On Terminal Coalgebras Derived from Initial Algebras

A number of important set functors have countable initial algebras, but terminal coalgebras are uncountable or even non-existent. We prove that the countable cardinality is an anomaly: every set functor with an initial algebra of a finite or uncountable regular cardinality has a terminal coalgebra of the same cardinality. We also present a number of categories that are algebraically complete and cocomplete, i.e., every endofunctor has an initial algebra and a terminal coalgebra. Finally, for finitary set functors we prove that the initial algebra mu F and terminal coalgebra nu F carry a canonical ultrametric with the joint Cauchy completion. And the algebra structure of mu F determines, by extending its inverse continuously, the coalgebra structure of nu F

Algebraic cocompleteness and finitary functors

A number of categories is presented that are algebraically complete and cocomplete, i.e., every endofunctor has an initial algebra and a terminal coalgebra. For all finitary (and, more generally, all precontinuous) set functors the initial algebra and terminal coalgebra are proved to carry a canonical partial order with the same ideal CPO-completion. And they also both carry a canonical ultrametric with the same Cauchy completion

Algebraic cocompleteness and finitary functors

A number of categories is presented that are algebraically complete and cocomplete, i.e., every endofunctor has an initial algebra and a terminal coalgebra. For all finitary (and, more generally, all precontinuous) set functors the initial algebra and terminal coalgebra are proved to carry a canonical partial order with the same ideal CPO-completion. And they also both carry a canonical ultrametric with the same Cauchy completion