Re: KT Chen's smooth CCC, a correction

Re: KT Chen's smooth CCC, a correction

[Tom Leinster]

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



-----
The University of Glasgow, charity number SC004401

Re: KT Chen's smooth CCC, a correction

[Bill Lawvere]

Dear Ronnie and Colleagues,

Your comments are extremely interesting.  Thank you very much for raising
in so striking a manner the question of the relation between general
monoidal structures and cartesian closed structures.
Below are some observations which show, I think,
that everybody should be interested in this relation because it is
manyfold and fruitful.

(1)	While cartesian closed structures have the virtue of being unique,
general monoidal closed structures have the virtue of not being unique.
Thus, for example, the cartesian closed presheaf toposes (with their
exactness properties and combinatorial truth object) often have a further
monoidal closed structure given by Brian Day's convolution with respect to
a pro-co-monoidal structure on the site. Cubical as well as simplicial
sets have both cartesian and non-cartesian closed structures, and that is
'true', not merely 'convenient'.

(2)	Another category having both cartesian and non-cartesian monoidal
structures is the real interval from zero to infinity with 'x dominates y'
as the morphism from x to y. (Actually, this category is derived by
collapsing a natural topos of dynamical systems in 'Taking categories
seriously' TAC Reprints.) Categories enriched with respect to the
non-cartesian structure here (see 'Metric Spaces' TAC reprints) arise
every day in analysis and the rich insights of enrichment theory (Functor
categories, bi-module composition, free categories, etcetera) should be
systematically applied to the advance of analysis and geometry, while on
the other hand metric examples inspire further developments of enrichment
theory. Cauchy (who never worked on idempotent splitting in ordinary
categories and additive categories in the way that Freyd and Karoubi did)
does not deserve to have his name brandished as a joke to scare one's
uncomprehending colleagues in analysis. The kind of completeness that is
inspired by two-sided intervals (unlike the one-sided intervals
inaccurately alluded to in common discussions of 'density') indeed reduces
to the one attributed to Cauchy in the particular example of Metric
Spaces. The author hoped that observation would contribute to the advance
of analysis and the development of enrichment theory, not to the supply of
buzzwords.

     In fact, there is an insufficiently known branch of analysis called
'Idempotent Analysis', which deals largely with composition of bi-modules,
or more precisely, with the relation between the two closed structures on
the infinite interval. Of course, that monoidal category is isomorphic to
the unit interval under multiplication (still cartesian closed too) which
induces many of the relations between probablility and entropy.

(3)	Perhaps the most common relation between non-cartesian monoidal
categories and cartesian categories arises when a structure such as vector
space is interpreted in a cohesive background. I am sticking to my story
that cohesive backgrounds are basically cartesian closed, due to the
ubiquitous role of diagonal maps and also due to the fact that, for
example, bornological vector spaces have an obvious monoidal closed
structure, whereas topological vector spaces have none. The rumor that
topological vector spaces might have a tensor with an adjoint hom is part
of the disinformation that makes functional analysis look more difficult
than it is. A more accurate account of the relation between non-Mackey
convergence and closed structure can be found in C. Houzel's paper on
Grauert finiteness, Mathematische Annalen, vol. 205, 1973, 13-54:
essentially, the topological categories are merely enriched in the
genuinely monoidal closed bornological ones. Similarly, the idea that not
all dual spaces are complete seems to be based on a misguided generality
in the notion of Cauchy nets (they should be bounded).

(4)	Although pointed spaces are somewhat entrenched in algebraic
topology, there is an improvement suggested by your own work, Ronnie.
Consider the category whose objects are arrows S ---> E where E is a space
(object of a cartesian closed cohesive background category) and S is a
discrete space. This category is even a topos if the category of E's was,
as is the larger category of arrows between general pairs of spaces. The
first category is actually an adjoint retract of the second, correcting
the discontinuity that arises from the traditional limitation S = 1.
Intuitively, in the case where the pair of spaces is a subspace inclusion,
the adjoint collapses the subspace to a point if the subspace is
connected, but if it is not connected, does not artificially merge its
components. There are many applications of this corrected construction of
the space which results from 'neglecting' a subspace, both in algebraic
topology and in functional analysis, too numerous to discuss here.

