## December 7, 2008

### Smooth Structures in Ottawa

#### Posted by John Baez

Here at the $n$-Café we’re trying to get to the bottom of some big questions — for example, the nature of smoothness. A manifold is a kind of smooth space — but more general smooth spaces have been studied by Chen, Lawvere, Kock, Souriau and others, and these are starting to find their way into mathematical physics.

That’s just the beginning, though! Smoothness has a lot to do with derivatives. The concept of derivative can be generalized in some surprising ways. For example, it’s important in Joyal’s work on combinatorics — he explained how we can take the derivative of a structure like ‘being a 2-colored finite set’ More recently, Goodwillie introduced a concept of ‘approximation by Taylor series’ for interesting functors in homotopy theory. Even more recently, Ehrhard and Regnier introduced derivatives in logic — or more precisely, the lambda calculus.

So, it’s a great idea to have a conference on all these concepts of smoothness:

Here’s what the organizers have to say:

Abstract categorical approaches and analogies with the differential calculus and the theory of smooth manifolds arise in a number of diverse areas of mathematics. For example, the well-known fact that the category of manifolds and smooth maps fails to be cartesian closed motivated both the theory of convenient vector spaces due to Froelicher, Kriegl, and Michor, and work on categories of smooth spaces initiated by Chen and Souriau. In topos theory, synthetic differential geometry, developed by Lawvere, Kock, Moerdijk, Reyes, and others, provides an appealing abstract setting for differential geometry using the theory of nilpotent infinitesimals. In logic, the differential lambda-calculus, due to Ehrhard and Regnier, was inspired by considerations from linear logic, differential calculus, and work on locally convex topological models of linear logic. This theory subsequently gave rise to the recent development of differential categories by Blute, Cockett, and Seely. In topology, the Goodwillie calculus, which also has connections with the study of smooth manifolds, is an example of a ‘calculus of functors’ drawing inspiration from differential calculus. And in theoretical physics, recent work by Baez and Schreiber on higher gauge theory exploits some of these more abstract versions of differential geometry in order to avoid technical difficulties implicit in the theory of infinite-dimensional manifolds.

The Logic and Foundations of Computing group at the University of Ottawa is happy to announces a workshop, supported by the Fields Institute, which aims to bring together researchers from these different areas in order to encourage further interaction in the study of smooth structures in logic, category theory and physics. In addition to the main invited lectures, several of the invited speakers will give tutorials on their areas of expertise in order to make the subject accessible to students and other new researchers in the area. The (confirmed) invited speakers are:

• John Baez (UC Riverside)
• Kristine Bauer (Calgary)
• Thomas Ehrhard (PPS Paris)
• Anders Kock (Aarhus)
• Andrew Stacey (NTNU Norway)

Some student support from the Fields Institute will be available. There will also be some time reserved in the schedule for a selection of contributed talks. Further details regarding student support and contributed talks can be found on the workshop webpage.

With best regards,

Richard Blute
Pieter Hofstra
Philip Scott
Michael A. Warren

Posted at December 7, 2008 12:40 AM UTC

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### Re: Smooth Structures in Ottawa

Papers on differential categories can be found here.

It would be good to have someone connected with generalised species at the workshop.

Fiore writes:

…generalised species encompass and generalise various of the notions of species used in combinatorics. Furthermore, they have a rich mathematical structure (akin to models of Girard’s linear logic) that can be described purely within Lawvere’s generalised logic. Indeed, I will present and treat the cartesian closed structure, the linear structure, the differential structure, etc. of generalised species axiomatically in this mathematical framework. As an upshot, I will observe that the setting allows for interpretations of computational calculi (like the lambda calculus, both typed and untyped; the recently introduced differential lambda calculus of Ehrhard and Regnier; etc) that can be directly seen as translations into a more basic elementary calculus of interacting agents that compute by communicating and operating upon structured data.

Posted by: David Corfield on December 8, 2008 10:12 AM | Permalink | Reply to this

### Discrete Smooth Spaces

This looks like fun! I wish I could attend.

Semantic question…

To what extent are the words “smooth” and “continuum” interrelated? For the longest time, I’ve tried to argue that the continuum was over-rated and that it was the differential algebra that was important. For example, this is one of my all-time favorite papers:

Einstein Algebras
Robert Geroch
Comm. Math. Phys. Volume 26, Number 4 (1972), 271-275

I’ve found the nLab page, which sends me to read a bunch of neat looking papers that, on first glance, seem to be well over my head.

