Homotopy Theory and Higher Algebraic Structures at UC Riverside

September 24, 2009

Homotopy Theory and Higher Algebraic Structures at UC Riverside

Posted by John Baez

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This year the Fall Western Section Meeting of the American Mathematical Society will be held here at UC Riverside. Julie Bergner and I are running a session on homotopy theory, $n$ -categories and related topics. If you’re anywhere nearby, I hope you drop by!

Special session on homotopy theory and higher algebraic structures, UC Riverside, November 7-8, 2009. Organized by Julie Bergner and John Baez.

If you’re interested in our session, you may also like this one:

Special session on algebraic structures in knot theory, UC Riverside, November 7-8, 2009. Organized by Alissa Crans and Sam Nelson.

It’ll include talks by Louis Kauffman, Mikhail Khovanov, Scott Carter, Masahico Saito, Scott Morrison and other people who live near the interface of topology, categories and physics.

As if that weren’t enough, there’s also another session on knot theory:

Special session on knotting around dimension three: a special session in memory of Xiao-Song Lin, UC Riverside, November 7-8, 2009. Organized by Martin Scharlemann and Mohammed Ait Nouh.

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The special session that Julie Bergner and I put together has a great lineup of talks. First, here are the vaguely $n$ -categorical talks, listed in no particular order:

Categorification via quiver varieties. Anthony Licata (with Sabin Cautis and Joel Kamnitzer). abstract.

Categorifying quantum groups. Aaron D Lauda. abstract.

2-Quandles: categorified quandles. Alissa S. Crans. abstract.

Group actions on categorified bundles. Weiwei Pan. abstract.

A categorification of Hall algebras. Christopher Walker. abstract.

A categorification of the Hecke algebra. Alexander E Hoffnung. abstract.

3-Categories for the working mathematician. Christopher L Douglas (with Andre Henriques). abstract.

Mapping spaces in quasi-categories. David I. Spivak (with Daniel Dugger). abstract.

String connections and supersymmetric sigma models. Konrad Waldorf. abstract.

As you can see, categorification is becoming a big business. Of course, a lot of this is due to the work of Mikhail Khovanov. And over in Alissa and Sam’s special session, he’s giving an hour-long talk entitled “Adventures in categorification”!

Second, here are the vaguely homotopy-theoretic talks. Of course there’s no sharp dividing line, and I’m not trying to create one… I’m just trying to avoid a single enormous list of talks which none of you will read:

Generating spaces for $S(n)$ -acyclics. Aaron Leeman. abstract

A homotopy-theoretic view of Bott–Taubes integrals and knot spaces. Robin Koytcheff. abstract.

On the $K$ -theory of toric varieties. Christian Haesemeyer (with Guillermo Cortinas, Mark E Walker, and Charles A. Weibel). abstract.

Homotopy colimits and the space of square-zero upper-triangular matrices. Jonathan W Lee. abstract.

An application of equivariant $\mathbb{A}^1$ -homotopy theory to problems in commutative algebra. T Benedict Williams. abstract.

String topology and the based loop space. Eric J Malm. abstract.

Unstable Vassiliev theory. Chad D Giusti. abstract.

The Atiyah–Segal completion theorem in twisted $K$ -theory. Anssi S. Lahtinen. abstract.

Monoids of moduli spaces of manifolds. Soren Galatius. abstract.

Relations amongst motivic Hopf elements. Daniel Dugger (with Daniel C. Isaksen). abstract.

Real Johnson-Wilson theories. Maia Averett. abstract.

Orbifolds and equivariant homotopy theory. Laura Scull (with Dorette Pronk). abstract.

Universal Bott Samelson resolutions. Nitu R Kitchloo. abstract.

There will also be a double talk on ‘blob homology’. I don’t completely understand blob homology yet, but it seems to be a practical way of computing homology with coefficients in an $n$ -category with duals. If so, it straddles the worlds of $n$ -category theory and homotopy theory so neatly that it makes a mockery of the distinction:

Blob homology 1. Scott Morrison (with Kevin Walker). abstract.

Blob Homology 2. Kevin Walker (with Scott Morrison) abstract.

If you want to know when the talks are actually taking places, check out the schedules here.

Posted at September 24, 2009 4:39 PM UTC

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Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

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The basic idea of blob homology should be that of factorization algebra, which in turn is not unsimilar to local nets in AQFT.

Posted by: Urs Schreiber on September 24, 2009 6:15 PM | Permalink | Reply to this

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

I agree there is significant similarity with ideas from factorization algebras and local nets, but I’m not sure that I would characterize this as the “basic idea”.

One way of thinking of the blob complex is as a generalization of the Hochschild complex to higher categories and higher dimensional manifolds. One thinks of the Hochschild complex as associated to a 1-category and a 1-manifold (the circle). It’s a fairly small complex, analogous to cellular homology. The blob complex for the same input data (1-category and circle) yields a quasi-isomorphic but much much larger chain complex, analogous to singular homology. It’s advantage over the Hochschild complex is that it is “local”. In higher dimensions this locality means that it is easy to (well-) define the blob complex of an n-category + n-manifold without choosing any sort of decomposition of the n-manifold.

