Recursive data types in algebraically \(\omega\)-complete categories (Q1892884)

A connection between the notions of ``algebraically complete'' and ``algebraically compact'' category is considered. The notions were introduced by \textit{P. Freyd}. In the first case, every functor should have a least fixpoint, and in the second, a least and the largest fixpoint, that canonically coincide. It is shown that 1) several interesting categories are algebraically \(\omega\)-complete and 2) categories enriched over complete partial orders are ``almost'' \(\omega\)-compact: each localy-continuous functor has a canonical fixpoint.