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January 25, 2007

Classical vs Quantum Computation (Week 11)

Posted by John Baez

Today in our course on Classical vs Quantum Computation we covered lots of examples of 2-categories, to show how widespread these gadgets are:

  • Week 11 (Jan. 25) - Examples of 2-categories. The 2-category of categories. The fundamental 2-groupoid of a topological space. The 2-category of topological spaces, maps, and homotopies between maps. The 2-category of topological spaces, maps, and homotopies between maps. The 2-category implicit in extended topological quantum field theories, due to Jeffrey Morton. The 2-category implicit in string theory, due to Stolz and Teichner. Monoidal categories as one-object 2-categories. The 2-category of rings, bimodules and bimodule homomorphisms. Monoidal categories as one-object 2-categories. The 2-category of rings, bimodules and bimodule homomorphisms.

    Supplementary reading:

Last week’s notes are here; next week’s notes are here.

The newest of these examples deserve a bit of advertisement.

Not enough people seem to have realized that Jeffrey Morton’s paper solves a tough problem in an elegant way. For work on extended topological quantum field theories, we really want a (weak) 2-category with

  • compact (n2)(n-2)-manifolds as objects,
  • (n1)(n-1)-dimensional cobordisms between these as morphisms,
  • and nn-dimensional cobordisms between those as 2-morphisms.

It’s an intuitive idea, but actually getting your hands on this 2-category seems annoyingly tricky. Jeffrey does it in an elegant by first building a bigger structure — a ‘double bicategory’ in the sense of Dominic Verity — and then finding the desired 2-category inside there. In fact, the double bicategory may ultimately be more useful than the 2-category, as I sketched in week242.

(In fact, there’s an important picture in this week’s notes that doesn’t really live in the 2-category I was describing — only in the double bicategory. As an exercise, try to spot it!)

In string theory we’d like some sort of 2-category like the above one for n=2n = 2, where the 2-morphisms are ‘string worldsheets’. But, to get this to work we really need the string worldsheets to be equipped with complex-analytic structures. The closest thing I know to a full treatment is section 4.2 in the paper by Stolz and Teichner cited above. Does anyone know a reference with more details? There are a lot annoyingly tricky issues.

Posted at January 25, 2007 10:06 PM UTC

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Re: Classical vs Quantum Computation (Week 11)

The 2-category implicit in string theory, due to Stolz and Teichner.

We did talk about this way back here. Is there progress on understanding this better?

Posted by: urs on January 26, 2007 9:43 AM | Permalink | Reply to this

Re: Classical vs Quantum Computation (Week 11)

The 2-category implicit in string theory, due to Stolz and Teichner.

Just to remind myself:

Somebody should really look into this: as Bruce mentioned here, people do consider categories of gradient flows. These follow precisely the same general idea as the Hilbert uniformization procedure in complex analysis.

I imagine it should be rather easily possible to get a 2-category of Morse flow and, correspondingly, a 2-category of “Hilbert-uniformized” little rectangular patches of a given complex surface.

Posted by: urs on January 26, 2007 5:13 PM | Permalink | Reply to this

Re: Classical vs Quantum Computation (Week 11)

Urs wrote:

Is there progress on understanding this better?

Not that I know of. The treatment in section 4.2 of Stolz and Teichner makes it look like they’ve solved the problem, and maybe they have, but this section is rather short on details, so I can’t tell.

Posted by: John Baez on January 26, 2007 7:48 PM | Permalink | Reply to this

Stolz and Teichner’s 2-category

John wrote :

The closest thing I know to a full treatment is section 4.2 in the paper by Stolz and Teichner cited above. Does anyone know a reference with more details? There are a lot annoyingly tricky issues.

I’ve always been interested in this viewpoint of Stolz and Teichner, because it shows that its possible to interpret nCob (the n-category of points, lines, surfaces, etc.) without using manifolds with corners.

Heresy, you say?

