gluing, lifting and partial maps

gluing, lifting and partial maps

[Paul Taylor]

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.

Re: gluing, lifting and partial maps

[Ronnie Brown]

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.