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April 7, 2009

Categorification and Topology

Posted by John Baez

On Friday I’m going to the workshop on Categorification and Geometrisation From Representation Theory in Glasgow. I plan to learn a lot of algebra and give this talk:

The weekend after that I’ll give a more leisurely version of this talk at the Graduate Student Topology Conference in Wisconsin.

I decided that with Jacob Lurie making serious progress on the Cobordism Hypothesis and Tangle Hypothesis, it’s a good time for everyone to learn the basic ideas.

  • Abstract: The relation between nn-categories and topology is clarified by a collection of hypotheses, some of which have already been made precise and proved. The “homotopy hypothesis” says that homotopy nn-types are the same as nn-groupoids. The “stabilization hypothesis” says that each column in the periodic table of nn-categories stabilizes at a certain precise point. The “cobordism hypothesis” gives an nn-categorical description of cobordisms, while the “tangle hypothesis” does the same for tangles and their higher-dimensional relatives. We shall sketch these ideas, describe recent work by Lurie and Hopkins on the cobordism and tangle hypotheses, and say a bit about how these ideas are related to other lines of work on categorification.

If you see typos or other problems, please let me know!

The Graduate Student Topology Conference will also have other talks of interest to nn-Café regulars. For example:

The workshop on Categorification and Geometrisation From Representation Theory has so many interesting talks that I won’t even try to list them here! So, take a look at the program.

I will explain this diagram in my talk:

Posted at April 7, 2009 1:42 AM UTC

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Re: Categorification and Topology

Grothendieck wrote to Larry Breen in 1975 about n-types and n-groupoids. See here for a scan of one of the letters concerning this. It is certainly true that in the opening pages of pursuing stacks the homotopy hypothesis is made explicit, but the idea goes back much further.

Posted by: David Roberts on April 7, 2009 3:25 AM | Permalink | Reply to this

Re: Categorification and Topology

I read Grothendieck’s letter to Breen when I first started trying to understand nn-categories. A kind soul sent me a copy of that along with Grothendieck’s 600-page letter to Quillen. This was in the Dark Ages, before the world was uploaded.

I’d forgotten the letter to Breen came so long before the letter to Quillen! Thanks — I’ve fixed the date.

Posted by: John Baez on April 7, 2009 7:29 AM | Permalink | Reply to this

Re: Categorification and Topology

On slide 33 you say:

There is also a simplicial approach to (,1)(\infty,1)-categories: ‘complete Segal spaces’.

Now complete Segal spaces are a mix of a simplicial and a topological definition. Asked for “a simplicial approach to (,1)(\infty,1)-categories” I would of course think of quasi-category.

Of course I see that on this slide you want to keep going to (,n)(\infinity,n)-category, for which complete Segal spaces lend themselves more than quasi-categories.

The pattern here could be stated as:

- an (,0)(\infty,0)-category is a (nice) topological space;

- an (,n)(\infty,n) complete Segal-space-type category is a category weakly enriched in (,n1)(\infty,n-1)-categories.

That seems to downplay the simplicial aspect. Of course it’s present in making the statement “weak enrichment” work.

Posted by: Urs Schreiber on April 7, 2009 6:24 AM | Permalink | Reply to this

Re: Categorification and Topology

Urs writes:

Now complete Segal spaces are a mix of a simplicial and a topological definition.

Julie Bergner, who proved that the model category of complete Segal spaces is Quillen equivalent to the model category of simplicial categories, defines a ‘space’ to be a simplicial set! This is actually quite common among homotopy theorists.

So, a complete Segal space, which is a kind of simplicial space, is for these people a kind of simplicial simplicial set — or ‘bisimplicial set’.

I agree that quasicategories are the most obvious simplicial approach to (,1)(\infty,1)-categories. However, in his work on the cobordism hypothesis, Lurie seems to use Segal spaces. Luckily the model categories of these are Quillen equivalent.

Posted by: John Baez on April 7, 2009 7:20 AM | Permalink | Reply to this

Re: Categorification and Topology

Julie Bergner, who proved that the model category of complete Segal spaces is Quillen equivalent to the model category of simplicial categories, defines a ‘space’ to be a simplicial set!