Well, think about it. Let’s strip down their definition, and get rid of all the conformal stuff. Define the 2-category 2Cob as follows:

Objects are 0-dimensional manfiolds xx.

A morphism γ:xy\gamma : x \rightarrow y is a 1d oriented manifold with boundary identifications xγyx \rightarrow \partial \gamma \leftarrow y.

If γ,γ:xy\gamma, \gamma' : x \rightarrow y are morphisms, then a 2-morphism σ:γγ\sigma : \gamma \Rightarrow \gamma' is an oriented 2d manifold with boundary (not with corners) together with identifications

σγ x,yγ\partial \sigma \cong \gamma \union_{x,y} \gamma'.

That is, the boundary of σ\sigma is γ\gamma glued with γ\gamma' along x,yx,y.

See? No manifolds with corners needed. And the process can be continued to arbitrary nn. It will always be a case of an nn-morphism being a nn-manifold with boundary, whose boundary is identified with two (n1)(n-1)-morphisms glued over their source and target (n2)(n-2)-morphisms.

However, there’s a caveat. If you work through the 2Cob example, you’ll see that it doesn’t give you what you might ‘want’ from 2Cob. Its the old old problem of it not distinguishing ‘D-branes’ from ‘open strings’. It should be pointed out, though, that neither does ‘traditional’ nCob - the one with points, lines, surfaces with corners, etc.

I’m not saying that this version of nCob is better than the manifolds-with-corners one. I’m just pointing out (1) it seems to be ‘no problem’ (give or take difficulties with n-categories :-)) to define a version of nCob without using corners - which is often not appreciated - and (2) whichever approach one uses (with corners or without) it is still confusing, and doesn’t give you intuitively what you might expect, or want.

One thing I’ll say about the traditional ‘with corners’ nCob : it’s perfect for tangles, as the Tangle Hypothesis shows!

I’d love to hear other thoughts on this, since its been confusing me for ages.

Posted by: Bruce Bartlett on January 26, 2007 5:53 PM | Permalink | Reply to this

Re: Stolz and Teichner’s 2-category

The “2-cobordisms” in the Stolz-Teichner-like description are all “globular”. All 0-morphisms have the topology of a (single!) point, all 1-morphisms that of the interval, all 2-morphisms that of the disk.

One would not want to interpret them as “global” total 2-cobordisms, but as “probes” of larger 2-cobordisms.

For instance, given a fixed 2-manifold Σ\Sigma, it may be useful to consider P 2(Σ)P_2(\Sigma), the 2-category of Stolz-Teichner-like 2-cobs inside Σ\Sigma.

No 2-morphism on P 2(Σ)P_2(\Sigma) will look exactly like Σ\Sigma itself. Albeit one can alway cut Σ\Sigma open to get a disk-shaped surface, this will involve choices.

Therefore, if one decides to replace 1-functors on ordinary 2-cobordisms with 2-functors on the Stolz-Teichner-like ones, and if one still wants to be able to say what the value of that functor on the total 2-cobordisms is supposed to be, in a way that does not depend on the choices involved in cutting that 2-cobordism, then one needs a notion of trace on the images of these 2-functors.

This is actually familiar from the use of traces one dimension below:

we can set up a theory of holonomy of circles in a bundle with connection by using something like 1Cob. But that’s not what one usually sees in the textbooks! Instead, in the textbooks people use (more or less implicitly) P 1(X)P_1(X), i.e. moprhisms that all look like intervals. The familiar trace in the definition of holonomy is there to glue these globular 1-paths to true 1Cobs from the empty set to the empty set.

One categorification step above this, one can see that traces on states of the open string do produce states of the closed string. Or so I think.

And I believe that also the issue of physical versus states boundaries (D-branes versus incloning/outgoing strings) can naturally be treated this way.

For globular 1-cobordisms, there is exactly one way to pair two “states”. The result is the usual “Wilson line between two quarks”, i.e. the 1-disk holonomy with two boundary insertions. No D-branes.

