Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

October 15, 2009

Seminar on Cobordism and Topological Field Theories at UCR

Posted by Alexander Hoffnung

This quarter Julie Bergner has begun a seminar at UC Riverside on cobordisms and topological field theories. The abstract on the seminar homepage states:

In this seminar we’ll work through recent notes of Lurie giving an outline of his proof of the Cobordism Hypothesis, relating cobordism classes of manifolds and topological field theories. This work brings together several areas of recent mathematical interest: topological field theories, cobordisms of manifolds, and homotopical approaches to higher categories. We’ll go over basic definitions and examples of all of the above and then work towards understanding Lurie’s proof.

The main reference for this seminar is Jacob Lurie’s paper on the classification of topological field theories.

You can also see the list of scheduled speakers on the homepage. I believe the plan is that Julie will give most of the talks while others fill in once in a while.

I will try to keep posting notes from this seminar here for anyone who is interested in following along. I will start by posting the introductory lecture by John Baez, which took place about two weeks ago, and I will post a lecture every few days until we are caught up to the seminar. Then I will try to post once a week after each seminar. Christopher Walker has been kind enough to draw pictures without which these notes would be terribly hard to follow.

John begins the story with the cobordism hypothesis and the birth of nnCob:

The Cobordism Hypothesis. nnCob is the free stable (,n\infty, n)-category on a fully dualizable object.

nCob began life as a category where:

  • objects are framed (compact smooth) (n1n-1)-dimensional manifolds and
  • morphisms are framed (compact smooth) nn-dimensional cobordisms.

… click here for more. Enjoy!

Posted at October 15, 2009 11:04 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2085

34 Comments & 1 Trackback

Re: Seminar on Cobordism and Topological Field Theories at UCR

We’re also having a seminar on the same subject here at Stony Brook.

Posted by: Eitan Chatav on October 16, 2009 12:58 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

That looks great. Would anyone there be able to share notes as well? (typed or hand-written) I can make a page on the nLab where notes from each seminar could be posted.

Posted by: Alex Hoffnung on October 16, 2009 1:10 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Thanks for this, Alex!

I can make a page on the nLab where notes from each seminar could be posted.

So far nnLab: cobordism hypothesis exists (but wouldn’t mind more attention) and I kept collecting links there.

But, yes, it might be a good idea to collect the links for your lecture notes on a separate nnLab page and then link to that page.

Posted by: Urs Schreiber on October 16, 2009 8:06 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Yes, this is great. Looking forward to it in a big way! Alex, thanks for tpying up the notes. The Tex notes are great; I was quite fond of your handwritten notes from previous seminars though (within ϵ\epsilon of the legendary quality of Derek Wise’s notes!). If we could also get some notes from Stony Brook… well that would be heaven.

Posted by: Bruce Bartlett on October 16, 2009 12:15 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

As a service to Bruce as requested on the nForum an announcement:

Bruce created a dedicated page

UC Riverside Seimar on Cobordisms and Topological Field Theories.

Posted by: Urs Schreiber on October 17, 2009 2:41 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Thanks Urs… I’m at work now so I can post.

Here is a question on the first introductory talk by John. I got a bit confused near the last part of the talk, the part about the “almost theorem 2” that “1Cob is the free symmetric monoidal category on one object xx with a dual”. Below that theorem, it was a bit confusing because it said that the first Reidemeister move didn’t hold in the free stable category on one object and a dual. Firstly, I don’t get that: I thought with higher codimensions to play with the first Reidemeister move would always hold. Secondly, I got confused: the almost theorem didn’t refer to “stable”, but the counterexample referred to “stable”, as if they were talking about different things?

Posted by: Bruce Bartlett on October 17, 2009 3:12 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Hi Bruce,

I will try to answer to see if I am following and if not, I am sure someone will jump in and help out.

Firstly, I don’t get that: I thought with higher codimensions to play with the first Reidemeister move would always hold.

I think the point is that John is just talking about something algebraic, the free symmetric monoidal category on one object, with a nice property, a dual. So, this is not a priori a topological issue. Instead it is just a fact about this category.

Or, am I missing your point, and you think framings don’t do the trick; that the first Reidermeister move still holds with framings with high enough codimension?

Secondly, I got confused: the almost theorem didn’t refer to “stable”, but the counterexample referred to “stable”, as if they were talking about different things?

