The n-Category Café

John writes, in this week’s lecture notes:

What’s really going on?

Is “quantization” some arbitrary trick, or does it have some deeper meaning? Let’s try to dig deeper! What sort of entity is the action?

Incidentally, when reading this I am in the middle of writing up a new refinement of the description of the cube.

I am going to post more details in a while, but I am also trying to condense the main idea into a single table, that organizes all the physics terms and tries to show what’s really going on.

The action functor is featured in the middle column:

I will describe this in more detail in a seperate post and discuss examples.

Two caveats:

(1) The above can be applied blindly only to the kinematical part of the quantization. Dynamics should follow the same pattern, but is more subtle.

But, on the other hand, we have a kind of holography at work, which says that the kinematics of the $n$-particle looks like the dynamics of the $(n-1)$-particle. I am gradually better understanding the details of the formalism behind that, but not sufficiently yet.

I do wonder, though, if, in the end, we want to turn that around and make it a definition: instead of directly saying what the quantum dynamics of the $n$-particle is, we define it as the quantum kinematics of the $(n+1)$-particle.

I’ll need to think about this and work through more examples.

(2) Where the above table says “transgression” I am slightly abusing common terminology.

Ordinary transgression is the composition of a pullback along $$\mathrm{conf}\times \mathrm{par} \stackrel{\mathrm{ev}}{\to} \mathrm{tar} \,,$$ as above, but then followed by “integrating out parameter space”, by pushing forward along the projection $$\array{ \mathrm{conf} \times \mathrm{par} \\ \downarrow \\ \mathrm{conf} } \,.$$ As you can see, I don’t use this push-forward in the prodecure indicated in the above table.

Instead, I retain the information of parameter space and get out an extended QFT, namely a functor on $\mathrm{par}$ in one step (instead of successively integrating out various parts of parameter space).

So, maybe I shouldn’t say “transgression” in the above. But it actually does, in the end, amount to the same sort of construction. (See this discussion for more on how “my transgression” relates to ordinary transgression)

Sorry to be nitpicky, but it always terribly irritates me when people confuse “principal” (meaning “main, chief, first” or “head of an educational institution”) and “principle”. The “principal of least action” would presumably be the director who does least…

Thanks, David, for prodding me to clarify the big picture here. I’m sorry not to have responded to all your questions! I’ve been really busy, and a lot of these questions require serious thought.

Anyway, here’s one where I’ve made some progress, thanks to you:

John said back in week 1 that (morally) the classical dynamics of point particles is the same as the statics of point particles, except that instead of using $X$, configuration space, you use $P X$, path space.

Hmm!

That’s one analogy:

particle statics                particle dynamics
configurations:  X              histories:  PX
energy:  V: X -> R              action:  S: PX -> R

Classically, on the left we minimize energy while on the right we minimize action.

But there’s also another analogy:

particle dynamics          string statics
histories:  PX             configurations:  PX
action:  S: PX -> R        energy:  V: PX -> R

Classically, on the left we minimize action while on the right we minimize energy.

In fact, the zig-zag pattern on first page of the first day’s notes arises when we alternately employ these two analogies! The first analogy moves us down a notch, while the second one moves us across a notch.

Categorification is an important part of this business, but I’m not sure which of these two analogies deserves to be called ‘categorification’. Maybe categorification happens only after we do both. I’ll have to think about this more.

David wrote:

Naturally, John has already answered my last question.

Naturally? Yeah, I guess this idea:

The dynamics of $p$-branes is the Wick rotated statics of $(p+1)$-branes.

has been heavily on my mind for quite a while.

Maybe I’ll take this chance to recall yet again how this idea impressed itself forcefully on my feeble brain. Some of you know this story all too well, but I’ve reached the age where I’m starting to enjoy saying the same thing over and over. So, here I go again:

The first way I met the above idea was more complicated than the second. (This is typical — in math, we often get interested in ideas because they seem complicated and impressively ‘sophisticated’, and only later realize they’re pathetically simple.)

I’d been studying elliptic functions, enjoying the fact that they’re doubly periodic in the complex plane, when I suddenly realized that this was related to how they show up in a famous classical mechanics problem: the motion of a pendulum. Note: not the wimpy old harmonic oscillator, but the full-fledged pendulum, which only reduces to the harmonic oscillator in the limit of small oscillations!

I realized that the pendulum’s motion is periodic not only with respect to real time $t$, but also with respect to imaginary time $i t$. The reason is simple: the Wick rotation

$$t \mapsto i t$$

turns Newton’s law

$$F = m a$$

into

$$F = -m a$$

This merely amounts to turning the pendulum upside-down! But, an upside-down pendulum is just another copy of the same pendulum. So, its motion is still periodic.

