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spaces (called smooth spaces there), in particular the algebraic topology of a special class of them, is in the unpublished PhD thesis of Andreas Cap http://www.mat.univie.ac.at/~michor/cap_diss.pdf