## September 3, 2008

### 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:

For a full-fledged textbook treatment, which may be gentler for beginners, try this:

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

Posted at September 3, 2008 7:37 PM UTC

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### Re: Comparative Smootheology, III

So I guess the real problem is that the site $S$ is concrete.

What can be done about that? Is there a useful non-concrete site to hand?

Posted by: David Corfield on September 4, 2008 2:48 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

Kock’s book Synthetic Differential Geometry (it takes a while to download) has a chapter on ‘Models’ — Chapter III. The book starts out with axioms you might want a topos of smooth spaces to satisfy, but in this chapter it constructs topoi satisfying these axioms. It says “all the models we consider are closely related to categories of the form $Set^R$ where $R$ is some small category of [commutative] rings.”

In other words: presheaves on $R^{op}$.

So, the basic idea is the familiar idea behind algebraic geometry: treat $R^{op}$ as analogous to a category of spaces.

And part of the point is that $R^{op}$ should contain the ‘infinitesimal arrow’ space

$k[x]/\langle x^2 \rangle$

where $k$ is your favorite field.

Let’s pick $k = \mathbb{R}$ and let $R$ be the category of finitely general algebras over $\mathbb{R}$.

Then as an object of $R^{op}$, the ‘infinitesimal arrow’ space $\mathbb{R}[x]/\langle x^2 \rangle$ has just one global point, but it’s not a point — i.e., it’s not $\mathbb{R}$.

So this choice of site, $R^{op}$, is not concrete.

This is just the beginning of Kock’s story…

Posted by: John Baez on September 4, 2008 5:33 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

A little diversion since you mention Kocks’s stuff…

I was recently writing some code and it reminded my of synthetic geometry. I had Kock’s book sitting on my desk for years hoping that its proximity would somehow allow the contents to seep into my brain, but very little osmosis actually occurred unfortunately. What I do vaguely remember is that there is some element $\epsilon$ that we can think of as being “really small” in the sense that $\epsilon^2 = 0$. I remember thinking of this as some first order approximation.

In Matlab (my favorite number crunching/modeling tool) there is a number $\varepsilon$ that is so tiny that its square is too tiny to be registered as anything other than 0, i.e. $\varepsilon^2 = 0$ as far as Matlab is concerned. This number is approximately $\varepsilon = 1.5458e-162$.

I doubt that has anything to do with anything, but who knows :)

Posted by: Eric on September 4, 2008 6:21 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

In a suitable programming language it’s straightforward to implement a class, like the complex numbers, in which there is an element whose square is zero. This then gives you an elegant, accurate and highly practical way to numerically compute derivatives without using finite differencing or symbolic algebra. This is one approach to what’s called automatic differentiation.

Infinitesimals can give a nice practical way to do more abstract stuff too. For example, if you’ve written code to implement a Lie group (eg. the group of 3D rotations), and you use that code with the reals extended by infinitesimals, then you get code for computing in Lie algebras for free, and a very down-to-earth way to see what Lie algebras are.

I wish I understood better the relationship between these techniques and toposes.

Posted by: Dan Piponi on September 4, 2008 7:28 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

Just to point out the probably well known fact that such a number doesn’t satisfy the expected rules for nilpotence in that $x \varepsilon \varepsilon$ evaluates differently depending whether the machine instructions actually used evaluates: $(x \varepsilon)\varepsilon$ will give nonzero if $x$ is big enough whilst $x(\varepsilon\varepsilon)$ will be zero regardless of $x$. Of course this doesn’t matter because as Dan points out you’d deal with various kinds of number as distinct program elements with their own rules, just like complex numbers. The issue does come up with other things like representing Clifford Algebra elements in terms of their real/complex/quaternion number matrix representation where nilpotent matrices often no longer extactly come to 0 if you do operations in a different order.
Posted by: bane on September 4, 2008 10:29 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

Eric wrote:

What I do vaguely remember is that there is some element $\epsilon$ that we can think of as being “really small” in the sense that $\epsilon^2 =0$.

That is perhaps the single most important idea behind synthetic differential geometry: setting up an approach to calculus and differential geometry that permits the rigorous use of ‘infinitesimal numbers’ satisfying equations like $\epsilon^n = 0$, where you get to choose $n$ to be whatever you want.

Topos theory provides a nice framework to do this sort of thing. Physicists and engineers have been using these ideas in an informal way since Newton. The idea is to do it carefully, beautifully and without shame.

