Comparative Smootheology, III
Posted by John Baez
Recently on the category theory mailing list, Bill Lawvere mentioned Chen’s work on smooth spaces. Jim Stasheff suggest that he post his remarks to our thread on comparative smootheology. I mentioned my recent paper with Alex Hoffnung where we showed that the category of Chen spaces is a ‘quasitopos’. Chen spaces are sets equipped with a collection of ‘plots’, a severe generalization of the ‘charts’ familiar from smooth manifolds. A ‘quasitopos’ is a nice sort of category that’s almost, but not quite a topos.
Lawvere replied saying that it’s a step backwards to treat smooth spaces as sets with extra structure. After all, his work together with Anders Kock and others gets an actual topos of smooth spaces — including ‘infinitesimal’ spaces that allow a beautiful approach to calculus with infinitesimals! — precisely by dropping the requirement that smooth spaces be sets with extra structure.
Jim Stasheff suggested that I post my reply to the $n$-Category Café. Since that reply makes little sense without the whole exchange, I’ll post the whole thing. There’s a certain amount of heat in Lawvere’s comments — but never mind: let’s focus on the math.
To understand Lawvere’s work on smooth spaces, I suggest starting here:
- F. William Lawvere, Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body.
- F. William Lawvere, Toposes of laws of motion.
For a full-fledged textbook treatment, which may be gentler for beginners, try this:
- Anders Kock, Synthetic Differential Geometry.
Lawvere 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
Stasheff wrote:
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 where there’s been an intense discussion of ‘smooth spaces’ in various incarnations.
jim
I wrote:
Hi -
Bill Lawvere mentioned that KT Chen had a cartesian closed category of smooth spaces. I’ve found this very useful in my work on geometry. I kept wanting more properties of this category, so finally my student Alex Hoffnung and I wrote a paper about it:
Convenient Categories of Smooth Spaces
Abstract: A “Chen space” is a set X equipped with a collection of “plots” - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace or quotient space of a Chen space is a Chen space, and the space of smooth maps between Chen spaces is again a Chen space. Souriau’s “diffeological spaces” share these convenient properties. Here we give a unified treatment of both formalisms. Following ideas of Dubuc, we show that Chen spaces, diffeological spaces, and even simplicial complexes are examples of “concrete sheaves on a concrete site”. As a result, the categories of such spaces are locally cartesian closed, with all limits, all colimits, and a weak subobject classifier. For the benefit of differential geometers, our treatment explains most of the category theory we use.
In particular, at some point we break down and admit we’re dealing with a “quasitopos”.
Best,
jb
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 functoriality with respect to general maps; in that way it provided tools to construct even those cohomologies (such as compact support and $L^2$ 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 $p:E \to U$; 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
I replied:
Bill Lawvere wrote:
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.
I chose Chen’s framework when Urs Schreiber and I were doing some work in mathematical physics and we needed a “convenient category” of smooth spaces. I decided to choose one that was easy to explain to people brainwashed by the “default paradigm”, in which spaces are sets equipped with extra structure. Later I realized I needed to write a paper establishing some properties of Chen’s framework. By doing that I guess I’m guilty of reinforcing the default paradigm, and for that I apologize.
If I understand correctly, one can actually separate the objections to continuing to develop Chen’s theory of “differentiable spaces” into two layers.
Let me remind everyone of Chen’s 1977 definition. He didn’t state it this way, but it’s equivalent:
There’s a category S whose objects are convex subsets $C$ of $\mathbb{R}^n$ ($n = 0,1,2,...$) and whose maps are smooth maps between these. This category admits a Grothendieck pretopology where a cover is an open cover in the usual sense.
A differentiable space is then a sheaf $X$ on $S$. We think of $X$ as a smooth space, and $X(C)$ as the set of smooth maps from $C$ to $X$.
But the way Chen sets it up, differentiable spaces are not all the sheaves on $S$: just the “concrete” ones.
These are defined using the terminal object 1 in $S$. Any convex set C has an underlying set of points $hom(1,C)$. Any sheaf $X$ on $S$ has an underlying set of points $X(1)$. Thanks to these, any element of $X(C)$ has an underlying function from $hom(1,C)$ to $X(1)$. We say X is “concrete” if for all $C$, the map sending elements of $X(C)$ to their underlying functions is 1-1.
The supposed advantage of concrete sheaves is that the underlying set functor $X \mapsto X(1)$ is faithful on these. So, we can think of them as sets with extra structure.
But this advantage is largely illusory. The concreteness condition is not very important in practice, and the concrete sheaves form not a topos, but only a quasitopos.
That’s one layer of objections. Of course, these objections can be answered by working with the topos of all sheaves on $S$. This topos contains some useful non-concrete objects: for example, an object $F$ such that $F^X$ is the 1-forms on $X$.
But now comes a second layer of objections. This topos of sheaves still lacks other key features of synthetic differential geometry. Most importantly, it lacks the “infinitesimal arrow” object $D$ such that $X^D$ is the tangent bundle of $X$.
The problem is that all the objects of $S$ are ordinary “non-infinitesimal” spaces. There should only be one smooth map from any such space to $D$. So as a sheaf on $S$, $D$ would be indistinguishable from the 1-point space.
So I guess the real problem is that the site $S$ is concrete: that is, the functor assigning to any convex set $C$ its set of points $hom(1,C)$ is faithful. I could be jumping to conclusions, but it seems to me that that sheaves on a concrete site can never serve as a framework for differential geometry with infinitesimals.
Best,
jb
Re: Comparative Smootheology, III
What can be done about that? Is there a useful non-concrete site to hand?