## May 20, 2008

### Hopkins-Lurie on Baez-Dolan

#### Posted by Urs Schreiber

I just heard at HIM from Mike Hopkins apparently the kind of talk that Jacob Lurie gave a while ago, as recounted here.

It’s about their work on formulating the Baez-Dolan tangle hypothesis/extended TQFT hypothesis (which essentially says that every extended TQFT is already fixed by the “$n$-space of objects” which it assigns to the point) within $\infty \;n$-categories and then proving it (proof done for low $n$ and sketched for all $n$).

The key ingredients (as presented in the talk) are these:

1) We need this definition: an $\infty$ 0-category is a topological space or something Quillen equivalent to it (a Kan complex aka $\infty$-groupoid, for instance). An $\infty\; n$-category is recursively defined to be a category enriched over $\infty\; (n-1)$-categories (enriched in the ordinary strict sense!).

then the point is: for all $n \in N$ framed $n$-manifolds naturally form an $\infty\; n$-category $\mathrm{Bord}_n^{fr}$ whose $(k \leq n)$-morphisms are framed cobordisms and whose $(k \gt n)$-morphisms are diffeomorphisms and higher homotopies between these.

A framed cobordisms here is a $(d \leq n)$-dimensional cobordism $\Sigma$ equipped with a trivialization of the $n$-stabilized tangent bundle $T \Sigma \oplus (\Sigma \times\mathbb{R}^{n-d}) \stackrel{\simeq}{\to} \Sigma \times \mathbb{R}^n$.

2) The notion of very dualizable objects (Jacob Lurie’s idea): to me this is the notion of $\infty$-equivalence with “invertible up to” everywhere replaced by “adjoint up to”.

recursive Def.:

an $\infty\; 2$-category with adjoints is one in which each 1-morphism has a left and right adjoint.

an $\infty\; n$-category with adjoints is an $\infty \;n$-category such that all Hom-($\infty\; (n-1)$-categories) are $\infty (n-1)$-categories with adjoints.

For every $\infty \;n$-category $C$ write $C^f$ for the largest $\infty \;n$-category with adjoints sitting inside. The “f” is for finite since having lots of adjoints is imposing lots of conditions which usually say that things are finite dimensional (simplest example: a vector space which has a dual such that it is the dual of its dual is finite dimensional).

This finiteness condition here ultimately is supposed to relate to things such as the “rational” in rational CFT (which says that certain 2-vector spaces are finite dimensional, i.e. that certain monoidal categories have finitely many simple objects).

Now finally: an object $V \in C$ for $C$ symmetric monoidal is called very dualizable if it is dualizable in $C^f$. That means that unit and counit 1-morphisms of the duality have to have 2-sideed adjoints whose structure 2-morpshisms have 2-sided adjoints, and so on.

Write $C^{fd}$ for the “space” of all very dualizable objects in the symmetric monoidal $\infty \; n$-category $C$. This turns out to be always an $\infty\; 0$-category, i.e. “really a space”.

(I asked afterwards if he knew about Eugenia Cheng’s result that An $\omega$-category with all duals is an $\omega$-groupoid, but he didn’t.)

Then we have their

Theorem (Baez-Dolan hypothesis). The space (i.e. $\infty\; 0$-category) of $\infty\; n$-functors $\mathrm{Bord}_n^{fr} \to C$ is the space (i.e. $\infty\; 0$-category) of very dualizale objects in $C$: $hom(Bord_n^{fr}, C) \simeq C^{fd} \,.$

Mike Hopkins says they have a detailed proof for $n \leq 2$ and a sketch for arbitrary $n$. He pointed out that Bartels, Douglas and Henriques have a similar statement for strict $n$-categories going up to $n \leq 7$.

This is in particular equivalent to saying that (hope I get this right now): $\mathrm{Bord}_n^{fr}$ is the free $\infty \; n$-category on a single very dualizable object.

Mike Hopkins mentioned a curious direct corollary of this:

there is an action of $\mathrm{GL}_n(\mathbb{R})$ on $\mathrm{Bord}_n^{fr}$ which rotates the framing everywhere. By the above theorem this now means there is a $GL_n(\mathbb{R})$-action on the space of of very dualizable objects.

Then came something which I now see I still don’t fully understand. He writes $\mathbf{B} G$ for the 1-object groupoid version of the group $G$ (well, the boldface is the notation we are using here on the $n$-Café, so I am happy with that) then uses $SO(2) \simeq B \mathbb{Z}$ to say that there is a functor from $\mathbf{B} \mathbb{Z}$ to the space of very dualizable objects, which means that the above $GL_n(\mathbb{R})$-action in particular picks one automorphism of every very dualizable object. In fact, that can be built easily from using the structure maps of the duality on it.

Sorry, I’ll straighten that last part out and get back to you then.

Posted at May 20, 2008 5:33 PM UTC

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### Re: Hopkins-Lurie on Baez-Dolan

This is fantastic! I saw Hopkins give a talk a while a go, it was sort of a dumbed down version of the first distinguished lecture talk he gave at the fields institute (which is a available in streaming audio) I am gonna have to go over this a lot more. I think this stuff is really pretty amazing!
thanks for posting on it.
sean

Posted by: Sean Tilson on May 20, 2008 7:23 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

I think this stuff is really pretty amazing!

What I find amazing about it – but that already pertains to the original Baez-Dolan hypothesis – that this statement is so fundamental on the mathematical side.

