Algebraic cocompleteness and finitary functors

TL;DR

This paper identifies categories that are algebraically complete and cocomplete, demonstrating that for certain set functors, initial algebras and terminal coalgebras have canonical order and metric structures with shared completions.

Contribution

It establishes algebraic completeness and cocompleteness for categories and characterizes the order and metric structures of initial algebras and terminal coalgebras for finitary and precontinuous set functors.

Findings

Initial algebra and terminal coalgebra carry canonical partial orders.

Both structures have the same ideal CPO-completion.

They also carry a canonical ultrametric with the same Cauchy completion.

Abstract

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.