Subject: categories: lifting and non-Artin gluing

From: Paul Taylor <[email-masked]>
Date: Mon, 12 Jun 2000 16:05:19 +0100

Non-Artin Gluing in Recursion Theory
                   and Lifting in the Abstract Stone Duality

                       for "Category Theory 2000", Como

                  http://hypatia.dcs.qmw.ac.uk/author/TaylorP

    The analogy between `open' and `recursively enumerable' sets has long
    been a mainstay of theoretical computer science, but the traditional
    axiomatisation of general topology involving arbitrary unions is an
    obstacle to the formal unification of these disciplines.
    
    In particular, whilst Artin showed how to glue an open subspace
    to its closed complement using a comma square of frames, we give
    a diagonalisation argument to show that this representation fails for
    any recursively enumerable subset of $\mathbb N$ that is not decidable.
    
    Abstract Stone Duality is a re-axiomatisation based on the monadic
    adjunction between the dual categories of `frames' and `spaces'. Despite
    the failure of Artin gluing, we show in this formulation that the lift
    or scone, constructed using a comma square, still provides the classifier
    for partial maps with open domain of definition.  As the axiomatisation
    is not given in terms of finite intersections and arbitrary unions,
    a careful study of the modular law is needed to prove this.

I would very much like to receive comments and references to relevant work on
 *  Artin gluing and the partial map classifier FOR LOCALES (not toposes),
 *  modular lattices and the (Goursat?) equation  ab=bab  for idempotents.

I will be preparing an abridged version for the Como proceedings during the
last week in June, and submitting the paper to "Theory and Applications of
Categories" at the same time.

The first paper on "Abstract Stone Duality" has been with the TAC referees
for nearly a year.  Both can be found linked from my Hypatia page.

Paul