But for 2-morphisms one finds “first order” and “second order” pairings of states. Working things out, it turns out that the first describe the state/field insertions on a 2-disk, while the latter describe the “physical boundaries”.

So here, again, the message is that the Stolz-Teichner 2-cobordisms is only a “probe” of the disk. By itself, it sees neither state boundaries nor physical boundaries. These come in only as we suitably pair states by the transport over that disk.

For all these reasons, I think that, in addition to the “extended quantum field theories” which are reps of extended cobordism categories, there should be “extended globular QFTs” # which probe full cobordisms on globular slices and which have a notion of state pairing and traces.

Posted by: urs on January 27, 2007 1:37 PM | Permalink | Reply to this

Re: Stolz and Teichner’s 2-category

Urs wrote :

The “2-cobordisms” in the Stolz-Teichner-like description are all “globular”. All 0-morphisms have the topology of a (single!) point, all 1-morphisms that of the interval, all 2-morphisms that of the disk.

I’m not sure thats the way I understand it. In the original Stolz-Teichner description, they say (stripped of spin stuff):

The objects of D nD_n are 0-dimensional spin manfiolds… i.e. a finite number of points…

Similarly in the stripped down description of extended 2Cob I gave above, objects are 0-dimensional manifolds (collections of points).

It’s easy to get surfaces of nontrivial topology in this setup. For instance, a torus can be viewed as a 2-morphism T:ϕ 1ϕ 1T : \phi_1 \Rightarrow \phi_1, where ϕ 1:ϕ 0ϕ 0\phi_1 : \phi_0 \rightarrow \phi_0 is the empty 1d-manifold which interpolates between the empty 0d-manifold (agreed, its a bit pathological to have such a plethora of empty manifolds around, but this seems an unavoidable consequence of the idea of ‘nCob’.)

Nevertheless, I do agree that it can be helpful to consider internal versions of nCob, such as the P 2(Σ)P_2 (\Sigma) you describe.

Posted by: Bruce Bartlett on February 1, 2007 8:20 PM | Permalink | Reply to this

Re: Stolz and Teichner’s 2-category

Bruce wrote:

I’m not sure thats the way I understand it.

You are perfectly right. I am sorry. I did not correctly remember the definition they give.

Posted by: urs on February 1, 2007 9:12 PM | Permalink | Reply to this

Re: Classical vs Quantum Computation (Week 11)

(In fact, there’s an important picture in this week’s notes that doesn’t really live in the 2-category I was describing — only in the double bicategory. As an exercise, try to spot it!)

I haven’t really read Jeff’s paper, but he’s briefly explained it to me.

I think that the naughty diagram is towards the bottom of page 19 (page 6 in the PDF file). If you compare that with its schematic counterpart directly to its right, you’ll see that the nature of x and x′ are a bit vague. In particular, is x′ the big circle at bottom, or the small one? To be fair, you ought to shrink all three outer big circles, as they approach the top or bottom, to meet the inner small circles, thereby specifying x and x′ precisely. (The next diagram, atop page 20, does this.) As it is, the surfaces linking the big circles to the small circles must be horizontal morphisms in Jeff’s double bicategory.

Things would be even worse if there were holes in those horizontal surfaces, so that it would be impossible to shrink the outer circles to meet the inner circles. To be fair, the picture as given is probably equivalent, in a suitable sense internal to a double bicategory, to the properly drawn picture in the bicategory. But a picture with holes in the horizontal surfaces would be tractable only in the full double bicategory.

Am I right?

Posted by: Toby Bartels on January 29, 2007 2:17 AM | Permalink | Reply to this
Read the post Classical vs Quantum Computation (Week 12)
Weblog: The n-Category Café
Excerpt: How to see computation as a process: rewrite rules.
Tracked: February 2, 2007 12:01 AM
Read the post Amplimorphisms and Quantum Symmetry, I
Weblog: The n-Category Café
Excerpt: Localized endomorphisms of quantum observables in arrow-theoretic terms.
Tracked: February 6, 2007 12:07 PM

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