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.