You can read the details here:

• John Baez, homework problem on The Pendulum, Elliptic Functions and Imaginary Time.
• Toby Bartels, Answers.

I found this exciting because I’d mainly thought of Wick rotation as a fancy thing you did in quantum mechanics, especially quantum field theory. I knew how Wick rotation turned the wave equation into Laplace’s equation. But, I’d never thought about it in the simple context of classical mechanics, so I’d never realized it just replaces $F = m a$ by $F = -m a$.

Later, I realized that you could turn any physics problem involving Newtonian gravity upside down by going to imaginary time!

The simplest of these is the thrown rock. When you consider a thrown rock in imaginary time, you get a hanging elastic spring — the Wick rotation flips the parabolic arc of the rock upside down to give the parabola traced out by the hanging elastic spring! You can read the details here:

• John Baez, homework on A Spring in Imaginary Time.
• Garett Leskowitz, Answers.
• Mike Stay, Answers.

This example made it clear that there was an simple relation between the dynamics of particles and the statics of springs — or strings.

So, by the time I wrote week225, I was just having fun taking a peek at how this works one dimension up. Everyone knows that the dynamics of strings simply minimizes their surface area in spacetime. And, everyone knows that the statics of soap films minimizes their surface area in space. So, a Wick rotated string is a soap film… and a Wick rotated ‘D-brane’ turns out to be precisely the mathematical analogue of the ‘wire frame’ we often use to hold a soap film in place!

Anyway, I think there’s a lot of mileage left in this idea… it’s simple but it hasn’t been fully exploited.

In fact, I predict that in 2030, Polchinski will usher in the Seventh String Revolution when he invents the Wick rotated analogue of those little pipes kids use to blow soap bubbles — and uses it to explain the creation of the universe.

I’m going to jump down here to continue discussing Lawvere’s topos-theoretic approach to mechanics… the tree of comments is getting oppressively complex!

Urs wrote:

But actually, what I understand best [of Lawvere and Kock’s work on physics and geometry] is that where the topos-theoretic underpinning is not directly visible.

Yeah, I feel the same way. Luckily, we don’t really need to know much topos theory to deal with this stuff. We can just imagine that we’re working in a category of smooth spaces that lets us do everything we might ever want to do — including work with infinitesimal spaces.

As you know, the most important infinitesimal space is what Jim Dolan calls ‘walking tangent vector’: an infinitesimal arrow. Lawvere calls it $T$, since maps from $T$ to any smooth space $X$ are tangent vectors in $X$:

$$T X = X^T$$

Chen’s category of smooth spaces, as defined in our paper, is not big enough to include such infinitesimal spaces. We defined this category as follows:

A smooth space is a set $X$ equipped with, for each convex set $C$, a collection of functions $\phi : C \to X$ called plots in $X$, such that:

1. If $\phi : C \to X$ is a plot in $X$, and $f : C' \to C$ is a smooth map between convex sets, then $\phi \circ f$ is a plot in $X$,
2. If $i_\alpha : C_\alpha \to C$ is an open cover of a convex set $C$ by convex subsets $C_\alpha$, and $\phi : C \to X$ has the property that $\phi \circ i_\alpha$ is a plot in $X$ for all $\alpha$, then $\phi$ is a plot in $X$.
3. Every map from a point to $X$ is a plot in $X$.

A smooth map from the smooth space $X$ to the smooth space $Y$ is a map $f : X \to Y$ such that for every plot $\phi$ in $X$, $\phi \circ f$ is a plot in $Y$.

In highbrow lingo, this says that smooth spaces are sheaves on the category whose objects are convex sets and whose morphisms are smooth maps, equipped with the Grothendieck topology where a cover is an open cover in the usual sense. However, smooth spaces are not arbitrary sheaves of this sort, but precisely those for which two plots with domain $C$ agree whenever they agree when pulled back along every smooth map from a point to $C$.

When we restrict to sheaves with the special property above, we exclude infinitesimal spaces and we don’t get a topos — we get something called a ‘concrete quasitopos’.

If we allow all sheaves, we get a topos. Unfortunately, I believe this does not include infinitesimal spaces.

(I wanted to exclude infinitesimal spaces in our paper since I wanted our smooth spaces to be sets with extra structure. In other words, I wanted a forgetful functor from smooth spaces to sets which was faithful — mainly so differential geometers and topologists wouldn’t freak out. That’s the point of condition 3 above, and that’s also what the word ‘concrete’ means above.)

When I started writing this post I hoped Chen’s smooth spaces would include the ‘walking arrow’ if we modified the definition to obtain a topos. Now I don’t believe this. But, since I spent a lot of time writing this post, I’ll describe the modification anyway!

