The n-Category Café

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

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 $\epsilon$ of the legendary quality of Derek Wise’s notes!). If we could also get some notes from Stony Brook… well that would be heaven.

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?

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 $[0,\infty)^k x R^{n-k})$ 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 $f :X \rightarrow Y$. 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 $X \times_{Z} Y$ 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.

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.

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 $x$ 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.

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.

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 $\mathbb{R}^n$ and then letting $n \to \infty$. 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 $n$-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.

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.