Subject: for the list
From: Johannes Huebschmann
Date: Wed, 12 Sep 2007 15:59:42 +0200 (CEST)
Here is a reaction from Peter Michor
(comments inserted) which he asked me to forward
to the list.
Best regards
Johannes
>
> Subject: Re: four postings
> From: Andrew Stacey
> Date: Mon, 10 Sep 2007 15:53:47 +0200
>
> You will probably get lots of replies about 'diffeological spaces', and
in
> particular references to Patrick Iglesias' book (see his homepage at:
> http://math.huji.ac.il/~piz/Site/Welcome.html). Another approach is
> Frolicher
> spaces. I'm not sure if there is a book about Frolicher spaces, but the
> book
> "A Convenient Setting for Global Analysis" (Kriegl and Michor, available
for
> download from the AMS) mentions them and can be viewed as being about
> "locally
> linear Frolicher spaces". There's quite a lot of geometry in that book.
There is a section on general Froelicher spaces there, section 23.
>
> I think that Frolicher spaces and diffeological spaces (Chen's spaces)
are
> one
> and the same. Frolicher spaces are specified by giving the smooth
curves
> and
> smooth functionals, Chen spaces by giving the plots. However, the
smooth
> curves completely determine the Frolicher structure, and a plot is
> completely
> determined by its restriction to the smooth curves that factor through
it.
> Thus both are completely determined by declaring a family of curves to
be
> "smooth".
This is not completely true. Any Froelicher space is a diffeological
space.
To any diffeological space one may associate a Froelicher space by taking
the curves, looking at the smooth functions, and then enlarging the set
of smooth curves to those which are determined by the functions: this
construction is a left adjoint functor.
The embedding of Froelicher spaces into diffeological spaces is a right
adjoint functor.
A lot about Froelicher spaces (called smooth spaces there),
in particular the algebraic topology of a special class of them, is in
the unpublished PhD thesis of Andreas Cap
http://www.mat.univie.ac.at/~michor/cap_diss.pdf