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.h...

wlawvere@buffalo.edu wrote:

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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

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.

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