This number is approximately $\epsilon = 1.5458 \times 10^{-162}$

Well, now that’s settled.

Posted by: John Baez on September 4, 2008 7:12 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

“That is perhaps the single most important idea behind synthetic differential geometry: setting up an approach to calculus and differential geometry that permits the rigorous use of ‘infinitesimal numbers satisfying equations like ϵ^n = 0, where you get to _choose n to be whatever you want_.

SH: Then if n
ϵ^1 = 0 so ϵ = 0 ?
ϵ^ϵ = 0 ?
ϵ^0 = 0 ?
————————————–

Classical vs Quantum Computation (Week 3)
“To use this trick, you need to know that

0^0 = 1

This is something they don’t teach in school! In analysis, X^Y can approach anything between 0 and 1 when X and Y approach 0. So, teachers like to say 0^0 is undefined. But X^X approaches 1 when X → 0.
———————————-

SH: But if ϵ^n = 0 in Comparative Smootheology III how can ϵ^1 = ϵ = 0
ϵ^ϵ = 0, or 0^0 = 0, (substituting 0 for ϵ^n)be consistent with Week 3, where 0^0 = 1 ? The axiomatic assumption appears to be different.

Posted by: Stephen Harris on September 6, 2008 7:56 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

John wrote:

infinitesimal numbers satisfying equations like $\epsilon^n = 0$, where you get to choose $n$ to be whatever you want.

Stephen wrote:

$\epsilon^\epsilon = 0$?

$\epsilon^0 = 0$?

Umm, you can’t choose $\epsilon$ to be whatever you want.

I meant you can choose $n$ to be $1,2,3,\dots$ etc. It’s a way of making rigorous the usual process of ‘discarding higher-order terms’ when working with Taylor series.

Posted by: John Baez on September 10, 2008 3:23 AM | Permalink | Reply to this

### Adjoining nilpotents (Was: Comparative Smootheology, III)

Steven Harris wrote in part:

ϵ^1 = 0 so ϵ = 0 ?
ϵ^ϵ = 0 ?
ϵ^0 = 0 ?

and then:

ϵ^1 = ϵ = 0
ϵ^ϵ = 0, or 0^0 = 0

The exponent n can be anything you like, but only one thing at a time! So you can’t take it to be 1 (to get ε = 0) and then take it to be ε or 0 (to get 00 = 0, hence the fundamental numerical contradiction 0 = 1). Actually, you can never take the exponent to be ε itself; n has to be a natural (or whole) number (so John’s phrase ‘whatever you want’ is an exaggeration).

That said, you’re right to be concerned about the possibility that n = 0; John cleverly didn’t allow that in his reply to your comment. After all, that 00 = 1 is just a special case of the fact that x0 = 1 for any value of x in any algebra. So if ε0 = 0, then you still get the numerical contradiction that 0 = 1.

This is not a logical contradiction; it just collapses the new algebra into triviality (through the argument that x = 1x = 0x = 0, so that every value in the new algebra is 0), so the result is not particularly useful. It’s worthwhile to prove, therefore, that for any other value of n (1, 2, 3, etc), there is no collapse. (To be precise: if x and y are ordinary numbers in the old algebra, and if x = y in the new algebra obtained by adjoining ε, then in fact x = y already in the old algebra.)

Posted by: Toby Bartels on September 20, 2008 9:51 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

I went through a phase where I was trying to decide which road to take for my application, which is nonabelian differential cohomology. I tried synthetic differential reasoning following Anders Kock’s nice expositions. But at some point a found myself having abandoned it in favor of a notion of smooth spaces in terms of sheaves on smooth test domains. Not for some grand philosophical reason. But just because it works nicely in the given application. For me.

I have seen others try to use synthetic differential reasoning for similar purpose. I saw indications that it can be useful to some extent, but for the time being I am more convinced by the non-synthetic approach. That might change again.

But in any case, it seems that what counts is which applications you have. Funny to see me with this attitude on two different sides of the two cultures at the same time, depending on who I talk to.

Just for the record I close by mentioning that I found the following general concept to be relevant:

write Spaces for the category of sheaves on, say, manifolds.

Write $\infty Cat$ for your favorite notion of infinity-categories internal to Spaces. Write DGCAs for the category of differential graded-commutative algebras.

