Re: KT Chen's smooth CCC, a correction

[Bill Lawvere <wlawvere@buffalo.edu>]

Dear Jim and colleagues,

By urging the study of the good geometrical ideas and constructions of
Chen and Froelicher, as well as of Bott, Brown, Hurewicz, Mostow, Spanier,
Steenrod, I am of course not advocating the preferential resurrection of
the particular categories they tentatively devised to contain the
constructions.

Rather, recall as an analogy the proliferation of homology theories 60
years ago; it called for the Eilenberg-Steenrod axioms to unite them.
Similarly, the proliferation of such smooth categories 45 years ago would
have needed a unification. Programs like SDG and Axiomatic Cohesion have
been aiming toward such a unification.

The Eilenberg-Steenrod program required, above all, the functorality with
respect to general maps;  in that way it provided tools to construct even
those cohomologies  (such as compact support and L2 theories) that are
less functorial.

The pioneers like Chen recognized that the constructions of interest (such
as a smooth space of piecewise smooth paths or a smooth classifying space
for a Lie group) should take place in a category with reasonable function
spaces. They also realized, like Hurewicz in his 1949 Princeton lectures,
that the primary geometric structure of the spaces in such categories must
be given by figures and incidence relations (with the algebra of functions
being determined by naturality from that, rather than conversely as had
been the 'default' paradigm in 'general' topology, where the algebra of
Sierpinski-valued functions had misleadingly seemed more basic than
Frechet-shaped figures.) I have discussed this aspect in my Palermo paper
on Volterra (2000).

The second aspect of the default paradigm, which those same pioneers
seemingly failed to take fully into account, is repudiated in the first
lines of Eilenberg & Zilber's 1950 paper that introduced the key category
of Simplicial Sets. Some important simplicial sets having only one point
are needed (for example, to construct the classifying space of a group).
Therefore. the concreteness idea (in the sense of Kurosh) is misguided
here, at least if taken to mean that the very special figure shape 1 is
faithful on its own. That idea came of course from the need to establish
the appropriate relation to a base category U such as Cantorian abstract
sets, but that is achieved by enriching E in U via E(X,Y) = p(Y^X),
without the need for faithfulness of
p:E->U;  this continues to make sense if E consists not of mere cohesive
spaces but of spaces with dynamical actions or Dubuc germs, etcetera, even
though then p itself extracts only equilibrium points. The case of
simplicial sets illustrates that whether 1 is faithful just among given
figure shapes alone has little bearing on whether that is true for a
category of spaces that consist of figures of those shapes.

Naturally with special sites and special spaces one can get special
results: for example, the purpose of map spaces is to permit representing
a functional as a map, and in some cases the structure of such a map
reduces to a mere property of the underlying point map. Such a result, in
my Diagonal Arguments paper (TAC Reprints) was exemplified by both smooth
and recursive contexts; in the latter context Phil Mulry (in his 1980
Buffalo thesis) developed the Banach-Mazur-Ersov conception of recursive
functionals in a way that permits shaded degrees of nonrecursivity in
domains of partial maps, yet as well permits collapse to a 'concrete'
quasitopos for comparison with classical constructions.

Grothendieck did fully assimilate the need to repudiate the second aspect
(as indeed already Galois had done implicitly; note that in the category
of schemes over a field the terminal object does not represent a faithful
functor to the abstract U). Therefore Grothendieck advocated that to any
geometric situations there are, above all, toposes associated, so that in
particular the meaningful comparisons between geometric situations start
with comparing their toposes.

A Grothendieck topos is a quasitopos that satisfies the additional
simplifying axiom:
        All monomorphisms are equalizers.
A host of useful exactness properties follows, such as:
         (*)All epimonos are invertible.
The categories relevant to analysis and geometry can be nicely and fully
embedded in categories satisfying the property (*). That claim arouses
instant suspicion among those who are still in the spell of the default
paradigm; for that reason it may take a while for the above-mentioned
45-year-old proliferation of geometrical category-ideas to become
recognized as fragments of one single theory.

There is still a great deal to be done in continuing
  K.T. Chen's application of such mathematical categories to the calculus
of variations and in developing applications to  other aspects of
engineering physics. These achievements will require that students persist
in the scientific method of alert participation, like guerilla fighters
pursuing the laborious and cunning traversal of a treacherous jungle
swamp. For in the maze of informative 21st century conferences and
internet sites there lurk fickle pedias and beckening bistros which, like
the mythical black holes, often regurgitate information as buzzwords and
disinformation.

Bill


On Sun, 17 Aug 2008, jim stasheff wrote:

> 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
>>
>>
>>
>>
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