One might have imagined that quantum field theory turns out to be an intrinsically intricate and complicated concept. But this theorem tells us, at least for TQFT, that it is one of the most “tautological” structures in math. As we discussed elsewhere, by some magic the free stable $\infty$-groupoid on a single dualizable object is just another incarnation of the sphere spectrum. It doesn’t get much more “deep” and “foundational” in math than the sphere spectrum, I suppose. If you know what I mean.

So the Baez-Dolan hypothesis is an amazing intersection of abstract nonsense with deep pure-math structure and with fundamental physics. Quite amazing.

Posted by: Urs Schreiber on May 20, 2008 10:45 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Can I give a bit of history on some aspects of this ETQFT story? About 12 years ago I put in a proposal to a research council which was refused. I had been following up and extending Dave Yetter’s ideas on defining 2D TQFTs from low dimensional models for 2-types. He had used categorical groups, I had made the link in a paper to the simplicialy enriched groupoid construction of Dywer-Kan (and also Joyal and Tierney). I had then given a construction of TQFTs from models of finite homotopy $n$-types, and an interpretation of the result in terms of $n$-gerbes-like objects. The proposal was to use Kan enriched categories (i.e. fibrant simplicially enriched categories = locally Kan categories = ? ($\infty$, 1)-categories (my numbering here may be wrong!) and also $A_\infty$-categories and to link the constuction with extended TQFTs using various coefficients, including various $A_\infty$-algebras, $L_\infty$-algebras etc. The idea was to test the construction to destruction’ and to use the ‘test results’ to try to find a classification theorem for TQFTs.

Almost needless to say I did not get the grant. Some referees liked it but one said it was rubbish and in no uncertain words said that even the idea of investigating TQFTs in this way using simplicially enriched techniques was ridiculous. It leaves me a bit bitter to see that I was right and (s)he was completely wrong.

There was a problem with the construction and that was that it was limited to finite discrete ‘coefficients’ or at least finite dimensional ones.

Given the relationships that have been revealed more recently, it is interesting to speculate what would have happened if I had got the grant and had a good postdoc working on that area. (In the event Bangor University closed down the Mathematics Department, and peer evaluation of the research in the department was not positive!)

On a positive note, my paper with Vladimir Turaev on his HQFTs with background a homotopy 2-type is to be published by JHRS very soon. I must admit that I do not quite see how this HQFT stuff relates to the Hopkins-Lurie ideas. Can anyone help on that?

Posted by: Tim Porter on May 21, 2008 8:01 AM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Sorry to hear that about the grant and the referee. Next time with a similarly visionary proposal maybe apply to FQXi which is designed to help in cases like that (at least that’s the idea).

I must admit that I do not quite see how this HQFT stuff relates to the Hopkins-Lurie ideas. Can anyone help on that?

HQFT is the equivariant case.

Baez-Dolan/Hopkins-Lurie consider bare topological QFT. No extra structure on the cobordisms (well, except for the framing, actually, so its “framed topological QFT”).

All other kinds of QFTs are supposed to be obtained from that picture by equipping the cobordisms with more structure. Such as spin structure, conformal structure, Riemannian structure, pseudo-Riemannian structure, String structure, Fivebrane structure, you name it.

Equipping the cobordisms with homotopy classes of maps into some pointed topological space as in HQFT is such a kind of extra structure. If that space is a $B G$, this should be addressed as “equivariant TQFT”, I think.

So: HQFT goes beyond the Baez-Dolan hypothesis. Somebody should try to conjecture the generalization of that hypothesis to the HQFT case.

Obvious guess:

Conjecture. Extended HQFTs over $B G$ with codomain $C$ are equivalent to very dualizable objects with $G$-action in $C$.

Posted by: Urs Schreiber on May 21, 2008 9:46 AM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Thanks, Urs.

As you know, the new’ HQFT paper looks at the case of BM for a crossed module M and thus the equivariance is more complex and related to the characteristic maps corresponding to some sort of M-bundles or similar. (The paper has been on the archive for years, but it was lost by the first journal we sent it to and sat hidden in their server without generating the necessary replies to the authors etc.)

One of my interesting examples’/ thought experiments’ from way back was the idea of lifting one structural map (from $G$-bundles say to $H$-bundles along a map $H\to G$). This sort of thing was simplicially important in the search for the relationships between triangulations and smoothing. $BTop$, $BDiff$ and $BPL$ kept on coming up in that work, and the homotopy fibres of the maps between them held the key to the problem of classification. Here $Top$, $Diff$ etc. were simplicial monoids, and some of the themes from back then (early 1970s probably) may be relevant now.

Of course, this same idea is behind the exact sequences in (non-Abelian) cohomology. I think the HQFTs may be too closely related to non-Abelian cohomology sets’, and that is why the extended TQFTs are more important. All your stuff on non-Abelian cohomology may be useful here. Perhaps one should formulate a general Baez-Dolan conjecture about lifting stuctured ETQFTs and relating them by exact sequences of triangles (as in triangulated category theory).

Posted by: Tim Porter on May 21, 2008 10:29 AM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Only a snatched moment in the middle of a workshop. So how does your extended conjecture relate to the Generalized Tangle Hypothesis + its extension to spin and complex structures?

