On Well-Founded and Recursive Coalgebras

TL;DR

This paper explores the relationship between well-founded and recursive coalgebras in category theory, providing new proofs and characterizations that deepen understanding of induction and recursion models.

Contribution

It offers new, shorter proofs that well-founded coalgebras form a coreflection and generalizes Taylor's Recursion Theorem, with novel characterizations of well-foundedness.

Findings

Every coalgebra has a well-founded part

Well-founded coalgebras are recursive

Well-foundedness characterized by coalgebra-to-algebra morphisms

Abstract

This paper studies fundamental questions concerning category-theoretic models of induction and recursion. We are concerned with the relationship between well-founded and recursive coalgebras for an endofunctor. For monomorphism preserving endofunctors on complete and well-powered categories every coalgebra has a well-founded part, and we provide a new, shorter proof that this is the coreflection in the category of all well-founded coalgebras. We present a new more general proof of Taylor's General Recursion Theorem that every well-founded coalgebra is recursive, and we study under which hypothesis the converse holds. In addition, we present a new equivalent characterization of well-foundedness: a coalgebra is well-founded iff it admits a coalgebra-to-algebra morphism to the initial algebra.