Posted by: Kevin Walker on September 28, 2009 9:59 PM | Permalink | Reply to this

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

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Thanks, Kevin.

Maybe “basic idea” was the wrong term and “significant similarity” captures it better.

In any case, we (whoever “we” is) should write an $n$ Lab entry on blob homology.

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

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

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I moved your comment to the wiki, for a start: $n$ Lab: blob complex.

What’s the canonical reference?

Posted by: Urs Schreiber on September 29, 2009 8:32 PM | Permalink | Reply to this

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

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Kevin Walker and Scott Morrison are writing a huge paper on blob homology. For a while the canonical reference has been this short abstract. A more detailed draft is available if you know where to look, but I hear a much better version will be released in a few weeks, so I’ll let Kevin and Scott decide if they want to publicize that draft in the interim.

By the way, I keep wanting to type the phrase ‘blog homology’ — maybe we should invent that sometime.

Posted by: John Baez on September 29, 2009 11:23 PM | Permalink | Reply to this

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

There’s a more recent abstract/announcement in this Oberwolfach report.

The current draft of the paper is too rough to be made public, but anyone who is motivated enough to track down my email address and send a personal request is welcome to a copy.

Urs: Thanks for making the nLab entry. Feel free to copy from or link to the above report. If you or anyone else has questions, I’ll be happy to attempt to answer them here or on the wiki.

Posted by: Kevin Walker on September 29, 2009 11:47 PM | Permalink | Reply to this

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

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Thanks, I hadn’t known that our Oberwolfach report had already appeared.

I added the link to $n$ Lab:blob complex.

I also wanted to add the link to the set of slides on your homepage, but couldn’t find them.

I’ll be happy to attempt to answer them here or on the wiki.

Great. Here are mine:

a) What’s the definition, precisely?

b) What are the first few interesting examples, in detail?

c) What can you say about the relation to Costello’s and to Lurie’s factorization algebras?

If you could add a bit of information on that to the wiki, that would be great (I have created dummy headlines for you already, you just need to paset in the text).

You’ll remember that I had to miss your talk in Oberwolfach, unfortunately. Which was/is a pity.

Posted by: Urs Schreiber on September 30, 2009 10:36 AM | Permalink | Reply to this

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

Sorry about this late response – I’ve been busy lately.

“a) What’s the definition, precisely?”

The Oberwolfach abstract contains a terse but fairly detailed/complete definition. I’d try to copy it here or at nLab, but I don’t have any experience with MathML.

“b) What are the first few interesting examples, in detail?”

There are not currently as many fully developed examples as I would like. The motivation for developing the general theory was to apply it to two examples: (1) Khovanov homology, which can be thought of as a 4-category (with strong duality), and (2) tight contact structures, which can be viewed as a 3-category (albeit with some complications coming from the smooth, as opposed to PL, structure). Both of these categories are non-semisimple, so one expects that blob homology might have something interesting to say.

Some special cases of blob homology provide interesting (but not new) examples. The degree 0 blob homology of an n-manifold is isomorphic to the (dual) hilbert space of the n+1-dimensional TQFT constructed from the same n-category. The blob homology of a circle is isomorphic to the Hochschild homology of the input 1-category. More generally, if one views a commutative algebra C as an n-category which is trivial in dimensions 0 through n-1, then the blob homology based on C is conjecturally isomorphic to the higher Hochschild homology of C. (This conjecture is due to Thomas Tradler.) In the case that C is a truncated [multi-variable] polynomial algebra, one can relate blob homology to the usual homology of [colored] configuration spaces of points in the n-manifold in question.

“c) What can you say about the relation to Costello’s and to Lurie’s factorization algebras?”

Not as much as I would like. This is on my list of things to think about, but it’s not at the top of that list at the moment. Given a 1-category, one can “ringify” it by taking the sum of all the morphism spaces and defining multiplication to be zero if range and domain do not agree. The resulting ring is Morita equivalent to the original 1-category. Factorization algebras make me think of the result of ringifying (or maybe E_n-algebra-ifying) an n-category. All of the lower order morphisms have been removed. When n=1 we know that rings are just as general as 1-categories from a Morita point of view. Is the same true for n>1? At least one knowledgeable person has told me he thinks the answer is “no”. If the answer is no then I would guess that factorization algebras can be viewed as yet another (strictly) special case of the blob complex. If, on the other hand, the answer is yes, then perhaps factorization algebras are an equivalent way of doing blob homology that is much easier to define. (I suspect, though, that to do concrete calculations one would want to have the lower order morphisms back in the picture.)

I am very far from an expert on factorization algebras, so perhaps the above paragraph is all nonsense. If anyone else has thoughts on this topic, I’m eager to hear them.

I’ll try to fill in some of the blank spots on the nLab wiki in the future.

