categories: Re: nice category of "smooth spaces"?

[Andree Ehresmann <Andree.Ehresmann@u-picardie.fr>]

In answer to John Baez

 >has anyone formulated a good notion of a "category internal to Diff"?
         Already in the fifties, Charles Ehresmann has introduced and
extensively studied categories internal to Diff (which he called
"categories differentiables") in his development of Differential Geometry.
His first paper essentially devoted to differentiable categories and to
their relation to locally trivial fibred spaces is
"Categories topologiques et categories differentiables", Coll. Geom. Diff.
Globale Bruxelles, CBRM (1959), 137-150.
This paper is reprinted in
"Charles Ehresmann: Oeuvres completes et commentees", Part I (ed. Andree C.
Ehresmann), Amiens 1983.
In this volume of "Oeuvres" several other papers by Charles develop this
question, and there are numerous comments by different authors giving more
information.

Remark that at that early time, the general notion of an internal category
did not yet exist, and differentiable categories were one of the examples
which motivated Charles to  introduce the notion of what he called then a
"structured category" (later renamed internal categories) in 1963 in the
 paper:
"Categories structurees", Ann. Ecole Norm. Sup. 80, Pqaris (1963), 349-426.
This paper, as well as the long series of his following papers where the
notion is refined and extensively studied, is reprinted in the same
"Oeuvres", Part III.
Though the papers are in French I have added many comments in English with
links to more recent papers of other authors.


 >the category Diff of smooth manifolds and smooth maps isn't complete
and >cocomplete.  Is there some category of "smooth spaces" that repairs
 these
 >defects?
         This is also a problem my husband and I have much studied. At this
effect, Charles has given some abstract constructions to extend a concrete
category into a concrete one with "enough" limits; in particular, in the
 paper:
"Prolongements universels d'un foncteur par adjonction de limites",
Dissertationes Math. LXIV Varsovie (1969), 1-72.
This paper is also reprinted in the "Oeuvres" Part IV, and I have added
comments in English.
His construction as well as some done by others have been unified .in a
short paper I have written after his death:
"Partial completions of concrete functors", Cahiers Top. et Geom. Diff.
XXII-3 (1981), 315-327.

The more strict problem of embedding Diff in a "good" cartesian closed
category has been handled by several authors; the first construction is in
the paper (written under my maiden name Bastiani):
"Applications differentiables et varietes de dimension infine", J. Ana.
Math. Jerusalem  XIII (1964), 1-114.
In the eighties, there are been several other constructions, e.g. the
"convenient spaces" of Frolicher.

         Sincerely
                         Andree C. Ehresmann