Final coalgebras in accessible categories

TL;DR

This paper establishes conditions under which finitary endofunctors in accessible categories admit final coalgebras, providing explicit constructions in locally finitely presentable categories and extending to broader contexts.

Contribution

It introduces new criteria for the existence of final coalgebras in accessible categories and offers explicit constructions, expanding applicability beyond locally finitely presentable categories.

Findings

Conditions for final coalgebras in accessible categories

Explicit construction in locally finitely presentable categories

Applicability to categories beyond l.f.p. categories

Abstract

We give conditions on a finitary endofunctor of a finitely accessible category to admit a final coalgebra. Our conditions always apply to the case of a finitary endofunctor of a locally finitely presentable (l.f.p.) category and they bring an explicit construction of the final coalgebra in this case. On the other hand, there are interesting examples of final coalgebras beyond the realm of l.f.p. categories to which our results apply. We rely on ideas developed by Tom Leinster for the study of self-similar objects in topology.