Posted by: David Corfield on May 21, 2008 1:27 PM | Permalink | Reply to this

Perhaps I should explain my intuition in a simple case (basically because I do not understand (i.e. feel), the Thom space theory needed for the generalised tangle conjecture). Turaev’s HQFTs have objects $X$-manifolds that is d-manifolds, $M$, together with a `characteristic map’ $g :M\to X$ for a fixed space $X$. $X$-Cobordisms are similar, = cobordism plus characteristic map to $X$, and their characteristic maps must agree on the two ends (domain and codomain). However one only is interested in homotopy classes of $X$-cobordisms, relative to the two ends.

One gets a symmetric monoidal category. If $X$ is a $BG$ then this is a $G$-equivariant form of a TQFT. (It would be fun to try to define an EHQFT, to absorb the homotopy in a better way. Has anyone done this, do you know?)

What I was looking at was when all the manifolds came with two compatible structures, a map to $BG$ and to $BH$ where there is an epimorphism, $H\to G$. This occurs with Spin structures etc. I was trying to get a relative form of the theory, really by mapping into the homotopy fibre of $BH\to BG$. It worked well as a means of constructing TQFTs in the finite group case. It counted the lifts of a given $G$-structure to an $H$-structure. I have not looked at the idea in detail in the general case nor for HQFTs, but it is significant that Breen’s 1992 Bitorsor paper looks at the Puppe sequence of a fibration of simplicial groups as a way to get longish exact sequences in non-Abelian cohomology, and given the conceptual link between that theory and the HQFTs there is perhaps a relative form of HQFT as I believe.

That is not very exact as a description, but as I said the motivation did come from classifying lifts of structures on manifolds. Is there a relative Thom construction or a way of looking at the Thom construction as a homotopy fibre or similar object? I have no idea.

Posted by: Tim Porter on May 21, 2008 4:07 PM | Permalink | Reply to this

### Re:

It would be fun to try to define an EHQFT, to absorb the homotopy in a better way. Has anyone done this, do you know?)

Just heard the talk by Dan Freed following up on Mike Hopkins’ talk yesterday.

He started by “recalling” the theorem Mike Hopkins gave yesterday but actually stated it in more generality, which seems to be just the E-HQFT picture for the case $X = B G$:

namely, they talk about the $\infty \;n$-category of manifolds “with $G$ structure” $Bord_n^G$ which is (as far as I understood and he seemed to have confirmed that when I talked to him after the talk) just $Bord_n$ but with everything in sight equipped with homotopy classes of maps to $B G$.

So in particular $Bord_n^{SO(n)}$ is oriented manifolds, that’s the case he explicitly mentioned.

The Baez-Dolan-hypothesis theorem is supposed to say in this case:

Thm. For $C$ some $\infty \; n$-category the space of monoidal $\infty \; n$-functors $Bord^G_n \to C$ is that of very dualizable objects in $C^{fd}$ which are “homotopy fixed points for the $G$-action on $C^{fd}$.”

I suppose that just means that there is a weak action of $G$ on these objects.

I’ll try to confirm this.

Posted by: Urs Schreiber on May 21, 2008 6:50 PM | Permalink | Reply to this

### Re:

This is very interesting and it sure looks like things are moving fast. It is great that Urs is there to keep us informed and to represent the “Baez-Dolan” side of things. I am sorry I have not been able to keep up the pace. Recently Jamie Vicary posted some related thoughts and questions about this stuff over here.

Posted by: Bruce Bartlett on May 21, 2008 7:17 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

So how does your extended conjecture relate to the Generalized Tangle Hypothesis + its extension to spin and complex structures?

I wish John would chime in and help me a bit here.

I am having trouble incorporating the Thom spaces appearing in the generalized tangle hypothesis into the part of the picture that I understood so far.

The followup talk by Dan Freed today seems to have confirmed my little conjecture (but be careful, I might be mixed up about things here).

If that’s all correct so far, it would seem to say that maps from the fundamental $(n+k)$-category of that space $(M G,Z)$ to some $C$ pick out the “very dualizable” objects with $G$-action in $C$.

Does that sound anywhere close to plausible? I have currently no access to $(M G , Z)$. :-)

Posted by: Urs Schreiber on May 21, 2008 6:59 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Urs wrote:

I wish John would chime in and help me a bit here.

I’m at Berkeley busy talking to Alan Weinstein and Jenny Harrison — but more importantly, the computer here seems not to have Java enabled, and is thus unable to post comments to the blog. Actually, let me test that hypothesis once more…

Posted by: John Baez on May 21, 2008 7:35 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Hmm, so it actually works! Weird — I thought yesterday it didn’t.

Okay…

they talk about the $(\infty,n)$-category of manifolds “with G structure” $Bord_n^G$ which is (as far as I understood and he seemed to have confirmed that when I talked to him after the talk) just $Bord_n$ but with everything in sight equipped with homotopy classes of maps to $B G$.

That sounds roughly right… let me start by reminding you of the usual story about cobordism theories and Thom spaces, which is lurking behind all this stuff.

Homotopy theorists usually study cobordisms where all manifolds and cobordisms between them have their ‘stable normal bundle’ equipped with the structure of a $G$-bundle. To intuit the concept of ‘stable normal bundle’, first imagine your $n$-manifolds being embedded in $\mathbb{R}^{n+k}$, and cobordisms embedded in $\mathbb{R}^{n+k} \times [0,1]$. Then your manifolds and cobordisms will have a ‘normal bundle’ in the usual sense: the orthogonal complement of the tangent bundle. This normal bundle will be a vector bundle with $k$-dimensional fibers.

