Online seminar on Well pointed endofunctors and recursive constructions

[Paul Taylor]

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/