WELL POINTED ENDOFUNCTORS AND RECURSIVE CONSTRUCTIONS IN CATEGORY THEORY Paul Taylor Online seminar, Thursday 19 June 2025, 10:00 UTC Zoom: bham-ac-uk.zoom.us/j/81873335084 code 217 Montreal 6am, Halifax & Buenos Aires 7am, Reykjavik 10am, UK 11am, CEST noon, Tallinn 1pm, Sydney 8pm. Apologies to Calgary, San Diego and places I've forgotten. Max Kelly (1980) identified generating an idempotent monad from a WELL POINTED ENDOFUNCTOR as a simple case to which other similar problems can be reduced. He then did the construction using transfinite recursion. I will show that, simply by developing the notion of well pointed endofunctor further than Kelly did, along with the use of a GALOIS CONNECTION, the free monad may be expressed as A SINGLE SPECIFIC DIRECTED COLIMIT. This is a development of my work on the order-theoretic fixed point theorem, in which Dito Pataraia provided one of the key ideas. We now have a constructive proof that is vastly simpler than any of the classical ones, completely eliminating the use of transfinite numbers, just as Kazimierz Kuratowski told us to do a century ago. I have posted the new version of the order-theoretic proof using a Galois connection on MATHOVERFLOW: www.mathoverflow.net/a/496177/2733 I would be grateful if my categorical and constructive colleagues could visit that posting (and my others on that website) and UPVOTE them. I am old enough to know that reactionary opposition shows that I am making an impact. I am nevertheless I am a human being and it is demoralising to have to stand up against it alone. Of course in the categorical case the colimit need not exist. This particular technique does not seem to help in that situation in the way that my extensional well founded coalgebras provide "partial" algebras, for example for the covariant powerset functor. However, the colimits do exist in special cases, notably for POLYNOMIAL FUNCTORS (aka species, analytic functors, stable functors and containers), where the free algebras are called W-TYPES. Recall that ORDINAL ADDITION is associative, NON-commutative and preserves non-empty joins in the second argument. COMPOSITION OF ENDOFUNCTORS has similar properties. Therefore in complex recursive situations, such as type theory and proof theory, I am advocating the investigation of the INTRINSIC ALGEBRAIC STRUCTURE of the situation, as an alternative to classical ordinal arithmetic. www.paultaylor.eu/ordinals/ You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>