Then, take the limit $k \to \infty$. You get the notion of ‘stable normal bundle’, which is an infinite-dimensional vector bundle.

Note that before we take the limit $k \to \infty$, we are really considering higher-dimensional tangles. In the limit $k \to \infty$, things ‘stabilize’ and we are considering cobordisms. Homotopy theorists often focus on the stable situation, but the generalized tangle hypothesis discusses the case of finite $k$ as well.

Homotopy theorists like to equip the stable normal bundle of their manifolds or cobordisms between these with the structure of a $G$-bundle. Here $G$ is some group that’s equipped with a homomorphism to $GL(\infty)$ —or, with no loss of generality, $O(\infty)$ (which is homotopy equivalent).

Let’s call a manifold a ‘$G$-manifold’ if its stable normal bundle is equipped with the structure of a $G$-bundle, and similarly for ‘$G$-cobordism’.

Let’s say two $n$-dimensional $G$-manifolds are ‘equivalent’ if there’s a $G$-cobordism between them. Equivalence classes of $n$-dimensional $G$-manifolds form an abelian group under disjoint union.

The big theorem in cobordism theory is that this group is the $n$th homotopy group of a certain space, the ‘Thom space’ $M G$. This is an infinite loop space — that is, a space of loops in a space of loops in a space…

But, this big theorem is just a pale shadow of a bigger idea: the generalized cobordism hypothesis. This says, first, that we can make a stable $\infty$-groupoid

$\infty Cob^G$

whose objects are $0$-dimensional $G$-manifolds, whose morphisms are $G$-cobordisms between these, whose 2-morphisms are $G$-cobordisms between these, and so on.

Note: $\infty Cob^G$ should a stable $\infty$-category with duals, but now I believe that’s the same thing as a stable $\infty$-groupoid. And that should be the same thing as an infinite loop space.

And, the generalized cobordism hypothesis says this infinite loop space is just the Thom space $M G$!

The generalized cobordism hypothesis is itself a pale shadow of an even bigger idea: the generalized tangle hypothesis. This bigger idea is actually simpler in some ways, since it doesn’t involve taking the $k \to \infty$ limit. It deals with tangles whose normal bundle is equipped with the structure of a $G$-bundle where $G$ is equipped with a homomorphism to $O(k)$.

I explained the generalized tangle hypothesis here. If anyone forgets what a ‘Thom space’ is, I also defined that there.

There’s a lot more to say, but the main danger in this business is saying so much that people think you must be talking about something complicated and scary. Pictures would help a lot…

Posted by: John Baez on May 21, 2008 8:25 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

There should also be a description of these Thom spaces that sounds like ‘the free blah-di-blah on a blah-di-blah’…

Posted by: John Baez on May 21, 2008 9:49 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

In the other thread, I was relating the stable normal bundles to immersions. The immersion picture pushes out the trivial pieces as far as possible. Since the 1st stable stem is of order 2, it might be good to see why. Consider the standard embedded circle in the plane. This clearly bounds a disk. It is cobordantly trivial. The figure 8 map of the circle is of order 2 in the oriented cobordism group. The Figure 8 map corresponds to the stably framed Lie framed circle. In order to see this draw (x,y) coordinates at every point of the immersion and the embedding. Now compute the winding number of the tangent vector with a unit vertical. On the embedded circle there is a winding number 1, on the immersed fig 8 it is 0.

The Lie framed 2-torus is represented by a twisted
figure 8 times circle. Banchoff’s Klein bottle gives an idea how this works, but that has a 1/2-twist in figure-8 factor, and the Torus (obviously) has to have a full twist.

So if you want to study stable homotopy of a Thom space, you can think of the underlying vector bundle of the Thom space as the classifying space of the normal bundle of an immersion. Then gradually lift the immersion to an embedding by adding trivial factors, or by taking Cartesean products with real space. A host of invariants are obtained by looking at the self-intersection sets. This was worked out in detail in the 1970-80 by Koschorke, Eccles, Koschorke and Sanderson, and Kaiser, for example.

Posted by: Scott Carter on May 22, 2008 1:01 AM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Urs: in your post it looks like you wrote $SU(2)\simeq B\mathbb{Z}$, but in fact $U(1) \simeq B\mathbb{Z}$.

I say ‘it looks like you wrote’, because this old computer is displaying math symbols in a really weird way.

Feel free to delete this comment if it’s wrong, or correct your post and delete this comment if it’s right.

Posted by: John Baez on May 21, 2008 8:32 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Oops. The $U$ there was supposed to be an $O$.

The point was that in $GL(n,\mathbb{R})$ we have in particular sitting $SO(2) \simeq U(1)$. But I got confused about the supposed consequences of this statement anyway. Still am.

Posted by: Urs Schreiber on May 22, 2008 9:41 AM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

This inductive definition of (∞,n)-category with adjoints doesn’t look right. If an (∞,2)-category just has adjoints for 1-morphisms, then this induction means that an (∞,3)-category will just have adjoints for 2-morphisms, and an (∞,n)-category will just have adjoints for (n-1)-morphisms. Am I missing something?

Posted by: Eugenia Cheng on May 22, 2008 9:15 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Eugenia,

thanks for asking. Let me check this again. I’ll try to get back to you on this tomorrow.

