On Well-Founded and Recursive Coalgebras

Abstract

This paper studies fundamental questions concerning categorytheoretic 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.

A Categorical Framework for Learning Generalised Tree Automata

“…Such a coalgebra is recursive, with the unique map into an algebra being defined as a restricted reachability map. In fact, under certain conditions that are satisfied in the DA setting, recursivity of a coalgebra is equivalent to having a coalgebra homomorphism into the initial algebra [5,Corollary 5.6]. This means that every recursive coalgebra is isomorphic to one given by a prefix-closed multiset of words.…”