansion of similar results in [AM89], where coalgebras of endofunctors on a category of classes are discussed.…”

“…They are also invited to make the generalizations that are left implicit there. Section 2.3 might be of particular interest since it, together with Section 2.4, provides a bridge between the construction of final coalgebras as given in [AM89] and the one in [Bar93]. Section 5 might be a good example of the generality of the final semantics approach.…”

“…Labelled transition systems correspond to the coalgebras of a certain functor (because of a well-known bijection between relations and non-deterministic functions). Similarly, bisimulations are coalgebras ( [AM89]). (See also [Ken87] for an early reference.)…”

A concrete final coalgebra theorem for ZF set theory

“…From this point of view, the general theorems of Aczel and Mendler [3] and Barr [4] yield final coalgebras for a great many functors.…”

“…This category-theoretic notion relates to the methods of bisimulation and coinduction, which are heavily used in concurrency theory [6], functional programming [1] and operational semantics [7]. Aczel and Mendler [3] and also Barr [4] have proved that final coalgebras exist in set theory for large classes of naturally occurring functors. This might be supposed to satisfy most people's requirements.…”