Posted by: Urs Schreiber on May 26, 2008 6:12 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

What is the answer to Eugenia’s question? I am shamefully bad with $\infty$-categories, but I have to say that those definitions look too simple:

an $\infty$2-category with adjoints is one in which each 1-morphism has a left and right adjoint.

an $\infty$n-category with adjoints is an $\infty$n-category such that all Hom-($\infty$(n−1)-categories) are $\infty$(n−1)-categories with adjoints.

Posted by: Bruce Bartlett on June 4, 2008 8:13 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

I haven’t forgotten about this. But I didn’t get a chance to discuss this with Mike Hopkins before I left for California, and there I still am. Will fly back tomorrow afternoon.

Generally, I am very much lagging behind with reporting on HIM activity. I promise to try my best to forward the questions raised and to keep you all informed. Promised. But it may take me still a few days to do so.

Posted by: Urs Schreiber on June 4, 2008 9:05 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

OK, Urs and I just looked at Lurie’s notes and his definition is a bit more subtle than described above. Essentially, he asks for adjoints for 1-morphisms separately in the induction process, and he defines these by quotienting the $(\infty,n)$-category down to an ordinary 2-category (by taking isomorphism classes of 2-morphisms) and looking for adjoints in there in the usual 2-categorical sense.

Adjoints for morphisms in dimensions above 1 are produced by induction from the hom-$(n-1)$-categories as above.

This does produce something resembling adjoints for all the correct definitions, but at first sight they look very non-algebraic and…does anyone see any coherence conditions?

[It is Definition 2.3.13 in the version of Lurie’s paper I just got from his website.]

Posted by: Eugenia Cheng on January 19, 2010 3:33 PM | Permalink | Reply to this

### The SO(2)-action

I now understood that statement about the $SO(2)$-action and how it defines automorphisms of objects. It’s actually very simple:

as I said, there is an obvious action of $GL(n,\mathbb{R})$ on the $\infty \; n$-category of framed $n$-manifolds, where the group simply globally acts on the framing that everything is equipped with.

So in particular, if $n \geq 2$, there is an $SO(2)$-action.

Using that $SO(2)$-action, we can build a framed 1-dimensional manifold giving a cobordism from the point with your preferred framing on it to itself, with the same framing, but such that the framing rotates “once around” along the cobordism.

The image of that particular framed cobordism singles out an automorphism of the fully dualizable object that defines the TFT. That’s essentially all there is to it.

I could say much more. If only there were time…

Posted by: Urs Schreiber on May 26, 2008 6:19 PM | Permalink | Reply to this
Read the post HIM Trimester on Geometry and Physics, Week 4
Weblog: The n-Category Café
Excerpt: Talk in Stanford on nonabelian differential cohomology.
Tracked: May 29, 2008 11:25 PM
Read the post Teleman on Topological Construction of Chern-Simons Theory
Weblog: The n-Category Café
Excerpt: A talk by Constant Teleman on extended Chern-Simons QFT and what to assign to the point.
Tracked: June 17, 2008 6:56 PM
Read the post Schommer-Pries on Classification of 2-Dimensional Extended TFT
Weblog: The n-Category Café
Excerpt: Chris Schommer-Pries classifies 2-functors from a 2-category of 2-dimensional cobordisms.
Tracked: June 18, 2008 7:09 PM

### Re: Hopkins-Lurie on Baez-Dolan

Jacob Lurie has now made available a pdf

On the Classification of Topological Field Theories

Abstract: Our goal in this article is to give an expository account of some recent work on the classification of topological field theories. More specifically, we will outline the proof of a version of the cobordism hypothesis conjectured by Baez and Dolan.

Posted by: Urs Schreiber on January 16, 2009 12:09 AM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Impressive!

Like the cobordism hypothesis, the tangle hypothesis can be generalized in many ways. For example, one can consider embedded submanifolds with tangential structure more complicated than that of a $k$-framing and embedded submanifolds with singularities. In these cases, one can still use the argument sketched above to establish a universal property of the relevant $(\infty, n)$-category.

Posted by: David Corfield on January 16, 2009 9:06 AM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Yeps, looks important!

Posted by: Bruce Bartlett on January 16, 2009 9:22 AM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

It’s time to break out the champagne!

Posted by: John Baez on January 16, 2009 5:06 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Darn it, I am going to have to put aside my other stuff and try to read Lurie’s paper now. Maybe it’s all over and we can all go home! If he has indeed cracked it, then what are the key ideas he has brought to the table that we’ve been missing for ages (perhaps I only speak for my ignorant self here)? Is it the fact that Lurie and others have indeed now developed a managable homotopical definition of $n$-category, which lends itself to actually be able to do stuff with (like proving the cobordism hypothesis)? I was too sick, lame and lazy to learn the formalism when I should have.

Posted by: Bruce Bartlett on January 17, 2009 2:53 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Some thoughts:

if one goes to the bare bones of it, an “extended cobordism” is a multispan in something like $Top^op$ or $Diff^{op}$.

For instance an ordinary cobordism $\Sigma$ from $\Sigma_{in}$ to $\Sigma_{out}$ is a cospan

$\Sigma_{in} \rightarrow \Sigma \leftarrow \Sigma_{out}$

(where it is useful to think of all of this as being at the same dimension, i.e. thinking of $\Sigma_{in/out}$ as collared boundaries).

An extended cobordisms is a multispan in that $\Sigma_{in}$ and $\Sigma_{out}$ may themselves have inputs and outputs, e.g.