Then forming fundamental $\infty$-groupoids and forming classifying spaces yields a setup

$\infty Groupoids \stackrel{\leftarrow}{\to} Spaces$

This has an infinitesimal counterpart, where we take the DGCA of differential forms on a space and send a DGCA to its smooth classifying space. This yields a setup

$Spaces \stackrel{\leftarrow}{\to} DGCAs \,.$

So in total we have

$\infty Groupoids \stackrel{\leftarrow}{\to} Spaces \stackrel{\leftarrow}{\to} DGCAs$

It seems to me that this setup is of great relevance in applications. In particular, going from right to left through this is $\infty$-Lie integration, while going from left to right is $\infty$-Lie differentiation.

I’d be interested in seeing discussed a useful synthetic way of looking at this.

Posted by: Urs Schreiber on September 5, 2008 6:16 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

Can you quickly explain again, when you pass to the infinitesimal counterpart: are we to think of a Space as an infinitesimal version of an $\infty$groupoid, and of a DGCA as an infinitsimal version of a Space? It’s a bit confusing. Naively I would have thought a DGCA would be the infinitesimal version of an $\infty$groupoid, because those gradings are the tangent vectors “in the direction of the $n$-arrows”, right? But then I don’t know what the infinitesimal version of a Space is.

Heh, I am amused by the combative notation:

(1)$Spaces \rightarrow \leftarrow DGCA.$

This is the mathematical version of what happens when an irresistable force meets an immovable object.

Posted by: Bruce Bartlett on September 5, 2008 8:25 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

Hi Bruce!

are we to think of a Space as an infinitesimal version of an $\infty$-groupoid, and of a DGCA as an infinitsimal version of a Space?

Not quite. It’s really nice and easy, this way:

think, in this context, of an $\infty$-groupoid as the fundamental $\infty$-groupoid of some space, the “$\infty$-path groupoid” of the space, i.e. the $\infty$-groupoid whose $k$-cells are $k$-dimensional things in a given space.

Think of the qDGCA in this context as the infinitesimal version of this (and then dualize): its degree $k$ generators are the (duals to the) tangents to the $k$-dimensional things in the space.

The standard example to keep in mind is:

let the space be the classifying space of flat Lie algebra valued forms. Then the corresponding $\infty$-groupoid is the one-object groupoid $\mathbf{B}G$, i.e. $Hom_{\mathbf{B}G}(\bullet,\bullet) = G$ for $G$ the corresponding simply connected Lie group. And the DGCA is the Chevalley-Eilenberg algebra of the Lie algebra.

Naively I would have thought a DGCA would be the infinitesimal version of an $\infty$groupoid, because those gradings are the tangent vectors “in the direction of the n-arrows”, right?

Yes, exactly. (For the DGCA itself it’s the dual tangent vectors that appear.).

But then I don’t know what the infinitesimal version of a Space is.

Did I say “infinitesimal version of a space”? If so, sorry, this is maybe not too descriptive.

Posted by: Urs Schreiber on September 6, 2008 3:32 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

Lawvere is a bad influence on you, John. I thought that paper was really terrific, and I learned a lot from reading it. If you had gone out of your way to make everything point-free, I doubt I would have learned as much. Mathematics is not the handmaiden of ideology. There is no privileged right point of view.

Posted by: Walt on September 8, 2008 2:24 AM | Permalink | Reply to this

### Re: Comparative Smootheology, III

I’m glad you liked that paper, Walt!

When I started working on higher gauge theory I specifically chose to work with Chen spaces because I knew most people would be able to understand those more easily — including me. They definitely have their charms. But so far, I don’t see any technical advantage that they have over the topoi of smooth spaces considered by Kock.

Lawvere is a bad influence on you, John.

Do you mean he already is a bad influence on me, or he would be if I let him? I’m curious! This thread started when I got the feeling Lawvere was criticizing me for not having absorbed his ideas.

Posted by: John Baez on September 8, 2008 2:57 AM | Permalink | Reply to this

### Re: Comparative Smootheology, III

To throw in my two cents: I can’t agree a whole lot with this idea of Lawvere being a “bad influence”. My own experience is that Lawvere can be rather difficult to follow, and he surely doesn’t have anything like the pedagogical talents of John, but: he is undeniably one of the most brilliant and profoundly insightful category theorists anywhere. I often worry that a lot of his insights will be lost, unless some effort is made to pay attention to him. My experience is that it is very much worth the effort.

I imagine his heat and frustration stem generally from knowing he possesses tremendous insight, and yet most people don’t “get it”, probably due to his peculiar modes of expression. He is almost mercilessly concentrated, both in his spoken and written expression. He needs hermeneuts.

