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 conditions which imply the converse. 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.
CITATION STYLE
Adámek, J., Milius, S., & Moss, L. S. (2020). On Well-Founded and Recursive Coalgebras. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 12077 LNCS, pp. 17–36). Springer. https://doi.org/10.1007/978-3-030-45231-5_2
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