categories: Re: gluing, lifting and partial maps

[Ronnie Brown <r.brown@bangor.ac.uk>]

Abd-Allah and I dealt with this in

 ``A compact-open topology on partial maps with open domain'', {\em J. London
Math Soc.} (2) 21 (1980) 480-486.

as a development of work with Peter Booth in

 ``Spaces of partial maps, fibred mapping spaces and the compact-open
topology'', {\em Gen. Top. Appl.} 8 (1978) 181-195.

We got the idea of representability of partial maps from Peter Freyd's
article on topos theory. Is there an earlier result on these lines?

Ideas on making spaces over B into a Cartesian closed category came initially
from a paper of Rene Thom, and were developed in Peter's work at Hull.  This
eventually suggested the topologisation of spaces of partial maps as a step
towards Top/B.

Ronnie Brown

Paul Taylor wrote:

> I have long regarded it as "well known" that
>     the partial map classifier for topological spaces or locales
> where
>     by "partial" I mean a continuous function defined on an open subset
> is
>     the Artin gluing, Freyd cover or scone (Sierpinski cone).
>
> Can anybody point me to a published proof of this, or even tell me who
> first proved it?
>
> The same construction, with frames replaced by the categories of contexts
> and substitutions (a.k.a. classifying categories) for theories in other
> fragments of logic,  has also been used with spectacular results to prove
> consistency, strong normalisation, etc.   I know of plenty of work on
> that application itself,   but I wonder whether anybody has investigated
> the connection between these two applications of the construction.
>
> Paul
>
> PS Thanks to everyone who wrote to me about 1970s calculators.  I will be
> writing back and summarising the responses for "categories" after the end
> of term.   When the students have sat my exam paper (sometime in May)
> I will also post to "categories" the actual question that I composed.