Re: KT Chen's smooth CCC, a correction

[jim stasheff <jds@math.upenn.edu>]

Bill,

    Happy to see you contributing to the renaissance in interest in
Chen's work.

It would be good to post your msg to the n-category cafe blog
whee there's been an intense discussion of `smooth spaces' i various
incarnaitons.

jim

http://golem.ph.utexas.edu/category/2008/05/convenient_categories_of_smoot.html

wlawvere@buffalo.edu wrote:
> In my review of Anders Kock's Synthetic
> Differential Geometry, Second Edition,
> there is a wrong statement that I want to correct.
> (This was in the SIAM REVIEW, vol. 49, No.2
> pp 349-350). The statement was that Chen's
>  category does not include the representability
>  of smooth function spaces. But from his paper
> In Springer Lecture Notes in Mathematics,vol
> 1174, pp 38-42 it is clear that it does. I thank Anders
> for pointing out this slip.
>
> This is a good opportunity to emphasize that
> the works of KT Chen and of Alfred Frolicher
> (that were referred to in the beginning of the
> above review) contain several contributions
> of value both to applications and to more
> topos-theoretic formulations.  For example,
> Frolicher's use of Lemmas by Boman and others
> reveals how little of the specific parameter "smooth"
> needs to be given to the very general machinery of
> adjoint functors and abstact sets in order to obtain
> smooth infinite dimensional spaces of all kinds.
> (Namely a suitable topos of actions by only unary
> operations on the line is fully embedded
> in the desired topos in such a way that the algebraic
> theory of n-ary operations that naturally exist in
> the small one determines the whole algebraic category whose
> sheaves include the large one.)
> And Chen's smooth space of piecewise-smooth
> curves can surely be further applied, as can his
> special use of convex models for  plots.
>
> Bill Lawvere
>
>
>
>