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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\u2019s 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.<\/jats:p>","DOI":"10.1007\/978-3-030-45231-5_2","type":"book-chapter","created":{"date-parts":[[2020,4,17]],"date-time":"2020-04-17T10:02:53Z","timestamp":1587117773000},"page":"17-36","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On Well-Founded and Recursive Coalgebras"],"prefix":"10.1007","author":[{"given":"Ji\u0159\u00ed","family":"Ad\u00e1mek","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2021-1644","authenticated-orcid":false,"given":"Stefan","family":"Milius","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lawrence S.","family":"Moss","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,4,17]]},"reference":[{"key":"2_CR1","unstructured":"Ad\u00e1mek, J.: Free algebras and automata realizations in the language of categories. 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