In fact, so did Charles Rezk, who originally invented complete Segal spaces. Lurie is the first one I’ve ever heard of to use complete Segal spaces based on topological spaces rather than simplicial sets.

This is actually quite common among homotopy theorists.

Among a certain type of homotopy theorist, at least. I believe that type tend to come from MIT. (-:

At lunch in the Eagle yesterday, I heard about a math department (which shall remain nameless) where the introductory course on algebraic topology is taught without giving a definition of topological space (it being considered too difficult). Currently they do everything with metric spaces, but simplicial sets are also under consideration. I believe most of those present were of the opinion that the definition of a simplicial set is more difficult than the definition of topological space!

Posted by: Mike Shulman on April 7, 2009 9:48 AM | Permalink | Reply to this

Re: Categorification and Topology

“This is actually quite common among homotopy theorists.”

There are spaces which do not have homotopy type of polyhedra (realizations of simplicial complexes or equivalently CW complexes) while all spaces have WEAK homotopy type of polyhedra. Thus any homotopy theorist who would identify spaces with simplicial sets would neglect existence of homotopy types which are not homotopy types of polyhedra. But then why calling their representatives (models) spaces and not polyhedra ? This is NOT a matter of model for homotopy theory (where I would be happy for taking any equivalent model) but matter of scope. Of course for weak-homotopy theorists (those classifying weak homotopy types) this is OK, as polyhedra suffice as weak homotopy types.

Posted by: Zoran Skoda on April 7, 2009 12:30 PM | Permalink | Reply to this

Re: Categorification and Topology

On the other hand, when one wants to regard a weakly TopTop-enriched category (aka complete Segal space) as an (,1)(\infty,1)-category, one will want to mean by “Top” the category of those well behaved topological spaces which is Quillen equivalent to the standard model structure on SimpSet.

But what if somebody insisted on using enrichment in a category of more general, possibly more badly behaved topolgical spaces? How would we think of that, conceptually? Something which is to shape theory as (,1)(\infty,1)-categories are to homotopy theory?

Posted by: Urs Schreiber on April 7, 2009 1:32 PM | Permalink | Reply to this

Re: Categorification and Topology

But what if somebody insisted on using enrichment in a category of more general, possibly more badly behaved topolgical spaces? How would we think of that, conceptually?

I would say: the same way that we think of enrichment in a 1-category that is not SetSet. The fact that it happens to be possible to identify \infty-groupoids with a full sub-(,1)(\infty,1)-category of the one they are enriching in is interesting, but not particularly novel either. There are plenty of interesting 1-categories to enrich in that include SetSet as a full subcategory, e.g. PosPos, CatCat, even TopTop (in 2 different ways!).

Posted by: Mike Shulman on April 7, 2009 10:03 PM | Permalink | Reply to this

Re: Categorification and Topology

Lurie in Higher Topos Theory, section 7.1.6 sketches out a version of strong shape theory in the setting of infinity topoi; he also quotes Toen-Vezzosi math.AG/0212330 for an equivalent definition in the framework of Segal topoi.

Posted by: Zoran Skoda on February 7, 2010 9:10 PM | Permalink | Reply to this

Re: Categorification and Topology

Way back in April, Urs wrote:

But what if somebody insisted on using enrichment in a category of more general, possibly more badly behaved topolgical spaces?

Today I noted recent Zoran’s reaction, and would add a point to that. It is not so much a question of ‘insisting’ but unless in very nice cases, the natural enrichment in a context may be by spaces whose homotopy type is not that of a CW-complex. (That might easily occur in functional analytic contexts.) Working with weak homotopy type is possible but will it give the ‘right’ answer. I suspect that strong shape theory is nice enough to be able to do more or less everything that is needed, but I have not looked. Possibly Michael Batanin’s version of strong shape would do the trick.

