Final Coalgebras are Ideal Completions of Initial Algebras

Abstract

For ω‐continuous endofunctors of Set an ordering of a final coalgebra T is exhibited which makes T a CPO. Moreover, an initial algebra, considered as a canonical subobject of T, hasT as its ideal completion. In more generality, for ω‐continuous endofunctors of locally finitely presentable categories the analogous result holds: here the ordering is considered on the hom‐sets hom(B, T) for all finitely presentable objects B.