# Algebraic cocompleteness and finitary functors

## TL;DR

This paper identifies categories that are algebraically complete and cocomplete, showing 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 with finitary and precontinuous set functors, linking algebraic and metric structures.

## Key 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.