Posted by: Tim Porter on February 8, 2010 4:53 PM | Permalink | Reply to this

Re: Categorification and Topology

I’ve never met any modern homotopy theorists who are at all interested in “strong” homotopy types. The ubiquity of this is why “homotopy type” has come to essentially mean “weak homotopy type” or “m-cofibrant homotopy type.” One can argue about whether or not this is a good thing, but in my experience it is true. Do you have a different experience?

I would argue that from a categorical perspective, the restriction to weak homotopy types is actually the right thing to do. This is because homotopy theory, understood in a general way, is secretly the study of (,1)(\infty,1)-categories, and the prototypical (,1)(\infty,1)-category is that of \infty-groupoids, which can be identified with weak homotopy types. Strong homotopy types form a different (,1)(\infty,1)-category, in which one may certainly be interested, but one shouldn’t expect it to have nearly the importance of the (,1)(\infty,1)-category of weak homotopy types.

Posted by: Mike Shulman on April 7, 2009 1:40 PM | Permalink | Reply to this

Re: Categorification and Topology

Shulman wrote:
I’ve never met any modern homotopy theorists who are at all interested in “strong” homotopy types.

Please to meet you! am I not ‘the very model of a modern’ homotopy theorist? ;-)

Notice also the important distinction (if not over a field) between quasi-iso and homotopy equivalence

Posted by: jim stasheff on April 7, 2009 8:31 PM | Permalink | Reply to this

Re: Categorification and Topology

Please to meet you! am I not ‘the very model of a modern’ homotopy theorist? ;-)

I stand corrected. Clearly I just hang out with the wrong crowd. Are there others like you? Can you point me to some references that study non-m-cofibrant homotopy types?

Posted by: Mike Shulman on April 7, 2009 9:59 PM | Permalink | Reply to this

Re: Categorification and Topology

Shulman wrote: I’ve never met any modern homotopy theorists who are at all interested in strong homotopy types.

Probably you are right that it is very rare that somebody tries to compute some details about the homotopy type of a bad space; shape theory mantra is that homotopy theory is not good for locally bad spaces, and weak=strong for good spaces. However, in shape theory one usually approximates in this way or another good spaces by bad spaces, and in treating the approximations they use in intermediate steps some tools of usual homotopy theory or some homotopy invariant functors. When doing that the maps between bad spaces and good approximations have to be controlled with care; in particular the difference between weak and strong homotopy equivalence, or a general map and a cofibration are significant.

Posted by: Zoran Skoda on April 7, 2009 10:54 PM | Permalink | Reply to this

Re: Categorification and Topology

Zoran wrote approximately:

But then why call their representatives (models) spaces and not polyhedra?

I can think of two very good reasons:

1) ‘Space’ has one syllable and ‘polyhedron’ has four. This is actually important: when talking about an entity all day long, one wants a very short informal name for it.

2) There’s no reason ‘space’ needs to mean ‘topological space’. Thanks to item 1, people studying many different entities wind up calling these entities ‘spaces’. Vector spaces, metric spaces, manifolds, diffeological spaces… they’re all ‘spaces’.

(Regarding 1): Joyal is still seeking a shorter name for ‘quasicategory’. Possibilities include ‘arena’ and ‘agora’, but I think we need something with one syllable, like ‘place’.)

Posted by: John Baez on April 7, 2009 5:13 PM | Permalink | Reply to this

Re: Categorification and Topology

Joyal is still seeking a shorter name for ‘quasicategory’. […] I think we need something with one syllable, like ‘place’.)

Then we could say: take colimits all over the place.

Seriously:

what makes me wonder a bit now is that while the concept quasicategory looks like the very nice™ model for category-up-to-homotopy, it’s not the model that is being used for climbing the (,n)(\infty,n)-categorical ladder.

I mean, I read a book on quasicategories only to learn in the next installment that another model is needed in order to proceed.

I am exaggerating a bit, but maybe you see what I mean.

(I wish I were born 50 years later when this \infty-business had been settled. It keeps using up my time where I feel I was supposed to be thinking about something else. I feel like Atréju at the Southern Oracle: the oracle will eventually provide the answer, but to get inside one has to forget why one once wanted to know this answer…)

Posted by: Urs Schreiber on April 7, 2009 9:03 PM | Permalink | Reply to this

Re: Categorification and Topology

Urs wrote:

I wish I were born 50 years later when this \infty-business had been settled.