Bill



On Wed, 27 Aug 2008, R Brown wrote:

> Dear Bill and Colleagues,
>
> I would like to explain my own interest in function spaces and function
> objects since it has a different origin to what Bill explains and a
> different direction which could be of interest for comment and
> investigation.

...

Re: KT Chen's smooth CCC, a correction

[R Brown]

Dear Bill and Colleagues, 

I would like to explain my own interest in function spaces and function objects since it has a different origin to what Bill explains and a different direction which could be of interest for comment and investigation. 

Michael Barratt suggested to me in 1960 the problem of calculating the homotopy type of the space X^Y by induction on the Postnikov system of X, in contrast to Michael's own work on Track Groups, where he used a homology decomposition of Y, and using Whitney's tube systems gave explicit description of some group extensions in examples of the Barratt(-Puppe) exact sequence. (amazing!!?) 

Now the first Postnikov invariant in its simplest form is a Sq^2 but the extension is described by a Sq^1. How did the one transform into the other? Clue: the Cartan formula for Sq^2 on a product. How did a product get into the act? Answer: the evaluation map! 

Trying to write all this down led to using a number of `exponential laws' in spaces, spaces with base point, simplicial sets, pointed simplicial sets, chain complexes, simplicial abelian groups, etc. So it was dinned into me that an exponential law depended on the product as well as the function object. So why not try the known weak product for topological spaces? Surprise, surprise, it all worked, and was part 1 of my thesis, submitted 1961, with a sketchy account of what we now call monoidal closed categories, exemplified,  but not developed in general terms. 

Subsequent work with Philip Higgins has continued to use monoidal closed categories in algebraic topology. Indeed the category of crossed complexes is cartesian closed, but the homotopy theory one wants is given by the (different) monoidal closed structure. So the category of filtered spaces is usefully enriched  over this monoidal closed category. 

My question is then: what is the potential influence of this need for monoidal closed? It clearly does not lead to topos theory as such. It does lead to the possibility of some not previously available calculations, even of nonabelian homotopical invariants, and is relevant to the study of local-to-global problems. (I first heard these words from Dick Swan in connection with sheaf theory.)  But in this work cubical sets became essential, for ease of discussing subdivision, multiple compositions,  and homotopies, and here the monoidal closed structure is crucial. Kan's initial cubical work was neglected in favour of the (convenient in many ways) cartesian closed category of simplicial sets. 

One specific problem for me was a general notion of symmetry (naively, and using buzz words (!), higher order groupoids should yield higher order notions of symmetry!). In a cartesian closed category C we have for a specific object x not only Aut(x), the isomorphisms of x, but also AUT(x), the internal group object of automorphisms of x. This has been developed for the topos of directed graphs, in John Shrimpton's thesis, but actually the unaccomplished aim was to understand Grothendieck's Teichmuller Groupoid, and his envisaged computations of this by gluing or clutching procedures, but which needed topos theory, he claimed! When I asked for any notes on this he just said nothing was written down, it was all in his mind. Baffling!  

In the monoidal case we can get only that END(x) is an internal monoid wrt tensor. But in some cases  we have a candidate for AUT(x), even if `internal group' wrt tensor makes no sense.  One example was worked out with Nick Gilbert (published 1989). In this case there is a forgetful functor U to a cartesian closed category, in this case Set, and you can make up the rest. It worked in this dimension, relevant to homotopy 3-types, but still did not lead by induction to even higher order notions of symmetry. Pity! 

My question is now: given this background, how should  we  match the beautiful ideas and insights of Bill with what seem to be some monoidal closed realities? Could this be important for geometry, and, better still, even for analysis, and dynamics? 

Ronnie


From: Bill Lawvere <wlawvere@buffalo.edu>
To: categories@mta.ca
Sent: Tuesday, 26 August, 2008 9:07:58 PM
Subject: categories: Re: KT Chen's smooth CCC, a correction

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.

...

Re: KT Chen's smooth CCC, a correction

[Bill Lawvere]

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
>>
>>
>>
>>
>
>
>
>
>

Re: KT Chen's smooth CCC, a correction

[jim stasheff]

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
>
>
>
>

KT Chen's smooth CCC, a correction

[wlawvere]

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