KT Chen's smooth CCC, a correction

[<wlawvere@buffalo.edu>]

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