## May 10, 2010

### Quinn on Higher-Dimensional Algebra

#### Posted by David Corfield

Frank Quinn kindly wrote to me to point out an essay he is working on – The Nature of Contemporary Core Mathematics (version 0.92). Quinn will be known to many readers here as a mathematician who has worked in low-dimensional topology, and as one of the authors, with Arthur Jaffe, of “Theoretical mathematics”: Toward a cultural synthesis of mathematics and theoretical physics.

I crop up in the tenth section of the article, which is devoted to a discussion of “a few other accounts of mathematics”, including those of Barry Mazur, Jonathan Borwein, Keith Devlin, Michael Stöltzner, and William Thurston.

One objective is to try to understand why such accounts are so diverse and mostly – it seems to me – irrelevant when they all ostensibly concern the same thing. The mainstream philosophy of mathematics literature seems particularly irrelevant, and the reasons shallow and uninteresting, so only two are considered here. Essays by people with significant mathematical background often have useful insights, and when they seem off-base to me the reasons are revealing. The essay by Mazur is not off-base. (p. 53)

I take it that “irrelevant” is being taken relative to Quinn’s interest in characterising ‘Core Mathematics’.

Section 10.4 is where my work is given a going over. I’ll postpone a discussion of other sections to a later date, but wanted to know what people thought about subsection 10.4.3 (pp. 61-63), which treats the tenth chapter of my book on higher-dimensional algebra. One of the main thrusts is that I have been lured by John into believing that higher-dimensional algebra is more important and powerful than it really is. Evidence is given as to why $n$-categories are unlikely to help resolving issues concerning low-dimensional manifolds. For example,

Topological field theories on 2-manifolds can be characterized in terms of Frobenius algebras. The modular ones (roughly the ones coming form 2-categories) correspond to semisimple Frobenius algebras. Semisimple algebras are ‘measure zero’ in unrestricted algebras and have much simpler structure. this indicates that requiring higher-order decomposition properties corresponding to higher categories enormously constricts the field theories. To get more power we apparently need to reduce the categorical order rather than increase it.

Given we’ve started to get a little more self-reflective at the Café about what (higher) category theory means for us, I’d be interested to hear views on this subsection. No doubt the younger me who wrote that chapter around 8 years ago believed that ‘quantum topology’ would readily extend to knotted spheres in 4-space the account that saw tangles in 3-space as a free braided monoidal category, and invariants cropping up through functors to categories of representations of the same kind. Just devise some candidate braided monoidal 2-categories, and all would be fine.

Posted at May 10, 2010 1:46 PM UTC

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### Re: Quinn on Higher-Dimensional Algebra

David wrote:

No doubt the younger me who wrote that chapter around 8 years ago believed that ‘quantum topology’ would readily extend to knotted spheres in 4-space the account that saw tangles in 3-space as a free braided monoidal category, and invariants cropping up through functors to categories of representations of the same kind. Just devise some candidate braided monoidal 2-categories, and all would be fine.

Well, by now that’s well underway. It didn’t happen ‘readily’: it took a lot of work by dozens of mathematicians. After all, you can’t ‘just’ devise interesting braided monoidal 2-categories: doing so requires deep ideas!

But while you may have been overoptimistic concerning the rate of progress, we now have invariants of 2-dimensional surface in 4d space, obtained by categorifying the representation theory of quantum groups. And this is indeed one of the hottest topics in low-dimensional topology.

So, I’d say your younger self was on the right track. For a quick sketch of where we are now, try the section on Khovanov in the prehistory of $n$-categorical physics that Aaron Lauda and I wrote.

And let’s not forget the revolutionary work on TQFT that Jacob Lurie is busy carrying out. Higher category theory is fundamental here — indeed, he’s having to build the foundations of higher categories hand-in-hand with work on this topic.

Quinn wrote:

Topological field theories on 2-manifolds can be characterized in terms of Frobenius algebras. The modular ones (roughly the ones coming form 2-categories) correspond to semisimple Frobenius algebras. Semisimple algebras are ‘measure zero’ in unrestricted algebras and have much simpler structure. this indicates that requiring higher-order decomposition properties corresponding to higher categories enormously constricts the field theories. To get more power we apparently need to reduce the categorical order rather than increase it.

Well, Quinn’s attempts to crack the Andrews–Curtis conjecture by using computers to find suitable non-semisimple 2d TQFTs seem to have stalled out. But maybe he’s still optimistic? Or maybe he’s hinting at something else? I’m not sure exactly what ‘more power’ he’s hoping to get, and what he hopes to do with it. It would be nice if he were more explicit.