Earlier in the lecture John decrees stable and symmetric monoidal to be synonymous.

Posted by: Alex Hoffnung on October 17, 2009 5:14 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Ok, thanks Alex.

Or, am I missing your point, and you think framings don’t do the trick; that the first Reidermeister move still holds with framings with high enough codimension?

Darn, it’s the latter. I am embarassed but put me out of my misery: why shouldn’t we expect the first Reidemeister move to hold even for a knot in 10 dimensions?

Posted by: Bruce Bartlett on October 19, 2009 12:49 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

I think the point is to treat the framing as a separate issue from the topologically obvious equality you are referring to. Something like untwisting the left side of R1 give a non-trivial path in the normal bundle…or something like this. I am just trying to guess the answers to your questions from intuition, without formalizing much.

Posted by: Alex Hoffnung on October 20, 2009 8:10 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

I know much about quantum field theories, but not enough about the connections to category theory.

Could someone please explain in which way I can get a field theory in physicist’s parlance from an instance of Definition 1.1.5 in Lurie’s paper?

What, in this setting, is a field? And what, in this setting, are the physical observables?

Posted by: Arnold Neumaier on October 17, 2009 5:46 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

What, in this setting, is a field? And what, in this setting, are the physical observables?

If we define a TQFT of dimension nn as a symmetric monoidal functor Z:nCobVectZ : nCob \rightarrow Vect, then a field on a closed (n1)(n-1)-dimensional manifold Σ\Sigma (thought of as “space”) is simply an element of the vector space Z(Σ)Z(\Sigma). The “functor” stuff encodes the observables: instead of thinking of the thing as a functor, you can think of it as a machine for calculating “correlation functions” (which is the viewpoint Atiyah actually took, it was Segal who brought in the “functor” viewpoint). Namely, if ϕ 1,,ϕ n\phi_1, \ldots, \phi_n are fields living in Z(Σ 1),,Z(Σ n)Z(\Sigma_1), \ldots, Z(\Sigma_n), then you can calculate a number from this by letting MM be a cobordism from Σ 1Σ n\Sigma_1 \sqcup \cdots \sqcup \Sigma_n to the empty manifold (i.e. you picture MM as having nn “inputs” and no “outputs”). Then the fact that ZZ is a “monoidal functor” means that you can basically calculate the number

(1)ϕ 1ϕ n=Z(M)(ϕ 1ϕ n) \langle \phi_1 \cdots \phi_n \rangle = Z(M) (\phi_1 \otimes \cdots \otimes \phi_n)

which is an “observable” in this setting. See Atiyah’s original paper referenced in Lurie’s article for a much better explanation than this one.

Posted by: Bruce Bartlett on October 18, 2009 12:38 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

What’s the shortest root from this sort of abstract `field’ to the kind physicists are `really’ thinking of - functions or sections of some vector bundle or… -
or vice versa?

Posted by: jim stasheff on October 18, 2009 3:01 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

I have some vague souvenirs from the little book of Atiyah where he retraces the ongoing (by then), emergence of the theory of topological quantum field theory. As far as I remember, exemples of such functors are build by means of geometric quantization. This means that fields, in this setting, are sections of a line bundle. The title of this very short book is “the geometry and physics of knots”

Posted by: yael fregier on October 19, 2009 12:24 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Unlike Bruce I wouldn’t use ‘field’ to describe a vector in the vector space Z(Σ)Z(\Sigma). I would call that a ‘state’.

In full-fledged quantum field theory, ‘states’ are often functions on some set of ‘fields’. ‘Fields’, in turn, are often functions on a manifold Σ\Sigma which we call ‘space’.

The curious fact that ‘states’ are functions on a set of functions on space is the origin of the term ‘second quantization’.

Everything I just said admits endless generalization. I was trying to keep things simple, but it’s usually more complicated. For example, ‘states’ are often sections of line bundles on the set of ‘fields’ — this is what Yael was actually trying to say. Moreover, ‘fields’ are often sections of bundles over ‘space’, or connections on bundles over ‘space’, or even bundles-with-connection over ‘space’.

For example, consider the Dijkgraaf–Witten model, which is one of the simplest TQFTs. Here we pick a finite group GG. In this case, a ‘field’ is a principal GG-bundle over Σ\Sigma, where Σ\Sigma is the manifold describing ‘space’. Then a vector in Z(Σ)Z(\Sigma) — that is, a ‘state’ — is an equivalence class of functions on the set of ‘fields’.