$\array{ \Sigma_{in,in} &&&& \Sigma_{out,in} \\ \downarrow &&&& \downarrow \\ \Sigma_{in} &\to& \Sigma &\leftarrow& \Sigma_{out} \\ \uparrow &&&& \uparrow \\ \Sigma_{in,out} &&&& \Sigma_{out,out} }$

In following the lore of groupoidification, we want to send this to a multispan in some coefficient category. For a $\sigma$-model QFT for instance coming from some target spac $X$ with an $n$-bundle $P \to X$ over it, the $\sigma$-model QFT would be just homming into $P$

$QFT := [-, P]$

and would send the above multi-co-span to a multispan

$\array{ [\Sigma_{in,in},P] &&&& [\Sigma_{out,in},P] \\ \uparrow &&&& \uparrow \\ [\Sigma_{in},P] &\leftarrow& [\Sigma,P] &\rightarrow& [\Sigma_{out},P] \\ \downarrow &&&& \downarrow \\ [\Sigma_{in,out},P] &&&& [\Sigma_{out,out},P] }$

This can be interpreted as computing the transgression of the background field to the mapping spaces

$\array{ [\Sigma_{in,in},X] &&&& [\Sigma_{out,in},X] \\ \uparrow &&&& \uparrow \\ [\Sigma_{in},X] &\leftarrow& [\Sigma,X] &\rightarrow& [\Sigma_{out},X] \\ \downarrow &&&& \downarrow \\ [\Sigma_{in,out},X] &&&& [\Sigma_{out,out},X] }$

and then computing the sections of these transgressed fields, which are spans into these total spaces, for instance a section of the transgression of $P$ to $[\Sigma,X]$ gives a span

$\array{ && \Psi \\ & \swarrow && \searrow \\ \Omega_{pt}F &&&& [\Sigma,P] } \,.$

So from this perspective it seems useful to think of extended QFT very abstractly as the application of a continuous functor

$S^{op} \to V$

to multispans in some $S^{op}$ (multi-co-spans in $S$, as above), thought of as extended cobordisms, to multispans in $V$, thought of as genralized higher linear maps.

$\sigma$-models would be the particular special cases where the continuous functor is some kind of internal hom, mapping multi-cospans representing worldvolumes into multi-spans, representing target space (with its branes (one-sided spans) and bi-branes/correspondence spaces/dualities etc.).

If from that perspective I were to think of the Baez-Dolan hypothesis I would maybe think:

well, let’s see if all extended topological cobordisms are somehow nicely generated as multi-co-spans from something simple.

So let’s maybe start with the interval object in $Top$, $I = [0,1]$, and the canonical cospan it comes with

$\array{ && I = [0,1] \\ & {}^{in}\nearrow && \nwarrow^{out} \\ pt &&&& pt } \,.$

This describes one point coming in, running through an interval, and coming out at the other end

$pt \to pt \,.$

Alternatively, since we have an initial object $\emptyset$ in $Top$, we can easily bend the interval around, consider the co-span

$\array{ && I = [0,1] \\ & \nearrow && \nwarrow^{in \sqcup out} \\ \emptyset &&&& pt \sqcup pt }$

which represents something like

$\array{ & \leftarrow pt \\ \swarrow \\ \searrow \\ & \to pt }$

We have a canonical symmetric monoidal structure on $Top$. So let’s maybe consider the category $S^{op}$ which is generated from the above interval cospan under composition of co-spans and tensor product.

For instance if we take the tensor product of the interval co-span with itself

$\left( \array{ && I = [0,1] \\ & {}^{in}\nearrow && \nwarrow^{out} \\ pt &&&& pt } \right) \otimes \left( \array{ && I = [0,1] \\ & {}^{in}\nearrow && \nwarrow^{out} \\ pt &&&& pt } \right)$

we get a multi-co-span

$\array{ pt &\to& I &\leftarrow& pt \\ \downarrow && \downarrow && \downarrow \\ I &\to& I^2 &\leftarrow& I \\ \uparrow && \uparrow && \uparrow \\ pt &\to& I &\leftarrow& pt }$

which describes the disk, regarded as an extended cobordism with four boundary components of codim 1 and four of codim 2.

Since we are closing up under pushouts, we can take two of such cospans and glue them at their common boundary (or just one and glue it along its own boundary by just forming the pushout over the entire co-span itself) to obtain the sphere.

Both results we can again tensor with the interval cospan, to get the 3-disk and the 2-cylinder. Which in turn may be glued to produce the 3-sphere and the 2-torus.

Etc.

It seems that the total collection of multi-cospans which one obtains thus by closing the interval cospan under pushouts and tensor products is pretty much a model for extended cobordisms of arbitrary dimension. (?)

Now, a continuous and monoidal $S^{op} \to V$ will preserve all these limits in $S^{op}$ and tensor products. This directly implies that it will already be fixed on all these multi-co-spans by the value it takes on the interval cospan.

Now suppose further, as is true in $Top$, that we furthermore have a notion of weak equivalences in $S$, that the interval itself is weakly equivalent to the point and that our functor respects weak equivalences. Then what that functor does to all our multispans should already be fixed by what it does to the point.

Just a thought.