Posted by: John Baez on May 10, 2010 8:10 PM | Permalink | Reply to this

### Re: Quinn on Higher-Dimensional Algebra

Well Khovanov homology is known to give trivial (or more precisely uninteresting) invariants for surfaces in 4-space. Jacob told me that there are interesting examples of braided monoidal 2-categories with duals. I haven’t gotten my hands dirty with any of them yet. Unless, of course, quandle 3-cocycles give such examples.

It is true that Aaron is trying to tie this to the categorification of quantum sl(2). I don’t yet see a complete example.

Then again, my recent life has been dealing with trivialities.

Posted by: Scott Carter on May 10, 2010 8:18 PM | Permalink | Reply to this

### Re: Quinn on Higher-Dimensional Algebra

Scott wrote:

Well Khovanov homology is known to give trivial (or more precisely uninteresting) invariants for surfaces in 4-space.

Maybe that’s true for closed surfaces… is that what you mean? But Khovanov homology definitely does say very interesting things about 2d discs embedded in 4d space. So I’m afraid nonexperts will get the wrong impression from what you wrote here!

From the Prehistory:

One exciting aspect of Khovanov’s homology theory is that it breathes new life into Crane and Frenkel’s dream of understanding the special features of smooth 4-dimensional topology in a purely combinatorial way, using categorification. For example, Rasmussen has used Khovanov homology to give a purely combinatorial proof of the Milnor conjecture—a famous problem in topology that had been solved earlier in the 1990’s using ideas from quantum field theory, namely Donaldson theory. And as the topologist Gompf later pointed out, Rasmussen’s work can also be used to prove the existence of an exotic $\mathbb{R}^4$.

In outline, the argument goes as follows. A knot in $\mathbb{R}^3$ is said to be smoothly slice if it bounds a smoothly embedded disc in $\mathbb{R}^4$. It is said to be topologically slice if it bounds a topologically embedded disc in $\mathbb{R}^4$ and this embedding extends to a topological embedding of some thickening of the disc. Gompf had shown that if there is a knot that is topologically but not smoothly slice, there must be an exotic $\mathbb{R}^4$. However, Rasmussen’s work can be used to find such a knot!

Before this, all proofs of the existence of exotic $\mathbb{R}^4$’s had involved ideas from quantum field theory: either Donaldson theory or its modern formulation, Seiberg–Witten theory. This suggests a purely combinatorial approach to Seiberg–Witten theory is within reach. Indeed, Ozsváth and Szabó have already introduced a knot homology theory called ‘Heegaard Floer homology’ which has a conjectured relationship to Seiberg-Witten theory. Now that there is a completely combinatorial description of Heegaard–Floer homology, one cannot help but be optimistic that some version of Crane and Frenkel’s dream will become a reality.

Posted by: John Baez on May 10, 2010 8:35 PM | Permalink | Reply to this

### Re: Quinn on Higher-Dimensional Algebra

John wrote, “Maybe that’s true for closed surfaces … is that what you mean? ” Yes indeed, I forgot to write close surfaces.

Posted by: Scott Carter on May 10, 2010 11:27 PM | Permalink | Reply to this

### Re: Quinn on Higher-Dimensional Algebra

Ozsvath-Manolescu-Thurston have had a combinatorial version of Ozsvath-Szabo
four-manifold invariants (mod 2). http://front.math.ucdavis.edu/0910.0078
for a year or so now.

Posted by: Tom Mrowka on May 28, 2010 2:45 PM | Permalink | Reply to this

### Re: Quinn on Higher-Dimensional Algebra

This sentence here is true:

higher-order decomposition properties corresponding to higher categories enormously constricts the field theories.

This conclusion here, sounds strange to me:

To get more power we apparently need to reduce the categorical order rather than increase it.

I may be missing the point, but this sounds to me analogous to a statement of the sort: “Requiring a map of topological spaces to be continuous enormously restricts it. To get more power, we need to remove the condition that the map is continuous.”

Posted by: Urs Schreiber on May 10, 2010 10:20 PM | Permalink | Reply to this

### Re: Quinn on Higher-Dimensional Algebra

I think Quinn would say the better analogy is “Requiring a map of topological spaces to be a homeomorphism enormously restricts it. To get more power, we need to consider the larger class of continuous maps.”

I think Quinn’s point is that requiring that TQFTs satisfy a large number of higher order decomposition axioms means there will be fewer examples of TQFTs. Weakening the axioms means there will be more examples or TQFTs, some of which might be more powerful in the sense that they distinguish/detect the objects we are interested in.