But beware: while this is one of the simplest examples in many respects, it’s also far removed from the examples that first made physicists get interested in quantum field theory! So folks seeking physical intuition will need to look further.

The wonderful thing about topological quantum field theory is that mathematicians can prove theorems about it without understanding anything I just said. Atiyah’s axiomatic approach frees the mathematician from needing to understand the underlying physics.

However, unsurprisingly, a lot of the most profound work on topological quantum field theory has been done by people who do understand the physics.

Posted by: John Baez on October 20, 2009 7:19 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Doh doh pic, you’re right, I got confused; every now and then I make this same blunder! To make myself feel better, I will try to confuse others as follows. A state is a function on the space of functions on the space. But by the functor of points approach to geometry, we are told that a space is a function on the space of spaces! Putting this together, a state is a function on the …

Posted by: Bruce Bartlett on October 20, 2009 8:06 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Could someone please explain in which way I can get a field theory in physicist’s parlance from an instance of Definition 1.1.5 in Lurie’s paper?

try nnLab:FQFT and references given there

Posted by: Urs Schreiber on October 19, 2009 10:03 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Thought I’d sneakily post this: two interesting papers on the arXiv today definitely relevant for the n-category cafe, especially the first one!

  • Jonathan Woolf, Transversal homotopy theory. Implementing an idea due to John Baez and James Dolan we define new invariants of Whitney stratified manifolds by considering the homotopy theory of smooth transversal maps. To each Whitney stratified manifold we assign transversal homotopy monoids, one for each natural number. The assignment is functorial for a natural class of maps which we call stratified normal submersions. When the stratification is trivial the transversal homotopy monoids are isomorphic to the usual homotopy groups. We compute some simple examples and explore the elementary properties of these invariants. We also assign `higher invariants’, the transversal homotopy categories, to each Whitney stratified manifold. These have a rich structure; they are rigid monoidal categories for n>1 and ribbon categories for n>2. As an example we show that the transversal homotopy categories of a sphere, stratified by a point and its complement, are equivalent to categories of framed tangles.
  • Dominic Joyce, On manifolds with corners. Manifolds without boundary, and manifolds with boundary, are universally known in Differential Geometry, but manifolds with corners (locally modelled on [0,) kxR nk)[0,\infty)^k x R^{n-k}) have received comparatively little attention. The basic definitions in the subject are not agreed upon, there are several inequivalent definitions in use of manifolds with corners, of boundary, and of smooth map, depending on the applications in mind. We present a theory of manifolds with corners which includes a new notion of smooth map f:XYf :X \rightarrow Y. Compared to other definitions, our theory has the advantage of giving a category Man^c of manifolds with corners which is particularly well behaved as a category: it has products and direct products, boundaries behave in a functorial way, and there are simple conditions for the existence of fibre products X× ZYX \times_{Z} Y in Man^c. Our theory is tailored to future applications in Symplectic Geometry, and is part of a project to describe the geometric structure on moduli spaces of J-holomorphic curves in a new way. But we have written it as a separate paper as we believe it is of independent interest.
Posted by: Bruce Bartlett on October 20, 2009 7:01 PM | Permalink | Reply to this
Read the post Cobordism and Topological Field Theories Week 2
Weblog: The n-Category Café
Excerpt: Week 2 of the cobordism and TFT seminar introduces oriented manifold and cobordisms in a more formal way.
Tracked: October 22, 2009 8:56 PM

Re: Seminar on Cobordism and Topological Field Theories at UCR

Hello all,

I just looked at Lecture 1. I don’t understand the last argument,
about the Reidemeister-I move. Could someone explain it in more
detail?

As it stands it seems to me it contains a syntax error, possibly
caused simply by a missing macro definition, but when I try to
provide that definition, the equation ends up holding instead of
being an nonequality.

I understand that we are in a symmetric monoidal category, that the
diagrams should be read from the top to the bottom, that x is a
downward string, that x* is an upward string, that i_x is a
counterclockwise upper semicircle, that e_x is a clockwise lower
semicircle, and that the the orientation-perserving combinations of
these two symbols yield identity arrows.

But in the Reidemeister-I nonequality as drawn in Lecture 1, there is
a new symbol, namely a clockwise upper semicircle.