Posted by: Todd Trimble on September 8, 2008 5:58 AM | Permalink | Reply to this

### Re: Comparative Smootheology, III

I often worry that a lot of his insights will be lost, unless some effort is made to pay attention to him. My experience is that it is very much worth the effort.

We had plenty of discussion here at the Café with the aim of understanding aspects of Lawvere’s work and incorporating it into applications that some of us are working on. We have had various blog entries titled by and interested in Lawvere’s lores, albeit surely just scratching the suface.

I still am very interested. Right now in the process of getting back to our (your, Todd, and mine) discussion of qDGCA versions of $C^\infty$-algebras aka monoidal co-presheaves on cartesian spaces. This is relevant for $\infty$-Lie algebroid theory.

I wonder if any of this would be appreciated.I am interested in thinking about it whether or not it is appreciated, but it makes me wonder.

I feel like sharing the following experience:

after I grew very fond of those aspects of Lawvere’s work which I think I have understood, I went and talked to other mathematical physicists about its application to physics. Various mathematical physicists I talked to regarded this with reservation, to say the least, quite similar in its tone to the reservation Lawvere himself expresses in the exchange reproduced above.

I find myself between two camps that have trouble communicating with each other to the extent that they simply don’t: on the one hand high-powered modern abstract higher categorical math, on the other hand modern mathematical higher-dimensional physics. There is immense potential in merging the two, applying the former to the latter and letting the latter provide examples, motivation and thus guidance for the former.

There are a handful of people working on this clearly desireable unification. That there are not more of them is to some extent due to the fact that leading figures on both sides regard unfavorably the activity on the respective other side, usually the more so the less they understand the other side. I am wondering whether the “buzzwords” that Lawvere complained about without specifying them are well-defined technical terms that happen to be of interest in fields currently but not in principle disjoint from his area of expertise. But of course without concrete details I can’t know.

Posted by: Urs Schreiber on September 8, 2008 12:36 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

Occupying boundary spaces between disciplinary approaches is a notoriously awkward place to be, “generally incurring the suspicion and misunderstanding of members of both of the contending parties” (MacIntyre). In a better world, one where people took “pleasure in having been shown to be mistaken, something notoriously difficult to achieve”, people would be encouraged to move into these spaces, to learn both languages as first languages, so as to be able to use the insights of one party to address the problems of the second party as they, the latter, see things themselves.

Posted by: David Corfield on September 8, 2008 1:25 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

Occupying boundary spaces between disciplinary approaches is a notoriously awkward place to be

I guess you are right. On the other hand, in this case I find it a bit more disturbing than in general, since here arguably the boundary is between theory and application, with one party having developed an awesome theory in search of its killer application and on the other hand a party having stumbled across the killer application with no decent theory in hands to handle it properly.

I noticed that Lawvere explicitly mentions the caluclus of variations and “engineering physics”. I know that this is where his research originated and have no doubt that abstract category-theoretic formalism will eventually, at some point, also influence people working in classical continuum and point particle mechanics and engineering. But the “killer application” in physics lies beyond that.

Posted by: Urs Schreiber on September 8, 2008 5:21 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

Is it possible for the application people to describe in their own language what they see as the current obstacles, frustrations, lack of resources in their own work?

People don’t seem to be very good at this. It requires work to achieve and focuses attention away from one’s successes, something hardly encouraged in the competitive world of research. And it is best done together with boundary dwellers, such as yourself.

Posted by: David Corfield on September 9, 2008 9:06 AM | Permalink | Reply to this

### Re: Comparative Smootheology, III

Is it possible for the application people to describe in their own language what they see as the current obstacles, frustrations, lack of resources in their own work?

Regarding that aspect there is a problem, too. On the application side it happens that first approximations to solutions of problems are taken after a while to be the full truth and interest is lost in attempts, often necessarily involving lots of work and machinery, to get the full solutions.

For instance, people have been talking for ages about 2-dimensional conformal field theory. But only rather recently has a simple subclass of these, the “rational” 2d CFTs been entirely constructed and classified as proper full 2dCFTs (instead of just their local “chiral” parts) using, guess what, higher category theory. But then it turns out that some practitioners show themselves rather unimpressed in talks concerned with this classification result. “2dCFT is an old hat. And you can’t even go beyond the rational case, which is uninteresting.” is a common kind of comment.