Really? I have the opposite attitude. For me the golden era was around 1995, when weak nn-categories and their applications to physics were just a wild dream. Now the subject is getting crowded with competent experts who are willing to gnaw away on hard problems — it’s getting tough for a dilettante like me to come up with simple ideas that seem new and exciting. So, Jim Dolan and I are working on something else.

50 years from now, \infty-categories will be just ‘business as usual’. The problem then will be to infuse the massive formalism with some sort of life: to explain it in a way that kids will enjoy.

Posted by: John Baez on April 7, 2009 9:36 PM | Permalink | Reply to this

Re: Categorification and Topology

For me the golden era was around 1995, when weak n-categories and their applications to physics were just a wild dream. Now the subject is getting crowded with competent experts

Sure, I understand that you enjoy bringing up germs of good ideas in areas that few people pay attention to, and then see how they are grown to stalks, sheaves, stacks and \infty-stacks of good ideas by a crowd of others. With the triple cobordism-, stabilization- and tangle hypothesis (did I miss one?) this looks quite impressive at the moment, and more to come.

Sometimes, such as in a recent similar discussion we had about whether or not to think about String-2-bundles, I wished your threshold for what counts as a crowd were a bit higher (as in “three’s a crowd”) :-). But that probably doesn’t counterweight nearly the wishes you may have about my thresholds for doing certain things or not. :-/

But, yes, regarding your question “Really?”, the answer is: Yes. We can’t possibly all have the same attitude towards the choice of topics for research.

50 years from now, \infty-categories will be just ‘business as usual’.

I wouldn’t mind that. For the same reason that I don’t mind that differential geometry and homological algebra are business as usual today: while it surely must have been fun to be involved in inventing these formalism, I am still mainly interested in unraveling some basics of what quantum physics is.

For one, I am still pretty much wrecking my brain each day thinking about aspects of higher connections and their role in physics and quantization. While this is not the kind of thing I’ll exactly have the fun of explaining to a Kindergarten child anytime soon, it is still very much motivated from the desire of simplifying concepts, and finding overarching simple structures, of extracting simple powerful mechanisms behind a plethora of Erscheinungen.

For that topic, \infty-category theory is a bit like differential geometry is to GR: it’s the natural languge to describe the phenomena in. One doesn’t exactly want to get bogged down with worrying about the language itself all day, exciting as that may be in itself. But here it’s a means to an end here, and I am grateful for every crowd of experts on the language tool that comes by and drops some hints. But the crowd size which does is moderate. I keep fishing for it here and on the nnLab (with success, though, thanks for everybody who helped me!).

Posted by: Urs Schreiber on April 7, 2009 10:20 PM | Permalink | Reply to this

Re: Categorification and Topology

I seem to have missed the discussion whether or not to think about String-2-bundles

what is the issue?

Posted by: jim stasheff on April 8, 2009 2:09 PM | Permalink | Reply to this

Re: Categorification and Topology

I seem to have missed the discussion whether or not to think about String-2-bundles

what is the issue?

Nothing of importance, I was just reminded of the discussion here which involved personal estimates of the density of research on String 2-group related issues.

Posted by: Urs Schreiber on April 8, 2009 3:31 PM | Permalink | Reply to this

Re: Categorification and Topology

Urs wrote:

Sometimes, such as in a recent similar discussion we had about whether or not to think about String-2-bundles, I wished your threshold for what counts as a crowd were a bit higher…

Well, I thought that you, Stephan Stolz, Peter Teichner and André Henriques were getting ready to work on StringString-2-bundles and their applications to elliptic cohomology. To me that feels like a crowd!

It’s like when you throw a party and elephants start walking into the room: it doesn’t take many before it seems crowded.

But, yes, regarding your question “Really?”, the answer is: Yes.

Okay. Maybe you were dreaming of a world where first people work out the foundations of nn-category theory for 50 years, and then you get to apply it to physics. I can see why that would be very fun.