Posted by: Kevin Walker on October 6, 2009 8:05 PM | Permalink | Reply to this

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

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I don’t have any experience with MathML.

No need to. None of us has. You type in ordinary LaTeX code. Have a quick look at the source code of a random entry. It’s very easy. Just dollar signs as usual and there you go.

Posted by: Urs Schreiber on October 6, 2009 8:39 PM | Permalink | Reply to this

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

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tight contact structures, which can be viewed as a 3-category

Kevin,

Would you be willing to say a few more words about that?

Posted by: Eugene Lerman on October 6, 2009 10:56 PM | Permalink | Reply to this

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

Sure.

The somewhat tautological thing to say is that there is a 3-category where

* a 0-morphism is a point equipped with a germ of a contact structure on a 3-dimensional neighborhood of it;

* a 1-morphism is an arc equipped with a germ of a contact structure on a 3-dimensional neighborhood of it;

* a 2-morphism is a disk equipped with a germ of a contact structure on a 3-dimensional neighborhood of it;

* and a 3-morphism is a tight contact structure on a 3-ball (i.e. we declare non-tight (over-twisted) contact structures to be zero).

The rest of the 3-cat structure is given by the obvious geometric operations (gluing, restriction to boundary, etc.).

I’ve swept a lot of details under the rug. One of the more important details is that for technical reasons one insists that the 1-morphisms be germs of contact structures near legendrian arcs, not arbitrary arcs.

One of the interesting things about this 3-cat is that it can be given a completely combinatorial description. An old result (I forgot who’s) says that the germ of a contact structure near a surface in a contact 3-manifold is completely determined by the induced “characteristic foliation” of the surface (which is the singular foliation given by the intersection of the contact 2-planes and the tangent space of the surface). This means we have an uncountable number of 2-morphisms, which might at first be disappointing, but of course what we really care about is the number isomorphisms classes of 2-morphisms. A result of Giroux says that these isomorphisms classes correspond bijectively to isotopy classes of “dividing curves” on the surface. A set of dividing curves is a subdivision of the surface into positive and negative regions; roughly speaking, the positive region is the one where the orientation of the contact plane and the orientation of the surface agree (or at least are closer to agreeing than to disagreeing).

This means for for a disk with fixed boundary condition, we have only finitely many (catalan number many) isomorphism classes of germ. (A closed loop in the dividing curves on the disk would imply that the germ of contact structure is not tight. This is not obvious.) In other words, the contact 3-category can plausibly claim to be a categorification of the Temperley-Lieb 2-category with loop value d=0. Personally, I think this is very cool.

Papers by Ko Honda contain good background information on this topic. I’ve omitted many details, including tricky issues related to smoothing and unsmoothing corners.

Posted by: Kevin Walker on October 7, 2009 6:32 AM | Permalink | Reply to this

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

Very interesting. I presume a goal is to understand tight contact structures?

Have you thought of doing something similar in dimension 5? I am thinking of Klaus Niederkruger’s
plastikstufe. I understand that you don’t have the same infrastructure, like Giroux’s theorem etc, but it would be nice to have another tool for understanding contact structures in higher dimensions besides contact homology.

Posted by: Eugene Lerman on October 7, 2009 7:41 PM | Permalink | Reply to this

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

Yes, one of the goals is to understand tight contact structures. Another goal is to understand the structure of an interesting non-semisimple TQFT (or rather, decapitated TQFT – I’m not proposing to define the 4-dimensional part of the theory).

I haven’t really thought about higher dimensions (too much remaining to do in dimension 3). I’ll take a look at the link you included.

Posted by: Kevin Walker on October 9, 2009 3:14 AM | Permalink | Reply to this

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

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re blob homology:

in his latest opus, Jacob Lurie has now a detailed discussion of his notion of topological chiral homology (see links there) and a quick remark

It should be closely related to the theory of blob homology studied by Morrison and Walker

(p. 107).

I am hoping that eventually these two and factorization algebras will merge into a single thing. Would be a pity otherwise.

Posted by: Urs Schreiber on November 4, 2009 9:18 PM | Permalink | Reply to this

blob homology

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There is now a set of notes available by Scott Morrison and Kevin Walker on blob homology, which Kevin kindly linked to here.

Posted by: Urs Schreiber on November 16, 2009 11:42 PM | Permalink | Reply to this

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

Do you know if the lectures will be recorded for those that can’t make it to Riverside?

Posted by: anon on October 6, 2009 12:23 AM | Permalink | Reply to this

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

I’m pretty sure we won’t. We could, but I kind of doubt any of us will want to spend the weekend standing behind a camera — and we don’t have any slaves to do that for us.

So, come to Riverside!

Posted by: John Baez on October 6, 2009 5:56 PM | Permalink | Reply to this

Re: Homotopy Theory and Higher Algebraic Structures at UC Riverside

what! no grad students who could take turns?

reminds me of JHC’s aphorism (approximately)
grad students are machines for cellulose (i.e. papers) into theorems

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September 24, 2009