Since we are in a symmetric monoidal category, I can only interpret
this new symbol as being actually the composite of i_x with the
braiding. But then the left-hand side looks like an ampersand, and
since the braiding is just a symmetry we can drag the e_x across the
other string to eliminate the crossings (Reidemeister II, if you
wish), and with one application of the zig-zag law we arrive at a
single x-string.

Obviously my interpretation is different from the intended one.
What is the nonequality supposed to mean?

PS: in case my prose is too convoluted, I have put some drawings at
http://mat.uab.cat/~kock/tmp/_1028144125_001.pdf

Cheers,
Joachim.

Posted by: Joachim Kock on October 29, 2009 1:36 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Hi Joachim,

So is your point that in both the free symmetric monoidal category on an object with a dual and 1Cob that R1 should hold?

So we don’t need to introduce framings?

Posted by: Alex Hoffnung on October 31, 2009 12:26 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Hi Alex,

> So is your point that in both the free symmetric monoidal category
> on an object with a dual and 1Cob that R1 should hold?

I don’t speak about 1Cob or framings at all (yet).

So far I only try to figure out what R1 could mean in a symmetric
monoidal category with duals. What I suggested is that the left-hand
side of R1 contains a symbol that has not formally been introduced
yet, and the meaning I have been able to attribute to this symbol
leads to the conclusion that R1 holds…

> So we don’t need to introduce framings?

I don’t really understand the framing picture on the last page, so I
thought I should understand the algebraic side of the argument first.
Let me come back to the question of framings afterwards.

Cheers,
Joachim.

Posted by: Joachim Kock on October 31, 2009 4:44 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

The issue of the first Reidemeister move is subtle and deeply connected with the concept of ‘framing’. I didn’t have time to explain this issue in my lecture. I’d prepared about twice as much material as I was actually able to cover, and by the time I got to this point I was charging ahead like a madman! So, Joachim is correct to note that my talk inaccurately glossed over a very important issue. I have written much better explanations of this stuff elsewhere.

If we work with oriented 1d corbordisms, we need the first Reidemeister move to hold, so we have to set up our category theory to make sure it does. If we work with framed oriented 1d cobordisms, it doesn’t hold, so we need to make sure it doesn’t. After you think about this stuff a while, you see that framed oriented cobordisms are the nicest ones to study — though it may seem counterintuitive at first.

Historically, this whole issue first arose when people were seeking an algebraic description of the category of 1d tangles in 3d space. If you set things up right — which involves working with framed oriented tangles — this is the ‘free braided monoidal category with duals on one object’. The category of framed oriented 1d cobordisms is very similar. Why? Because it’s equivalent to the category of framed oriented tangles in 4d space, which is the ‘free symmetric monoidal category with duals on one object’. So, the only difference is the difference between braiding and symmetry.

I’m not trying to explain anything here, just give some pointers. For an actual explanation, I suggest reading the whole section entitled Freyd–Yetter in the paper ‘A prehistory of nn-categorical physics’, and then the section entitled Atiyah, and then the section entitled Doplicher–Roberts, and then the section entitled Baez–Dolan, and especially page 108.

Posted by: John Baez on November 4, 2009 12:38 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Hi,

Ok, I see your point. I would say that for a good notion of the word “dual”, we should expect that this symbol has been formally introduced, at least implicitly.

That is, if xx and x *x^* are dual to each other, then unless they each come equipped with a unit and counit, then our notion of duality has a preferred direction. I guess this would not be a good thing.

Posted by: Alex Hoffnung on November 1, 2009 3:37 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

> I would say that for a good notion of the word “dual”, we should
> expect that this symbol has been formally introduced, at least
> implicitly.

> That is, if x and x* are dual to each other, then unless they each
> come equipped with a unit and counit, then our notion of duality has
> a preferred direction. I guess this would not be a good thing.

I am sorry, I don’t understand these reflections. Does it mean that
the “almost theorem” is not based on definitions but on “almost
definitions”? Who wrote this Reidemeister nonequality in the first
place? Is there a reference I can look at?

Surely I am missing something, but I suspect more and more that the
missing bit is an essential point, and that it would be useful to
get it straight.

I would really be grateful if somebody can help me out. I am
confident I can provide the proof myself, if just I understand
the statement and the involved definitions.

Cheers,
Joachim.

Posted by: Joachim Kock on November 3, 2009 10:13 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

I guess I will back off trying to answer and ask a question, since I am still confused also.