But unfortunately, even before \infty-category theory is nicely developed, a crowd of smart people will try to apply it to physics — and everything else. So, I think you’re stuck trying to apply it while it’s being developed. And that of course means you have to spend some time developing it yourself — which can become quite tiring.

I am still mainly interested in unraveling some basics of what quantum physics is…

Good! Luckily it seems there are always more smart people trying to do impressive fancy things than smart people trying to understand basic things. Maybe it’s so hard to understand basic things that most smart people soon realize there are easier ways to get tenure.

But seriously: most experts on nn-category theory are more interested in topology and algebraic geometry than getting to the bottom of quantum mechanics — even if they know the definition of a TQFT. So, you will surely continue to discover many wonderful things, despite the growing crowd.

For one, I am still pretty much wrecking my brain each day thinking about aspects of higher connections and their role in physics and quantization.

I hope you’re racking your brain, not wrecking it. Be nice to your brain.

Posted by: John Baez on April 9, 2009 8:37 PM | Permalink | Reply to this

Re: Categorification and Topology

Zoran wrote:

But then why calling their representatives (models) spaces and not polyhedra ?

Because the word `polyhedra’ already has too much baggage, cf. the discussion of
polyhedra versus polytopes

Posted by: jim stasheff on April 7, 2009 8:27 PM | Permalink | Reply to this

Re: Categorification and Topology

True, in the end all four models for (,1)(\infty,1)-categories are simplicial in nature.

I am wondering: given those (n+1)(n+1)-fold simplicial sets obtained as nn-fold iterated weak enrichments in the complete Segal-space description of (,n)(\infty,n)-category, does it make sense to consider their nn-fold diagonals, i.e. their realization as (ordinary 1-fold) simplicial sets, and can we characterize these? Do the simplicial sets obtained this way satisfy the axioms for weak \infty-categories of, who was it, Street?

Posted by: Urs Schreiber on April 7, 2009 8:06 AM | Permalink | Reply to this

Re: Categorification and Topology

Urs wrote:

all four models for (,1)(\infty,1)-categories…

I listed 7 models here.

According to Julie, just one of these has not been rigorously related to the rest: namely, A A_\infty-categories. Here I mean gizmos that have a set of objects and for any pair of objects x,yx,y a topological space or simplicial set hom(x,y)hom(x,y), together with a composition that’s only weakly associative, in a manner governed by a many-object version of the A A_\infty operad. (I don’t mean the similar sort of thing where hom(x,y)hom(x,y) is a chain complex.)

I find it a bit sad and also a bit surprising that A A_\infty categories haven’t been organized into a model category that’s Quillen equivalent to the category of simplicial categories — since I’d expect that the ‘strictification’ theorem turning any A A_\infty space into a topological monoid could be generalized to A A_\infty categories.

Speaking of long lists of equivalent formalisms, Julie has a nice talk on thirteen ways of looking at a topological group.

Posted by: John Baez on April 7, 2009 5:54 PM | Permalink | Reply to this

Re: Categorification and Topology

all four models for (∞,1)-categories…

I listed 7 models here.

Hey, no fair!

First you declare # Top-enrichment counts as the same as SSet-enrichment, now here you count them as different!

;-)

Seriously: concerning sources for SSet-enriched categories such as model categories: maybe you or somebody else reading this can help us with this question:

among the light-weight siblings of model categories, Waldhausen categories induce a well-studied SSet-enrichment. Same should be true for their almost-duals, Brown categories, but I am not aware of any literature on this?

Did maybe Julie Bergner have a closer look at these cases, too?

Posted by: Urs Schreiber on April 7, 2009 8:05 PM | Permalink | Reply to this

Re: Categorification and Topology

I find it a bit sad and also a bit surprising that A A_\infty categories haven’t been organized into a model category that’s Quillen equivalent to the category of simplicial categories

Peter May and I are going to get around to doing this any day now. (-:O

Posted by: Mike Shulman on April 7, 2009 9:52 PM | Permalink | Reply to this

Re: Categorification and Topology

Mike wrote:

Peter May and I are going to get around to doing this any day now.

