categories: embeddability

[Ronnie Brown <mas010@bangor.ac.uk>]

I am embarrassed that I did not reply to kind and helpful messages to a
query I put on July 10 about embedability of a cartesian closed category in
a topos. The problem was that I got too many distractions (health,
holidays, family, other jobs) but really I did not explain the origin of
the question. So here goes. 

I am writing an invited article on k-spaces for an Encyclopaedia of general
topology to  be published by Elsevier, and of course k-spaces and also
sequential spaces form a cartesian closed category. I want to make a nod in
the direction of topos methods in order to be able to indicate how to deal
with spaces of  partial maps, and fibred exponential laws. There is quite a
bit of literature on these in algebraic topology but so far they do not go
the whole hog and use toposes. Again in analysis,  there is the book by
Moerdijk and Reyes, and also the later book by Kriegl and Michor `A
convenient setting for global analysis' which has the former book in its
bibliography but does not have the word topos in the index. There is a lot
of general topology work on hyperspaces, and also from a different approach
on spaces of partial maps, but one would like to know if this can all be
subsumed under topos work, or at least suggest it as a topic for
investigation. 

I realise as Peter Jonstone wrote that he has embedded the category of
sequential spaces in a topos, and I think Kock and Reyes have some work on
on embedding convenient vector spaces in a topos, but does this work for
k-spaces, or for the Kriegl and Michor situation? 

All this is not really my area but I would like to give helpful remarks in
this direction. For those interested I attach a dvi file of the current
draft and comments would be welcomed. 

Many thanks in anticipation and apologies again for not getting back on
this earlier. 

Best wishes

Ronnie