(I’m not necessarily endorsing Quinn’s view. Personally I’m quite fond of the higher-order TQFT axioms.)

Posted by: Kevin Walker on May 11, 2010 3:47 AM | Permalink | Reply to this

### Re: Quinn on Higher-Dimensional Algebra

It seems to me that there are a number of possible points of view on what the theory of TQFTs is “for”.

1) It could be for producing invariants that can be used to prove and disprove theorems about knots, manifolds, and the like.

2) It could be starting point for understanding a much more ambitious theory of nontopological QFTs.

3) It could be a development entirely within category theory, linking category-theoretic objects with combinatorial descriptions with geometry by means of “graphical calculi”.

I’m really only familiar with 3), where passing to “extended” TQFTs really shows its merits (only in the fully extended case do you get to formulate the Baez-Dolan cobordism/tangle hypotheses).

But it seems like Quinn’s position is perfectly reasonable from perspective 1). As far as I know, theories which are known to give interesting manifold invariants have begun life as “non-extended” theories which can be understood without higher category theory. It seems to me to be an interesting problem to take such a theory and “extend it down to points” (and also difficult problem, which I think is still not understood even for Chern-Simons theory, though perhaps this is now changing). But I’m not sure if finding such an extension would tell you anything concrete you didn’t know about the invariants assigned in the top dimension, which are of primary interest from perspective 1).

Of course, if someone managed to construct new manifold invariants by means of a TQFT which was only known to exist because of the cobordism hypothesis, then there would be a pretty compelling counterargument to Quinn.

Posted by: Jacob Lurie on May 15, 2010 12:57 AM | Permalink | Reply to this

### Re: Quinn on Higher-Dimensional Algebra

I think Jacob’s viewpoints 1-3 are very closely connected, and I don’t find Quinn’s position about extended TFT reasonable in a long-term sense even from POV 1, whether or not one can point yet to concrete statements about 3-manifold topology coming say from extended TFT.

The way I understand the cobordism hypothesis is as an attempt to capture something close to the richness of structure in a physicist’s TFT (specifically in a topologically twisted supersymmetric QFT). The assignments of traditional TFT capture a mere shadow of this structure, as is already evident in the 2d case (as Kontsevich discovered in homological mirror symmetry). I (learning from Hopkins and Freed) think of what one assigns to a point as a version of the physicists’ action - or more precisely of the action integrated over all fields on a small disc. The cobordism hypothesis teaches us how to perform this integration on larger manifolds - and in particular gives us criteria to tell if this integration is possible. In order to do so one needs higher category theory. Thus points 2,3 are very closely tied - the higher category is a powerful way to make sense of the local structure of path integrals in the topologically twisted SUSY setting.

As to 1, one of the reasons we should think hard about the structures physicists have in place (and thus model them as we do eg with extended TFT) is that they are potentially much more powerful than what we have available. The classic paradigm for this is dualities – physicists can identify apparently very different looking theories, drawing powerful conclusions. (My favorite of these being S-duality). By working with extended TFTs one can hope to get much closer to understanding these dualities, by separating the complexity of the theory and of the topological spaces on which we study the theory.

Witten once described what mathematicians tend to do with ideas trickling in from string theory as “in vitro QFT” - we take the beautiful rich structures of the physics, cut off a piece, let it shrivel and die in the lab, and see what conclusions we can draw. The Seiberg-Witten revolution in topology (among many) teaches us that these structures are much richer and smarter than the ones we used before – in particular in vivo Seiberg-Witten theory is much richer than what mathematicians usually mean. So I don’t think it’s the case that the theories that have given rise to important applications in topology began life as nonextended theories – they may have first made their appearance within mainstream math that way, but underlying them is something much more extensive and living. I think of extended TFT as an attempt to study in vivo QFT, and (to the extent that this language succeeds) I expect it will make an impact in topology as well.

Posted by: David Ben-Zvi on May 15, 2010 10:24 PM | Permalink | Reply to this

### nontopological QFT

a much more ambitious theory of nontopological QFTs.

What can we say about cobordisms equipped with (suitably well behaved) maps to a fixed smooth/topological space $X$?

Here is a simple low-dimensional observation, that smells a bit like it might be the indication of something. Probably of something obvious, but I feel like mentioning it anyway.

for $X$ a smooth manifold one can define a smooth category (i.e. 1-category valued stack on $CartSp$ or the like) $P_1(X)$ of smooth paths in $X$, whose morphisms are (certain classes of) certain smooth maps $[0,t] \to X$ for $t \in \mathbb{R}_+$. This is a bit like $Bord_1(X)$, but without the symmetric monoidal structure.