Really? Cool! How about this weekend?

On a vaguely related note: I found out yesterday that Julie Bergner is going to have Bernard Badzioch visit UCR soon. He wrote a paper called Algebraic theories in homotopy theory where he shows that any ‘up-to-homotopy’ model of an algebraic theory can be rigidified to a strict simplicial model.

I don’t know if anyone has tried to cook up a model category of these ‘up-to-homotopy’ models and show that it’s eQuillevant to some model category of strict simplicial models.

And, I don’t know if it’s useful to try to describe A A_\infty-categories with a fixed set of objects as ‘up-to-homotopy’ models of an algebraic theory. Algebraic theories seem like overkill here: hasn’t someone proved a result like Badzioch’s for multi-typed operads? I vaguely seem to recall the answer is yes.

Anyway: it just seems that there’s a painfully general idea lurking around here somewhere, yearning to be expressed in full generality.

Posted by: John Baez on April 9, 2009 8:56 PM | Permalink | Reply to this

Re: Categorification and Topology

And, I don’t know if it’s useful to try to describe A A_\infty-categories with a fixed set of objects as ‘up-to-homotopy’ models of an algebraic theory. Algebraic theories seem like overkill here: hasn’t someone proved a result like Badzioch’s for multi-typed operads?

Are you thinking of these three papers?

Posted by: Mike Shulman on April 9, 2009 10:33 PM | Permalink | Reply to this

Re: Categorification and Topology

According to Julie, just one of these has not been rigorously related to the rest: namely, A A_\infty-categories.

David Ben-Zvi kindly emphasizes by private email that at least for the usual case of linear A A_\infty-categories the connection to stable (,1)(\infty,1)-categories is understood:

when the Hom-spaces are linear over a field kk of characteristic 0, then the following are supposed to be equivalent, in the suitable sense

- kklinear dg-categories

- kk-linear A A_\infty-categories

- kk-linear stable (,1)(\infty,1)-categories.

This is mentioned also for instance on the bottom of p. 11 here.

Posted by: Urs Schreiber on April 10, 2009 6:06 PM | Permalink | Reply to this

Re: Categorification and Topology

What is the definition of a “kk-linear stable (,1)(\infty,1)-category,” and where can a proof of this equivalence be found?

Posted by: Mike Shulman on April 11, 2009 10:52 AM | Permalink | Reply to this

Re: Categorification and Topology

By a stable oo-category linear over k I would mean a module category over Perf_k, the symmetric monoidal stable oo-category of perfect k-modules - or one can phrase this in terms of enrichment over k, but I don’t know that formulation so well.

The equivalence Urs discusses is meant to apply to different enhanced versions of triangulated categories. It’s not true that any dg or A_oo category over k corresponds to a stable oo-category – for example dg categories with one object are not stable. But one can always add some cones (pass to the pre-triangulated hull, or category of twisted complexes, ie make the homotopy category triangulated) to get a stable category.

I don’t know a great reference for this equivalence of the three theories, though there are obvious functors between them (for the equivalence of the homotopy theories of Aoo and dg categories I would look for references in the writings of Keller, the construction is a form of bar-cobar). One thing that is easy to see is the equivalence for categories with a compact generator – in this case all three versions (dg, Aoo, stable) are given by module categories for a dga or Aoo algebra or Aoo ring spectrum (by theorems of Keller and Schwede-Shipley, or by Lurie’s Barr-Beck theorem), but over the rationals these theories are equivalent.

Posted by: D. Ben-Zvi on April 12, 2009 3:18 AM | Permalink | Reply to this

Re: Categorification and Topology

where can a proof of this equivalence be found?

I am currently visiting Zoran Škoda at MPI in Bonn. He kindly points out to me that there is a book by

Yu. Bespalov, V. Lyubashenko, O. Manzyuk, Pretriangulated A A_\infty-categories

which has lots of detailed information on the A A_\infty-dg connection.

It’s available online. You can find the link now at nnLab: A A_\infty-category.