An analogous remark applies to quantum field theory in its totality. There are two ways to give a comparatively exhaustive, at the conceptual level, answer to “What is quantum field theory?” using, guess what, higher category theory. I have seen it often happen that working QFT experts can’t relate to these approaches. Which would be fine by itself, were it not for the conceptual confusion one regularly sees displayed here and there, which is precisely supposed to be clarified by these formulations.

In String theory the clash used to be quite pronounced in a common attitude which I heard from several string theorists and mathematicians: string theorists were complaining about mathematician’s work fleshing out the conjectures and hints the former came across: “We know this for decades. Those mathematicians are only reproducing our work in weird language [read: in language we don’t understand].” I could cite concrete names and examples, but maybe I shouldn’t.

Rather recently one could see this happen in a quite extreme form when Langlands duality was connected to string theory: one string theorist in a seminar asked: “Who is learning something here from whom? It seems it’s just the mathematician’s finally learning about S-duality.”

All this is probably an evolution of the old phenomenon of the quantum gauge theorist (now becoming rare, but still existant) who is unaware of the concept of a fiber bundle and unwilling to appreciate its need.

Posted by: Urs Schreiber on September 9, 2008 12:00 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

Urs wrote:

But then it turns out that some practitioners show themselves rather unimpressed in talks concerned with this classification result.

Some of this is just the physicists’ bravado concerning math, nicely illustrated by Feynman in a joke that went something like this:

Some people say the work of mathematicians is useless for physics. That’s not true! Without mathematicians, progress in physics would have been set back by…

Posted by: John Baez on September 9, 2008 4:54 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

Urs wrote:

We had plenty of discussion here at the Café with the aim of understanding aspects of Lawvere’s work and incorporating it into applications that some of us are working on. We have had various blog entries titled by and interested in Lawvere’s lores, albeit surely just scratching the suface.

Oh, absolutely! Perhaps I gave the wrong impression – I was neither taking anyone here to task for failing to understand Lawvere, nor was I (or am I) taking sides in the case Baez v. Lawvere. I was writing more in reaction to putdowns like “bad influence” and “handmaiden of ideology”, which strike me as at least somewhat unfair to Lawvere.

after I grew very fond of those aspects of Lawvere’s work which I think I have understood, I went and talked to other mathematical physicists about its application to physics. Various mathematical physicists I talked to regarded this with reservation, to say the least, quite similar in its tone to the reservation Lawvere himself expresses in the exchange reproduced above.

I find myself between two camps that have trouble communicating with each other to the extent that they simply don’t: on the one hand high-powered modern abstract higher categorical math, on the other hand modern mathematical higher-dimensional physics. There is immense potential in merging the two, applying the former to the latter and letting the latter provide examples, motivation and thus guidance for the former.

Something I keep coming back to, when I hear things like the reactions of these mathematical physicists to Lawverean ideas, is that most people have a hard time absorbing or confronting new ideas unless they have a pretty serious “need to know”. People are proud and jealous of their own hard-won insights, and don’t necessarily smile when someone offers to show how it could be “thought better” (no matter how gentle and well-intended the offer is). That is, not until it becomes obvious that the new ideas help to solve problems that people are interested in, or make certain calculations considerably easier and clearer – typical ways in which that need to know is created.

The history of category theory seems to be full of such instances. A typical narrative [cf. Rota in Indiscrete Thoughts] seems to be that the category theorists of the 60s and early 70s, with Lawvere at the forefront, were a loud and brash group, bent on showing everyone how to “think it better”. I have some doubts about how accurate that narrative is; surely it was helped along by all those legends of political rabble-rousing and preaching of Marxist precepts by Lawvere. But I have a feeling that category theorists as a community weren’t so much brash, as legitimately enthusiastic about some really great ideas like topos theory, which have seeped into the broader mathematical culture only gradually as the need to know about them has grown. Similarly, SDG might be a great development, whose time has not yet come.

People are beginning to discover for themselves the need for ideas like categorification, various techniques of 2-categories, Day convolution, and so on. But in general, it’s hard being a Grassmann, or Lawvere, or Urs Schreiber – someone ahead of his time. :-)

Posted by: Todd Trimble on September 9, 2008 4:28 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

People are proud and jealous of their own hard-won insights, and don’t necessarily smile when someone offers to show how it could be “thought better”

And is it any wonder? We spend years cowering under the sword of damocles, in perpetual fear of being “scooped” and feeling like we have to defend our own viewpoints against all comers or our careers will be summarily executed.