Smooth functors $tra : P_1(X) \to Vect$ correspond to smooth vector bundles with connection on $X$.

Now, if this vector bundle is finite rank, then there is a unique extension of this data to a symmetric monoidal morphism $Bord_1(X) \to Vect$, where the value assigned to a circle in $X$ is the trace of the value of $tra$ on the path obtained by cutting open the circle to obtain a path.

Apologies for this lengthy description of the obvious. The simple observation I am meaning to get at is that this looks like the beginning of a free/forgetful adjunction

$\array{ \underline{P_1(X) \to U(Vect_{\otimes})} \\ Bord_1(X) \to Vect_{\otimes} } \,,$

where $U$ is the forgetful functor from symmetric monoidal (smooth) categories to bare (smooth) categories.

It kind of makes me want to think of $Bord_1(X)$ as $F(P_1(X))$, the “free smooth symmetric monoidal category with duals” on $P_1(X)$.

For $X$ the point we have essentially $P_1({pt}) = {pt}$ (not quite, possibly, depending on details that i want to be glossing over for the time being). So this kind of perspective might connect to the bare cobordism hypothesis, which, too, one would still hope to be the restriction to the point of a genuine free/forgetful adjunction.

Posted by: Urs Schreiber on May 25, 2010 9:20 PM | Permalink | Reply to this

### Re: nontopological QFT

How close would this idea get to Turaev’s HQFTs? In some parts of his work it looks as if one could encode extra structure, such as smoothness, with maps to a classifying space. It is a fascinating possibility and the theory is ready there for adapting.

Posted by: Tim Porter on May 26, 2010 6:25 AM | Permalink | Reply to this

### Re: nontopological QFT

it looks as if one could encode extra structure, such as smoothness, with maps to a classifying space.

I have been playing around with the thought of inducing extra structure such as notably (pseudo)Riemannian metric structure by such maps. (Smoothness in the setup I tend to take for granted.)

For instance for a sigma-model theory on a target space $X$ with pseudo-Riemannian parameter space $\Sigma$, the technology of multisymplectic geometry suggests that we regard the action as a functor

$\exp(S) : P_n(X \times \Sigma) \to n Vect$

on $n$-dimensional paths in the “extended cofiguration space” $X \times \Sigma$. The part of the path in $\Sigma$ picks up, by pullback, metric information from $\Sigma$ and hence may model what in physics-speak would be the relation between the affine parameter on the worldvolume and the worldvolume metric. At least up to some subtleties.

In fact, I would tend to allow $\Sigma$ here (also $X$ of course) to be not just a space but a smooth (higher) category itself, notably a smooth poset, encoding causal structure.

Then somehow the quantization step should provide a push-down along the projection $\Sigma \times X \to \Sigma$ so that we end up with something like

$Z_S : P_n(\Sigma) \to n Vect \,.$

For the example of $\Sigma$ a flat 2-dimensional worldsheet I enjoy the observation that such 2-functors on causal 2-paths in $\Sigma$ do encode 2-dimensional AQFT in that forming local endomorphisms of them produces a local net of observables on $\Sigma$.

So this is how I am currently trying to see if we can bring nontopological structure into the game by having cobordisms with maps into extended configuration spaces $\Sigma \times X$ where $\Sigma$ carries the extra worldvolume metric structure.

This perspective is of course a little different to a perspective, where $\Sigma$ itself is regarded as a cobordism with structure. If one does this in addition , one seems to arrive with the above at a rough picture of possibly a similar smell to it as topological chiral homology (though immensely less developed, of course). The difference to chiral homology and factorization algebras etc on the cobordisms being that this algebraic structure comes itself from “cobordisms inside the cobordism”. This reminds me a bit of what in string theory is called Green’s concept of “worldsheets for worldsheets”, where for instance the string’s worldvolume 2dCFT theory is itself regarded as the effective QFT of a string theory inside that worldsheet.

Not sure, this are just some thoughts.

Posted by: Urs Schreiber on May 26, 2010 8:37 AM | Permalink | Reply to this

### Re: nontopological QFT

Urs wrote

It kind of makes me want to think of…

What more needs to be done to see this unequivocally as a free/forgetful adjunction (2-adjunction?)?