(That entry needs somebody who expands it drastically…)

For instance corollary 1.4 on previewed on p. 10 gives for any (unital) A A_\infty-category the corresponding A A_\infty-equivalent dg-category.

Posted by: Urs Schreiber on April 15, 2009 4:11 PM | Permalink | Reply to this

Re: Categorification and Topology

Is an A A_\infty-category the same as a homotopically enriched category over chain complexes ChCh?

More precisely, if I take the definition of complete Segal space and replace the enrichment category TopTop by that of chain complexes ChCh, what’s the result? How close is it to A A_\infty?

Posted by: Urs Schreiber on April 15, 2009 5:24 PM | Permalink | Reply to this

Re: Categorification and Topology

Urs writes:

More precisely, if I take the definition of complete Segal space and replace the enrichment category Top by that of chain complexes Ch, what’s the result? How close is it to A A_\infty.

I feel sure that these should be ‘morally the same’.

But, I think we should start by getting Mike Shulman and Peter May to find a model structure on the simplicial-set-enriched A A_\infty-categories that gives a model category Quillen equivalent to all the formalizations of (,1)(\infty,1)-categories that are already known to be equivalent. For example, complete Segal spaces.

Then we should pay someone to generalize these results to handle the kk-linear case, which you are asking about.

Posted by: John Baez on April 16, 2009 2:14 PM | Permalink | Reply to this

Re: Categorification and Topology

I feel sure that these should be ‘morally the same’.

But, I think we should start by getting Mike Shulman and Peter May to find a model structure on the simplicial-set-enriched A A_\infty-categories

Okay, if the “simplicial-set-enriched A A_\infty-categories” are different from the “weakly SSet-enriched categories” given by “complete Segal simplicial sets” then chances are that the analogy I asked about holds not very directly.

I just (tried to) polish the nnLab entry on A A_\infty-category a bit.

After getting some feedback from others, I wrote this now in a way which emphasizes that the more conventional use of the term of “A A_\infty-catgegory” does exclusively concern the linear case.

This is true for pretty much all the literature that one finds with “A A_\infty” in the title.

I am wondering if it would make sense to agree that one should find a slightly different term for the non-linear case.

What do you think?

Posted by: Urs Schreiber on April 16, 2009 3:15 PM | Permalink | Reply to this

Re: Categorification and Topology

URS wrote:
After getting some feedback from others, I wrote this now in a way which emphasizes that the more conventional use of the term of Ainfty-catgegory does exclusively concern the linear case.
This is true for pretty much all the literature that one finds with Ainfty in the title.

You forget that originally Ainfty concerned topological spaces. Indeed, as a category theorist, surely you realize that Ainfty makes sense in any category with a good notion of homotopy.

Yes, the A refers to and should continue to refer to associativity. For any other operad, call it O and speak of Oinfty or ambiguously as an infty-cat.

Posted by: jim stasheff on April 17, 2009 3:01 PM | Permalink | Reply to this

Re: Categorification and Topology

You forget that originally Ainfty concerned topological spaces. Indeed, as a category theorist, surely you realize that Ainfty makes sense in any category with a good notion of homotopy.

I am wondering how the term is commonly used today. It seems that the vast majority of literature which claims to be about A A_\infty-categories takes for granted that the context is a linear one.

Maybe that shouldn’t worry me, but maybe it does. I’d like to know from you and others how you think the terminology ought to be used.

For instance, this here is the first paragraph from the book by Bespalov, Lyubashenko and Manzyuk on pretriangulated A A_\infty-categories:

Monoids associative up to homotopy appeared first in topology in works of Stasheff in early sixties. Simultaneously he started to study their differential graded algebraic analogues called A A_\infty-algebras. In nineties mirror symmetry phenomenon discovered by physicists was spelled by Fukaya in a new language of A A_\infty-categories. They combine features of categories and of A A_\infty-algebras. The binary composition operation in A A_\infty-categories is associative up to homotopy which satisfies an equation that holds up to another homotopy, etc. If these homotopies are trivial, we deal with a differential graded category.

It seems clear that these authors mean the dg-linear version when saying A A_\infty and use “associative up to homotopy” for the general version.

