Re: KT Chen's smooth CCC, a correction

["Tom Leinster" <t.leinster@maths.gla.ac.uk>]

On Tue, 26 Aug 2008, Bill Lawvere wrote:

> fickle pedias and beckening bistros which, like the mythical
> black holes, often regurgitate information as buzzwords and
> disinformation.

Disinformation is *deliberate* false information, false information
*intended* to mislead.  As I understand it, Bill's statement says,
among other things, that disinformation often appears on the
n-Category Cafe.

I don't know whether Bill really meant to say this.  I very much hope
not.  I can't think of a single instance where someone at the
n-Category Cafe has intended to mislead.

Best wishes,
Tom


On Tue, 26 Aug 2008, Bill Lawvere wrote:

> 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



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