Posted by: Urs Schreiber on January 19, 2009 1:29 PM | Permalink | Reply to this

### (co)-traces

Here is further thought. I’ll expand on my remarks above on the multi-(co)-span description of cobordisms and QFTs above in that

a) I talk about the gluing of common boundaries of a single cobordism which I mentioned above in more detail;

b) describe how it maps to the groupoidified trace operation (it’s obvious, this must have been discussed explicitly somewhere?);

and

c) prove the Willerton-Bartlett crossing-with-the-circle equation in this context, which asserts that an extended TQFT $Z$ assigns to a cobordism of the form $\Sigma \times S^1$ the endomorphisms on the identity on what it assigns to $\Sigma$ itself:

$Z(\Sigma \times S^1) \simeq End_{Id_{Z(\Sigma)}} \,.$

Points a) and b) by way of concrete example:

Let $\array{ && I \\ & {}^{in}\nearrow && \nwarrow^{out} \\ pt &&&& pt }$

be the standard co-span version of the interval $I = [0,1]$. To be thought of as a “straight” interval.

The “co-trace” over this is obtained by

- first dualizing on the output to arrive at the co-span

$\array{ && I \\ & {}^{in \sqcup out}\nearrow && \nwarrow \\ pt \sqcup pt &&&& \emptyset } \,,$

to be thought of as a curved/bended interval, and then composing on the left with the result of taking the identity

$\array{ && pt \\ & {}^{Id}\nearrow && \nwarrow^{Id} \\ pt &&&& pt }$

and similarly dualizing the input to an output to arrive at

$\array{ && pt \\ & {}^{}\nearrow && \nwarrow^{Id \sqcup Id} \\ \emptyset &&&& pt \sqcup pt }$

The compsite span is the pushout

$\array{ &&&& S^1 \\ &&& \nearrow && \nwarrow \\ && pt &&&& I \\ & {}^{}\nearrow && \nwarrow^{Id \sqcup Id} && {}^{in \sqcup out}\nearrow && \nwarrow \\ \emptyset &&&& pt \sqcup pt &&&& \emptyset }$ namely the circle. The co-trace over the interval cobordism.

Dually, if we have a span

$\array{ && P \\ & {}^{x}\swarrow && \searrow^{y} \\ X &&&& X }$

describing a “geometric function” or “groupoidified linear map” or whatever we’ll eventually agree on calling it, in that for instance $X$ is a finite set with $n$ elements and $P \to X \times X$ is a bundle of sets, i.e. a set-valued $n \times n$ matrix, thought of as a categorified $\mathbb{N}$-valued $n\times n$-matrix, then the co-version of the above procedure produces the trace of $P$

$\array{ &&&& tr(P) \\ &&& \swarrow && \searrow \\ && X &&&& P \\ & {}^{}\swarrow && \searrow^{Id \times Id} && {}^{x \times y}\swarrow && \searrow \\ pt &&&& X \times X &&&& pt }$

in that $tr(P)$ is, in the simple example mentioned, the set $tr(P) = \sqcup_{x \in X} P_{x,x}$.

So, now consider a $\sigma$-model QFT given by a background field which is given by an $n$-bundle $P \to X$ over some target space $X$. By general argument mentioned before, this is the multi-span map given by homming termwise into $P$

$QFT_P := [-, P]$

where the resulting spans, for instance

$QFT_P\left( \array{ && \Sigma \\ & {}^{in}\nearrow && \nwarrow^{out} \\ \Sigma_{in} &&&& \Sigma_{out} } \right) = \array{ && [\Sigma, P] \\ & {}^{in^*}\swarrow && \searrow^{out^*} \\ [\Sigma_{in},P] &&&& [\Sigma_{out},P] }$

is regarded as a groupoidfied linear map which acts on generalized sections $\Psi \in \Gamma([\Sigma_{in}, P])$ which are themselves spans

$\array{ && \Psi \\ & \swarrow && \searrow \\ \Omega_{pt}F &&&& [\Sigma_{in},P] } \,,$

(where $\Omega_{pt}F$ is the “ground monoid”, i.e. the endomorphism monoid of the point in the fiber $F$ of $P$ – but that’s not relevant for the argument right now)

by composition of spans.

So we can check what our $QFT_P$ does to cobordiusms of the form $\Sigma \times S^1$ by evaluating this prescription for that case.

For ease of notation assume $\Sigma$ has no in- and output. So that $\Sigma \times I$ is given by a co-span

$\array{ && \Sigma \times I \\ & {}^{in}\nearrow && \nwarrow^{out} \\ pt &&&& pt }$

So the point then is that we know what $QFT_P$ does to this co-span and, since it respects composition in multi-spans (from co-multispans to multispans) we know that to $\Sigma \times S^1$ the $QFT_P$ assigns the trace

$\array{ &&&& tr[\Sigma \times I, P] \\ &&& \swarrow && \searrow \\ && [\Sigma, P ] &&&& [\Sigma \times I, P] \\ & {}^{}\swarrow && \searrow^{Id \times Id} && {}^{d_0 \times d_1}\swarrow && \searrow \\ pt &&&& [\Sigma,P] \times [\Sigma, P] &&&& pt }\,.$

Using the hom-adjunction this is the same as the pullback

$\array{ &&&& tr[\Sigma \times I, P] \\ &&& \swarrow && \searrow \\ && [\Sigma, P ] &&&& [I,[\Sigma , P]] \\ & {}^{}\swarrow && \searrow^{Id \times Id} && {}^{d_0 \times d_1}\swarrow && \searrow \\ pt &&&& [\Sigma,P] \times [\Sigma, P] &&&& pt }\,.$

Now, maps into $[I,[\Sigma,P]]$ are homotopies (morphisms) between two maps into $[\Sigma,P]$. The pullback diagram says that maps into $tr[\Sigma \times I, P]$ are homotopies (morphisms) between one maps into $[\Sigma,P]$ and itself.

