categories: Seminar talk by Paul Taylor on Thursday March 2 2022

["Niccolò Veltri" <niccolo.veltri@gmail.com>]

Hi everyone,

On Thursday March 2 2023 at 4pm EET (2pm UTC), Paul Taylor will
give a presentation on well founded coalgebras. This is a shared
event in the seminar series of the "Logic and Semantics" group at
Tallinn University of Technology and the "Theory Group" of the
School of Computer Science at the University of Birmingham.
Everyone is invited to participate!

Details on how to access the talk on Zoom are available on any of
the following webpages:

https://www.paultaylor.eu/ordinals/
http://talks.bham.ac.uk/talk/index/5159
https://niccoloveltri.github.io/tsem22/taylor0203.html

Title and abstract of the talk are below.

Best,
Niccol??

---

"Well Founded Coalgebras"

Paul Taylor

Categorical set theory explores ideas taken from set theory
to develop mathematics using category theoretic tools.
It began in the 1970s when Mikkelsen and Osius interpreted
recursion and epsilon-structures in an elementary topos.
Well founded coalgebras generalise epsilon-structures to give
approximations to the free algebra for a functor even when
this does not exist.

The main recursion theorem is based on the one of von Neumann
for ordinals. Originally that was based on fixed points in
complete lattices, but in order to consider more general
categories and functors, we must use Pataraia's Theorem for
dcpos instead.

However, for our more complicated constructions, we need
to find a scalpel not a sledgehammer, so a more subtle form
of Pataraia's Theorem is developed.

The paper develops analogues of the recursion theorem and
Mostowski extensional quotient potentially in much more
general categories, with factorisation systems intead of
1-1 functions.

The obvious first application of this generalisation
replaces Set with Pos to study the different forms of
intuitionistic ordinals that were introduced in the 1990s.
This in turn leads to a formulation of transfinite iteration
of functors, based on a categorical axiom instead of the
set-theoretic axiom-scheme of replacement.

See?? www.paultaylor.eu/ordinals/?? for full details.



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