Relatively terminal coalgebras

Dana Scott’s model of λ-calculus was based on a limit construction which started from an algebra of a suitable endofunctor F and continued by iterating F. We demonstrate that this is a special case of the concept we call coalgebra relatively terminal w.r.t. the given algebra A. This means a coalgebra together with a universal coalgebra-to-algebra morphism into A.We prove that by iterating F countably many times we obtain the relatively terminal coalgebras whenever F preserves limits of ωop-chains. If F is finitary, we need in general ω+ω steps. And for arbitrary accessible (=bounded) set functors we need an ordinal number of steps in general. Scott’s result is captured by the fact that in a CPO-enriched category, assuming that F is locally continuous, ω steps are sufficient for algebras given by projections.