This in turn says that regarded as the corresponding groupoidied space of sections, the value $QFT_P(\Sigma \times I)$ is the collection of spans

$\array{ && \Psi \\ & \swarrow && \searrow \\ \Omega_{pt}F &&&& tr([\Sigma \times I,P]) } \,.$

By the universal pullback property of $tr([\Sigma \times I,P])$ just mentioned every such span is indeed an endomorphism of a section of $[\Sigma,P]$.

To get the full Willerton-Bartlett statement play with the hom-adjunciton a bit more…

Posted by: Urs Schreiber on January 22, 2009 3:58 PM | Permalink | Reply to this

### Re: (co)-traces

After checking with John, I made the above definition of span trace and co-span co-trace into $n$Lab entries:

$n$Lab: span trace

$n$Lab: co-span co-trace

Posted by: Urs Schreiber on January 24, 2009 3:24 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

A further remark on span trace:

it would be good to see how the “span trace” for spans of categories relates to the notion of trace on an endofunctor described by Ganter-Kapranov # and by Bartlett-Willerton #.

So let $C$ be a category and $F : C \to C$ an endofunctor. Bartlett-Ganter-Kapranov-Willerton (alphabetical! I am trying to attribute this correctly this time…) say that the categorical trace of $F$ is the collection of transformations $Id_C \Rightarrow F$.

On the other hand, $F$ defines the span

$\array{ && C \\ & {}^{Id}\swarrow && \searrow^{F} \\ C &&&& C }$

and by the general nonsense of span trace the trace of this span is the pullback over

$\array{ C &&&& C \\ & {}_{Id \times Id} \searrow && \swarrow_{Id \times F} \\ && C \times C } \,.$

One should take the suitable pseudo or lax pullback. What is it? (I am asking the same question here).

I am not fully sure yet, but let me approximate this question a bit.

Since I want to compute something like a homotopy limit, I should maybe replace my diagonal $C \stackrel{Id \times Id}{\to}C \times C$ by its factorization

$C^I \stackrel{d_0 \times d_1}{\to}C \times C$

for $I = \{a \to b\}$ the interval category. (Thinking of this as coming from homming out of the co-span co-trace this corresponds to making duality on objects not an operation at a point but smeared over an interval).

So let me for the moment consider the ordinary (1-categorical) limit over

$\array{ C^I &&&& C \\ & {}_{d_0 \times d_1} \searrow && \swarrow_{Id \times F} \\ && C \times C } \,.$

That limit, say $R$ for need of a symbol, should be the full subcategory of the arrow category $C^I$ on objects (morphisms in $C$) of the form $a \to F(a) \,,$ for $a \in Obj(C)$. I.e. the “lax fixed points of $F$”.

So how does that relate to Bartlett-Ganter-Kapranov-Willerton? Well, this $R$ is the object which represents their categorical trace.

Posted by: Urs Schreiber on January 28, 2009 9:47 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

I have now included a discussion of loop space objects $\Lambda B$ and how they arise as the homotopy trace on the identity span on $B$ at span trace.

This should be the homotopy-theoretic analog of the definition of loop space object in David Ben-Zvi’s work.

More later, have to get some food now…

Posted by: Urs Schreiber on January 29, 2009 7:48 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Jacques Distler blogs about (what is going to be) a lecture series by Jacob Lurie on this stuff here.

Posted by: Urs Schreiber on January 21, 2009 10:38 AM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Just for the record, I’ll collect the links here:

Distler on Lurie on Baez-Dolan, part I: fundamental $\infty$-groupoids.

Posted by: Urs Schreiber on January 23, 2009 6:07 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

(Feel free to delete this.)

Posted by: John Baez on January 23, 2009 6:53 PM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

Yet more discoveries about $(\infty, n)$-categories, this time from Charles Rezk giving A cartesian presentation of weak $n$-categories.

Rezk mentions a research statement of Clark Barwick, but the link ought to be to here. I’m generally quite partial to a research statement, but sometimes, as here, they show you how much you will probably never understand, e.g., the Quantum Geometric Langlands correspondences.

Posted by: David Corfield on January 26, 2009 11:31 AM | Permalink | Reply to this

### Re: Hopkins-Lurie on Baez-Dolan

The abstract of Ulrike Tillman’s talk at the upcoming workshop Categorification and Geometrisation from Representation Theory in Glasgow, to be found here ,sounds a bit as if she gives a strict $\infty$-category of cobordisms:

Cobordism categories are at the foundations of topological quantum field theory. We will discuss how to define a strict higher dimensional version of the cobordism category, associate a topological space to them, and explain how these spaces relate to classical spaces in cobordism theory as studied by Thom and others in the middle of the last century.

Does anyone know more?

Posted by: Urs Schreiber on January 30, 2009 1:54 PM | Permalink | Reply to this
Read the post Symmetric Monoidal Bicategories and (n×k)-categories
Weblog: The n-Category Café
Excerpt: Monoidal bicategories can be constructed easily from "fibrant" monoidal double categories, and likewise for higher n-categories.
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