So there seem to be two options for the defn of free symmetric monoidal category on an object with a dual:

1) Have the definition include all units and counits for the object and dual. (I do not know if this is the traditional definition or not, but from Joachim’s questions I am guessing it is not.)

2) Have the definition include a unit and counit only for the original object and not its dual. In this case Joachim has provided what seems to me to be candidates for the unit and counit of the dual object.

So the questions I have which may be almost the same are:

a) are these 2 categories equivalent?

b) are the new cap and cup Joachim defines the unit and counit for the dual object?

Posted by: Alex Hoffnung on November 3, 2009 5:36 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

The word ‘dual’ means right dual (it seems from Lecture 1). When in
(1) you suggest to include all units and counits for both x and x*, I
suppose it is meant that there is a third object, x**, the right dual
of x*. Then the ‘missing symbol’ involves x* and x**, not x, and the
R1 still does not make sense.

However, the presence of the braiding implies that also x can serve as
right dual of x* (that’s an easy fact, answering YES to question (b)
— I did indeed take this for granted in my first post), and
therefore, by uniqueness of right duals up to unique isomorphism,
there is a canonical identification of x with x**. But this
identification also identifies the unit and counit for x* -| x** with
the unit and counit for x* -| x that was constructed using the
braiding, and again the R1 of Lecture 1 ends up being valid — unless
I misunderstand some basic point.

For option (2) I already argued that R1 holds.

It seems to me that either way R1 will hold. (And it also looks
like the answer to question (a) will be YES, with appropriate
rigourous definitions.)

I fear I am not coming any closer to understanding the R1
non-equality.

Cheers,
Joachim.

Posted by: Joachim Kock on November 4, 2009 12:27 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Ok, thanks!

It seems to me that either way R1 will hold.

In my last comment I was not trying to say anything about R1. The proof you originally drew was fine for me. It was also nice because I hadn’t know this “ampersand trick”. It seems generally useful.

I was backing up to just try to understand the definitions for the algebraic structure. You cleared up a few things regarding this that I was very confused about.

I fear I am not coming any closer to understanding the R1 non-equality.

I fear I am in that boat with you.

Posted by: Alex Hoffnung on November 4, 2009 12:43 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Meanwhile, I also have some trouble with the description of the
framing, and in particular with the picture on the last page. This
trouble was already expressed by Bruce, but I am afraid the answers
he got are not enough to help me out.

The notes say:

> In this case, framings are a choice of smooth field of normal
> vectors along a 1-morphism.

Already this sentence I don’t understand. The notion of normal
vector, usually, is a relative notion and needs some ambient space to
make sense. Is the 1-morphism embedded somewhere?

Usually a framing is defined rather to be a trivialisation of the
tangent bundle (an intrinsic notion). In that case, as far as I
understand, a framing of a smooth 1-manifold is nothing more than
an orientation. But then I can’t make any sense of those Reidemeister
drawings :-(

More technically, the notion of framed 1-cobordism should also involve
the notion of 1-framing of a 0-manifold: this should be a
trivialisation of ‘tangent bundle oplus extra copy of R’. This extra
copy can perhaps be thought of as a normal bundle of the 0-manifold
once we stick it onto the 1-manifold…

I am suspecting that in dimension 0+1 all the framing formalism is
already encoded in the ‘tangle’ picture on page 1 (just after
‘Consider n=1’), and that it amounts to nothing more than the sign-
and-arrow convention. But then again, I don’t understand the caption
of that figure: ‘Picture of 1-tangle in 2-dimensions…’ Which 2
dimensions? (And the final Reidemeister drawing looks like it makes
reference to 3 dimensions…)

In the best interpretation I can come up with at the moment, the
framings are already there, R1 holds already in both 1Cob and the free
symmetric monoidal category with duals (insofar as I can make sense of
it), the almost theorem is a real theorem, and then I just accept not
to understand anything of the remaining remarks of the lecture :-(

Cheers,
Joachim.

Posted by: Joachim Kock on November 4, 2009 12:38 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

The issue of the first Reidemeister move is subtle and deeply connected with the concept of ‘framing’. I didn’t have time to explain this issue in my lecture. I’d prepared about twice as much material as I was actually able to cover, and by the time I got to this point I was charging ahead like a lunatic, handwaving madly! So, Joachim is correct to note that my talk inaccurately glossed over some very important issues. I have written much better explanations of this stuff elsewhere.