Posted by: John Armstrong on September 9, 2008 5:01 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

I don’t necessarily want to prolong this discussion about who is not listening to whom. Personally I have learned to happily arrange myself with the way things are. I just mentioned a couple of observations as a kind of reply to what seems to be a kind of accusation or at least complaint in between the lines of what Lawvere writes. As with a similarly toned discussion a year or two ago on the category theory mailing list, I don’t really know what precisely the complaint really is and am puzzled why one cannot just voice it openly and directly so that there is a chance of properly replying and maybe making some progress.

But anyway.

not until it becomes obvious that the new ideas help to solve problems that people are interested in

Yes, certainly. One should not forget, however, that before you can solve a big problem with a new machinery you need to get that machinery under control. That leads to time delays between the point where a new machinery is embraced by some and the bigger results proven using it.

But I guess we are working on it…

Posted by: Urs Schreiber on September 9, 2008 6:36 PM | Permalink | Reply to this

### Re: Comparative Smootheology, III

Todd wrote:

To throw in my two cents: I can’t agree a whole lot with this idea of Lawvere being a “bad influence”.

If I started prefacing all my papers with quotes from Lenin you might change your mind. Ditto if I stopped doing what I was good at, namely explaining stuff in simple terms. But when it comes to Laweve’s mathematical insights, I agree with you: we could really use a lot more people understanding those insights and being influenced by him.

Posted by: John Baez on September 8, 2008 7:01 AM | Permalink | Reply to this
Weblog: The n-Category Café
Excerpt: What is the right definition of generalized smooth differential graded-commutative algebras?
Tracked: September 9, 2008 11:13 AM

### Re: Comparative Smootheology, III

Lawvere posted this response to the final comment in John’s post above on the categories list:

There are no

objections to continuing to develop Chen’s theory of “differentiable spaces”.

Indeed on 8/17, 8/26, and 8/27 I urged the continuation of the development of Chen’s theory (for example the smooth space of piecewise smooth paths), making use of recent experience of the range of possible categories.

It is possible that

sheaves on a concrete site can never serve as a framework for differential geometry with infinitesimals.

But a proof would require a definition of what is meant by infinitesimals, as well as the constraint on the framework that the maps $1 \to \mathbb{R}$ and $\mathbb{R} \to \mathbb{R}$ are the standard ones. Otherwise nonstandard analysis might fit. The nilpotents or germs capture the Heraclitian nature of motion in a way that abstract sets do not directly.

The misplaced concreteness, according to which

spaces are [single] sets equipped with extra structure

is only a “second aspect of the default paradigm. The first aspect, successfully overcome by the named pioneers, is the one generalizing the default category of topological spaces (or locales). Here “default” refers to the habitual response to the frequently occurring need to specify a background category of cohesion in which to interpret our algebra. The generalization from Sierpinski-valued functions (open sets) to real-valued, has also been proposed, but that sort of attempt never succeeded in yielding a simple theory of map spaces.

In contrast to this “function-algebra $X/R$ as primary” paradigm, the semi-dual “figure-geometry $S/X$ as primary” has led to good map spaces (including internal function algebras) for many authors (Sebastiao e Silva, Fox, Hurewicz 60 years ago and several more recent). I believe that attempting to force nearly-perfect duality has in general not led to good results, but of course one studies the extent to which a monad (presumed identity on models $S$) approximates the identity on general spaces. For example, Froelicher’s duality condition applies not only to the line $\mathbb{R}$ but to the function space $\mathbb{R}^{\mathbb{R}}$, a non-trivial fact about the smooth case, derived by LSZ from a study of distributions of compact support (so citing it is not just name-dropping).

Posted by: David Corfield on September 19, 2008 12:44 PM | Permalink | Reply to this
Read the post Bär on Fiber Integration in Differential Cohomology
Weblog: The n-Category Café
Excerpt: On fiber integration in differential cohomology and the notion of generalized smooth spaces used for that.
Tracked: November 26, 2008 7:54 AM
Read the post Smooth Structures in Ottawa
Weblog: The n-Category Café
Excerpt: There will be a conference on Smooth Structures in Logic, Category Theory and Physics at the University of Ottawa on May 1-3, 2009.
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Read the post Comparative Smootheology, IV
Weblog: The n-Category Café
Excerpt: Take a look at Martin Laubinger's thesis on smooth spaces.
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Excerpt: A survey of Jacob Lurie's "Structured Spaces".
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