Posted by: David Corfield on May 26, 2010 9:05 AM | Permalink | Reply to this

### Re: nontopological QFT

In this case the category $Bord_1(X)$ is extremely simple, so the only thing to come from $P_1(X)$ to $Bord_1(X)$ is to allow disjoint unions…

But what would you say in the 2-dimensional case? The Path-2-groupoid $P_2(X)$ has just very simple 2-morphisms whereas the ones in $Bord_2(X)$ are more complicated (higher genus, more inputs etc.). How would the 2-dimensional case look like?

Posted by: Thomas Nikolaus on May 26, 2010 9:40 AM | Permalink | Reply to this

### Re: nontopological QFT

What more needs to be done to see this unequivocally as a free/forgetful adjunction

One would need to figure what the would-be free functor, left adjoint to the one that forgets symmetric monoidal structure with duals, would do to a general object.

Jacob Lurie’s statement of the cobordism hypothesis might be thought of (I’d think) as indicating what such a would-be free functor does to the point.

What I indicated above is what looks to me like a hint for what that would-be free functor would do to a category of the form $P_1(X)$.

If this is at all on the right track, then what would need to be done is to construct this free functor completely. In some context.

Posted by: Urs Schreiber on May 26, 2010 9:46 AM | Permalink | Reply to this

### Re: nontopological QFT

But what would you say in the 2-dimensional case? The Path-2-groupoid $P_2(X)$ has just very simple 2-morphisms whereas the ones in $Bord_2(X)$ are more complicated (higher genus, more inputs etc.).

For comparison, notice that the analogous statement holds and is of interest in an even simpler situation:

The point $*$ has just very simple $n$-morphisms, whereas the ones in $Bord_n$ are more complicated.

In fact, as the cobordism-hypothesis-theorem shows, the morphisms in $Bord_n$ are precisely all those that encode higher dimensional trace information, namely all operations obtained by bending things around, using duality. And nothing else.

On the other hand, if there is a target space $X$, then $P_n(X)$ encodes (or that’s at least the way I am thinking about it) all information of $Bord_n(X)$ that is not related to tracing, but just to the local behaviour of the cobordisms maps to $X$.

I think one can see the following (though I realize it’s been quite a while since I really thought about what I say now.)

For a 2-bundle/gerbe with connection on $X$ let $P_2(X) \to 2 Vect$ be the corresponding parallel transport, where $2 Vect$ is the category whose objects are algebras over the ground field, morphisms are bimodules. This is symmetric monoidal.

Then this extends to $Bord_2(X)$ essentially uniquely by mapping the duality morphisms that $Bord_2(X)$ has on top of those in $P_2(X)$ to the corresponding morphisms in $2 Vect$.

For instance a circle $S^1 \to X$ the functor $Bord_2(X) \to 2 Vect$ has to send to the composite of a path $\gamma : [0,1] \to X$ regarded as a morphism $\emptyset \to \gamma(0) \sqcup \bar \gamma(1)$ composed with the constant path $const_{\gamma_0} : [0,1] \to X$ regarded as a morphism $\gamma(0) \sqcup \bar \gamma(1) \to \emptyset$.

The former in turn comes by composing with a duality from a morphism $\gamma : \gamma(0) \to \gamma(1)$ in $P_2(X)$, which has a value under $tra : P_2(X) \to 2 Vect$, being some bimodule

$A_{\gamma(0)} \stackrel{N}{\to} A_{\gamma(0)} \,.$

All the non-topological parallel-transport information is in this assignment, all the topological/trace-information in where the trace/duality morphisms get mapped to.

The trace operation will send the above example to the vector space

$N \otimes_{A_{\gamma(0)}^{op} \otimes A_{\gamma(0)}} \otimes A_{\gamma(0)}$

regarded as a $k$-$k$ bimodule ($k$ the ground field) and I think this is effectively fixed by functoriality.

So that’s the general idea: $P_n(X)$ encodes exactly all the nontopological information, $Bord_n$ encodes exactly all the topological information, and their fusion into $Bord_n(X)$ encodes both.

Posted by: Urs Schreiber on May 26, 2010 10:12 AM | Permalink | Reply to this

### Re: Quinn on Higher-Dimensional Algebra

Just to say that I have written to Frank Quinn about his comments on groupoids on p. 43 of his draft book, and he has replied promptly that he would revise these comments in the light of the references I gave.

I am happy to send my comments to anyone who wants them.

This is relevant to his comments on higher dimensional algebra, of which a special case is higher dimensional group(oid) theory!

Ronnie Brown

Posted by: Ronnie Brown on September 1, 2010 9:20 PM | Permalink | Reply to this

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