Is there a such thing as a “discrete smooth space” or “abstract smooth space”, i.e. a smooth space whose underlying set is finite or countable?

I like the abstract simplicial complex example (Definition 23 on page 22) in this paper, which makes me think there could be an easier way, i.e. something even I could understand, if you confined yourself to the discrete world.

Posted by: Eric on December 8, 2008 3:50 PM | Permalink | Reply to this

### Re: Discrete Smooth Spaces

I’ve been fascinated by the interplay between discreteness and smoothness for a long time. My first grad student, James Gilliam, did his thesis on “discrete mechanics”, a generalization of classical mechanics in which both the phase space and time are discrete. There’s a lot of work on mechanics in which time takes integer values, so the new thing was adapting the use of calculus in physics to situations where the phase space is also discrete. It turns out that the Euler-Lagrange equation, Noether’s theorem, and the symplectic structure on phase space all generalize to this context! This required some ideas from algebraic geometry — but don’t worry, these are explained from scratch in his thesis:

• James Gilliam, Lagrangian and Symplectic Techniques in Discrete Mechanics, PhD thesis, U. C. Riverside, 1996. In PDF and Postscript.

Interesting examples include discrete versions of the harmonic oscillator, the rigid rotating body in n dimensions, and a variety of cellular automata including the Wess–Zumino model. Some but not all of this material was also published as a paper:

• John C. Baez and James Giliam, An algebraic approach to discrete mechanics, Lett. Math. Phys. 31 (1994), 205-212. In PDF and Postscript.

There’s more in the thesis than in this paper, though.

Our work would have been better if we’d known more algebraic geometry — but it may be easier to read because we didn’t.

Posted by: John Baez on December 8, 2008 5:37 PM | Permalink | Reply to this

### Re: Discrete Smooth Spaces

Thanks for reminding me about this paper. I started grad school just as Gilliam was finishing up, so I spent some time studying his thesis, but could never really absorb as much as I felt I should have been able to.

Could this material be considered examples of “discrete smooth spaces”?

By the way, I never noticed this before, but the rotation of your discrete WZW model seems to turn the square lattice into a 2-diamond. It would fun to resurrect this stuff.

Posted by: Eric on December 8, 2008 7:12 PM | Permalink | Reply to this

### Re: Discrete Smooth Spaces

Eric wrote:

Could this material be considered examples of “discrete smooth spaces”?

Yes, that’s why I mentioned it: it’s all about taking derivatives of functions between certain nicely structured finite sets.

By the way, I never noticed this before, but the rotation of your discrete WZW model seems to turn the square lattice into a 2-diamond.

Yes, the really nice way to think about the discretized WZW model is in terms of a 2-diamond lattice. The WZW model is an example of a conformal field theory, meaning a 2d field theory where all information travels at the speed of light (not slower). In conformal field theory, people call the 2-diamond idea ‘lightcone coordinates’.

Posted by: John Baez on December 9, 2008 1:29 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa

John wrote

Goodwillie introduced a concept of ‘approximation by Taylor series’ for interesting functors in homotopy theory.

That squares with this:

In the beginning of 90’s Goodwillie developed Calculus of Homotopy Functors, which provides powerful tools to interpolate between stable and unstable homotopy theory. According to Goodwillie, the category of spaces is like a manifold, the category of spectra are like the real numbers and homotopy functors from spaces to spectra are like real valued functions.

How do the category of spectra (your $\mathbb{Z}$-groupoids) and the real numbers resemble each other? Do they each have a similar universal property?

Posted by: David Corfield on December 8, 2008 5:05 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa

I don’t know. There’s some weird stuff about the Goodwillie calculus that I don’t understand, which is why it’s good Kristine Bauer is speaking at this conference — I got a lot out of earlier conversations with her on this subject.

Most importantly, it’s weird that the identity functor from spectra to spectra has a complicated and interesting Taylor series. The identity function, otherwise known as $x$, should have a simple Taylor series. But in the Goodwillie calculus it doesn’t:

This fascinating paper gives lots of clues as to what’s really going on, but I think the subject is still more obscure than it needs to be. I think something analogous to a ‘change of variables’ is going on, which makes even the identity functor have an interesting Taylor series.

André Joyal has a theory of ‘analytic functors’ which is much more intuitive than this Goodwillie calculus business — and closely related to things I like, such as ‘species’.

For example, the functor from $Set$ to $Set$ sending a set $X$ to the set

$1 + X + X^2/2! + \cdots$

is an analytic functor in Joyal’s theory. I think Joyal understands much better than I do how the Goodwillie calculus is related to analytic functors. I need to ask him about this sometime.