The idea is that we stop saying that a smooth space is a set $X$ equipped with a set of plots $\phi: C \to X$ for each convex set $C$. Instead, we simply say that a smooth space is a magical gizmo $X$ which assigns to each convex set $C$ a set $X(C)$ of ‘$C$-shaped plots in $X$’. We also need a way to ‘pull back’ $C$-shaped plots to $C'$-shaped plots whenever we have a map $f: C' \to C$.

More precisely, we say:

An abstract smooth space is a functor $X$ from convex sets to sets. So, for any convex set $C$ we have a set $X(C)$ of $C$-shaped plots in $X$, and if $f: C' \to C$ is a smooth map between convex sets, we get a pullback map $$X(f) : X(C) \to X(C')$$ such that $$X(f g) = X(g)X(f)$$ and $$X(1_C) = 1_{X(C)}.$$

Moreover, we demand that $X$ satisfy the sheaf condition: if $i_\alpha : C_\alpha \to C$ is an open cover of a convex set $C$ by convex subsets $C_\alpha$, and $\phi_\alpha \in X(C_\alpha)$ are plots in $X$ that agree when pulled back to the intersections $C_\alpha \cup C_\beta$, there is a unique $C$-shaped plot $\phi$ in $X$ that restricts to all the $\phi_\alpha$: $$X(i_\alpha): \phi \to \phi_\alpha.$$

This is a topos of sheaves on a category with a certain Grothendieck topology. But, I’m pretty sure it doesn’t include the ‘walking arrow’ $T$, since I can’t imagine what the nontrivial plots in $T$ would be.

I now think we need to generalize Mostow’s approach to smooth spaces instead of Chen’s if we want to include infinitesimal spaces. Or, we could just read Kock’s book — he has a construction that works too.

Now, as for how Lawvere uses such topos to describe ‘laws of motion’, maybe I’ll talk about that in a separate post. I heard him discuss these things at his 60th birthday conference in Florence, and — somewhat to my shock — I felt for the first time that I actually understood them! But, I’ll need to dig up my notes or look at his paper again before I can say anything intelligent.

I now think we need to generalize Mostow’s approach to smooth spaces instead of Chen’s if we want to include infinitesimal spaces.

Darn! I really liked the way that smooth spaces could be defined in a sheafy way.

Any chance of defining the infinitesimal space $T$ as a smooth space by declaring that $T(C) := T^*C$ , the underlying set of the cotangent bundle of $C$? At least this is contravariant, and it probably glues together to form a sheaf

If $X$ is a manifold, which we think of as a sheaf $X : convex sets \rightarrow Set$ in the obvious way, one would like to have that

[Ed : Is this what one would like to have?]. Sadly I couldn’t get this to work :-)

I’ll follow John down the tree as well, even if our heads still seem to be up in the clouds on this particular topic.

Previously, on The Amazing Right Adjoint, John wrote:

Jim was also wondering how bizarre the Grothendieck topology was that these guys use to get this amazing right adjoint. An example of a locus is a finite set, so it seems any ‘symmetric set’, i.e. functor $$F: FinSet^{op} \to Set,$$ gives a presheaf on the category of loci. He was wondering if these are actually sheaves. If they are, we’ve got some pretty weird stuff going on!

[Gives a presheaf on loci how? By left Kan extension along the inclusion $FinSet \to Loci$, maybe?]

These guys [who develop SDG – Synthetic Differential Geometry] consider lots of Grothendieck topologies, and not just on the category of “loci”. Near the beginning of his talk, Lawvere quickly mentions a number of contexts relevant to SDG, including algebraic geometry. Note that a finite set is also an example of a spectrum of a commutative ring. So you could apply Jim’s question equally well to all the various topologies that algebraic geometers consider on the category of affine spectra, and wonder how bizarre their sheaf toposes are from that point of view, although I’m still not sure what that point is.

I don’t think these guys set out to create a topos where this amazing right adjoint exists; I think it was probably more like, “hey, didja ever notice that in all these SDG toposes, the tangent bundle functor ( )$^T$ has a right adjoint ( )$_T$? Coool… so this means we can do all these other things” [that the traditionalists can’t]. I’d have to think about it to be sure, but I wouldn’t be at all surprised if this right adjoint $(-)_T$ crops up all over the place in algebraic geometry as well.

It certainly exists in the topos of presheaves on affine spectra over $R$, where $T$ is the spectrum of $R[x]/(x^2)$, just as it does in any presheaf topos $[C^{op}, Set]$ when $T$ belongs to $C$. I strongly suspect it exists in the case of the Zariski topos, and I’d bet it’s there in the etale topos as well. So?