Among other things, I didn’t have time to explain cobordism theory. Cobordisms come in various flavors, which come from equipping their stable normal bundle with various amounts of extra structure. The ‘stable normal bundle’ is defined by treating the cobordism as embedded in n\mathbb{R}^n and then letting nn \to \infty. It doesn’t require any metric for its definition.

Framed cobordisms have the most extra structure: a ‘framing’ of a cobordism is a trivialization of its stable normal bundle. Oriented cobordisms have less: just an orientation on the stable normal bundle, which is equivalent to an orientation on the tangent bundle. A framed cobordism is automatically oriented.

Knot theorists: note that this use of the term ‘framing’ differs from its use in knot theory!

If we work with oriented 1d cobordisms, we need the first Reidemeister move to hold, so we have to set up our category theory to make sure it does. If we work with framed 1d cobordisms, it doesn’t hold, so we need to make sure it doesn’t. After you think about this stuff a while, you see that framed oriented cobordisms are the nicest ones to study — though it may seem counterintuitive at first.

Historically, this whole issue first arose when people were seeking an algebraic description of the category of 1d tangles in 3d space. If you set things up right — which involves working with framed oriented tangles — this is the ‘free braided monoidal category with duals on one object’. The category of framed (oriented) 1d cobordisms is very similar. Why? Because it’s equivalent to the category of framed oriented tangles in 4d space, which is the ‘free symmetric monoidal category with duals on one object’. So, the only difference is the difference between braiding and symmetry.

I’m not trying to explain anything here, just give some hints about what I would need to explain for anything to make sense. For an actual explanation, I suggest reading the whole section entitled Freyd–Yetter in the paper ‘A prehistory of nn-categorical physics’, and then the section entitled Atiyah, and then the section entitled Doplicher–Roberts, and then the section entitled Baez–Dolan, and especially page 108.

Posted by: John Baez on November 4, 2009 12:40 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Thanks for the pointers, John. I would really like to understand this well. Thank you Bruce and Joachim for keeping this conversation going. It is kind of like taking the class in slow motion (a perfect pace for me.)

Posted by: Alex Hoffnung on November 4, 2009 12:51 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

For more on the cobordism theorist’s notion of ‘framing’ and how it differs from the knot theorist’s, try this.

Posted by: John Baez on November 4, 2009 2:19 AM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Unless I’m mistaken, the category of tangles in R^3 with trivialized normal bundles is the free -ribbon- category on one dualizable generator. The free braided monoidal category on one dualizable generator is instead given by tangles in R^3 with a trivialization of their Gauss map to RP^2 = O(1)\O(3)/O(2). Taking the limit as 2 -> infinity should give a description of the free symmetric monoidal category on one dualizable generator as a bordism category whose morphisms are 1-manifolds whose tangent bundles are equipped with trivializations (i.e., oriented 1-manifolds), rather than stable trivializations.

Posted by: Jacob Lurie on November 4, 2009 2:27 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

As soon as I get a chance I will try to use this conversation to expand and fix-up the notes to take into account the various issues people have pointed out.

And now that the AMS conference at UCR is over, I will get back to posting new notes.

Posted by: Alex Hoffnung on November 9, 2009 5:29 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Looking forward to it!

Posted by: Jamie Vicary on November 10, 2009 8:50 PM | Permalink | Reply to this

Re: Seminar on Cobordism and Topological Field Theories at UCR

Hi John,

thanks a lot for the explanation, that was enough to help me
out.

I am not too worried about not understanding all the details of
framing — geometry is a can of worms! — but I thought I
ought to be able to understand those symmetric monoidal
categories.

The Doplicher-Roberts section in the ‘Prehistory’ indeed made it
clear what is meant by duals. In retrospect, the confusion
comes from the fact that the Lecture (notes) only defines (and
only talks about) the notion of dual for objects, and the Almost
Theorem talks about ‘one object with a dual’ (not in plural).

I am now reassured that the Almost Theorem is actually a
Theorem, with the definitions I had first understood.

By the way the ‘Prehistory’ seems to be a very interesing
manuscript — I look forward to read the rest of it. But it’s
a greedy prehistory that goes as far as the Summer of 2009!

Cheers,
Joachim.

Posted by: Joachim Kock on November 4, 2009 2:51 PM | Permalink | Reply to this

Post a New Comment