Posted by: John Baez on December 8, 2008 5:53 PM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa

John wrote:

Most importantly, it’s weird that the identity functor from spectra to spectra has a complicated and interesting Taylor series. The identity function, otherwise known as x, should have a simple Taylor series. But in the Goodwillie calculus it doesn’t:

A homogeneous linear functor is defined to be one sending coproducts to products, so it is like an exponential. Compared to an exponential, the identity functor is like a logarithm, so it has a non-trivial Taylor series.

I think something analogous to a ‘change of variables’ is going on, which makes even the identity functor have an interesting Taylor series.

I think that’s all that is really going on. Everything is secretly composed with something analogous to a logarithm.

Posted by: Dan Christensen on December 9, 2008 2:46 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa

Dan wrote:

Everything is secretly composed with something analogous to a logarithm.

Yay! But: is there any to present this material in a way that’s not complicated by this trick? I want a ‘Goodwillie calculus for idiots’ where the Taylor series of the identity functor is $x$, and more complicated functors have complicated Taylor series.

Are there actually functors lurking around here that play the role of ‘exponential’ and ‘logarithm’ — converting sums into products or vice versa? If so, what are they? If not, well, maybe I forgive people for being so damned devious.

Posted by: John Baez on December 9, 2008 3:00 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa

But: is there any to present this material in a way that’s not complicated by this trick? I want a ‘Goodwillie calculus for idiots’ where the Taylor series of the identity functor is x, and more complicated functors have complicated Taylor series.

I don’t know enough about it to be sure, but I suspect that if there was an easier way, it would be the way that people thought about it. The issue may be that towers of fibrations, which are analogous to infinite products, are a common way to filter between known and unknown things, so one naturally wants something landing in a multiplicative world. Greg Arone would be a good person to ask.

Are there actually functors lurking around here that play the role of ‘exponential’ and ‘logarithm’ — converting sums into products or vice versa? If so, what are they?

The functor from spaces to spaces which sends $X$ to $\Omega^{\infty} \Sigma^{\infty} X = \colim \Omega^n \Sigma^n X$ sends coproducts to products and is supposed to be like $e^{x-1}$. (The “-1” comes about from issues to do with basepoints.)

If not, well, maybe I forgive people for being so damned devious.

Even though that functor exists, it isn’t invertible in any sense, so you can’t simply use it to get rid of the complexity.

Posted by: Dan Christensen on December 9, 2008 3:16 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa

Can we see why $\Omega^{\infty} \Sigma^{\infty}$ and $e^{x - 1}$ are similar in simple cases?

I guess for $X$ trivial, we have $e^{1 - 1} = 1$, which seems right.

How about for $X = S^0$? Then might $\Omega^{\infty} S^{\infty}$ be like $e$?

I can only remember some ideas of John and Jim on $\Omega^{\infty} S^{\infty}$. That its $\omega$-groupoid is the initial stable $\omega$-groupoid. And that it’s the group completion of the pointed space of all finite sets of distinct points in the infinite dimensional cube, with empty set as base point.

Hmm, shades of the groupoid cardinality of FinSet being $e$?

Posted by: David Corfield on December 9, 2008 9:29 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa

From the Arone-Kankaanrinta paper, it looks like you’d be better off not thinking in terms of Taylor series:

It is the point of this paper that the Goodwillie tower is the homotopy theoretic analog of logarithmic expansion, rather than of Taylor series. (p. 6)

What’s going on, they say, is like finding a function of the form $a^{x - 1}$ which best approximates $x$. This is when $a = e$. Page 3 explains why you’d want a function of that form.

Posted by: David Corfield on December 9, 2008 9:51 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa

To clarify one thing, it’s the identity function from spaces to
spaces that has an interesting Taylor tower. The identity
function from spectra to spectra is already linear, since (finite)
coproducts and products of spectra are the same up to homotopy.

By the way, the Arone-Kankaanrinta paper you cited is I think a
great source of interesting ideas about Euler characteristic and
homotopy cardinality.

Posted by: Dan Christensen on December 9, 2008 3:00 AM | Permalink | Reply to this

### Re: Smooth Structures in Ottawa

Abstracts for the conference are now available.

Posted by: David Corfield on April 24, 2009 1:42 PM | Permalink | Reply to this
Read the post Smooth Structures in Ottawa II
Weblog: The n-Category Café
Excerpt: A summary of some talks at the Fields Workshop on Smooth Structures in Logic, Category Theory and Physics.
Tracked: May 9, 2009 9:07 PM

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