I can live with either convention. I am just wondering, since people made the point that it is unfortunate to say nowadays “A A_\infty-category” without meaning the widely understood linear version.

(?)

Posted by: Urs Schreiber on April 17, 2009 4:28 PM | Permalink | Reply to this

Re: Categorification and Topology

I agree with Jim. A A_\infty makes sense in any homotopical situation, referring to any A A_\infty operad. If you feel the need to disambiguate for some particular audience, you could say “topological A A_\infty category.”

I would argue that for some other operad OO we should just say OO-categories, since in general there might not be anything “infinity” about OO. For instance, ordinary (strictly associative) categories as well as completely non-associative “categories” (that is, directed graphs with an arbitrary “composition” operation) are both OO-categories for suitable choices of OO.

Posted by: Mike Shulman on April 19, 2009 4:39 PM | Permalink | Reply to this

Re: Categorification and Topology

O_infty cats should be cats over an operad
resolving O

Posted by: jim stasheff on April 20, 2009 3:04 AM | Permalink | Reply to this

Re: Categorification and Topology

O O_\infty cats should be cats over an operad resolving OO.

Sure.

Posted by: Mike Shulman on April 20, 2009 5:17 PM | Permalink | Reply to this

Re: Categorification and Topology

There is a categorification step from A-infty spaces/A-infty algebras to A-infty categories. So due respect the discoverer of A-infty spaces, by original case of A-infty categories we should take 1990s works of Fukaya-Kontsevich; before then just A-infty spaces and algebras. In that horizontal categorification step one typically does NOT require A-infty categories to be (strictly unital) as Fukaya category as the main example isn’t.

Now we can form algebras over arbitrary opperad, but can we form categories over an algebraic operad ? I mean is the horizontal categorification for algebras over an operad automatic ? It seems that Mike’s post implicitly says this! Second, I still think that Urs’s post is right: apart from the terminology in Lurie’s work I do not know any case in standard references which
talks about A-infty categories in nonlinear case as a main subject of a paper, or in any extensive detail.

If I understood Markl’s paper right he calls strong homotopy algebras over some operad just in the case under the existence of minimal model assumption; so it seems it is a bit stronger then just doing any cofibrant replacement of any operad (say model structures from Hinich), or general “resolution”.

But I read that a while ago, so please correct me.

Posted by: Zoran Skoda on April 20, 2009 9:46 PM | Permalink | Reply to this

Re: Categorification and Topology

Yes, the definition of OO-category, for any (non-symmetric) operad OO, is automatic. If you’ve seen the definition of algebras for an operad, and of an enriched category, you can write it down directly.

Regarding the term “A A_\infty-category,” the point isn’t necessarily about historical precedence but about the logic of the concept. The natural home for the concept of “A A_\infty-category” is any place with homotopy, so you can talk about OO-categories where OO is some ‘cofibrant replacement’ of the associativity operad. Names can outgrow the place of their original use when their natural home is more general. For example, May’s original definition of ‘operad’ was in the topological case only, the symmetric case only, and moreover required O(0)O(0) to be a point—but we don’t hew to those restrictions today since the natural home of the concept ‘operad’ is more general.

I think it’s kind of unfortunate that people use “A A_\infty-category” to mean something without units. Given the pre-existing notions of A A_\infty-space and A A_\infty-algebra, the natural meaning of A A_\infty-category would include units; they should have called the ones without units “A A_\infty-semicategories” or something.

Posted by: Mike Shulman on April 21, 2009 12:06 AM | Permalink | Reply to this

Re: Categorification and Topology

non-unital would suffice
cf. non-symmetric operad

Posted by: jim stasheff on April 22, 2009 1:05 PM | Permalink | Reply to this

Re: Categorification and Topology

the point isn’t necessarily about historical precedence but about the logic of the concept.

So do we, or do we not agree on the way I phrased it a few days ago at A A_\infty-category?

It seems clear that while it may be justified on absolute grounds to say without qualification “A A_\infty-category” for a category over a non-linear operad, this is likely to baffle the entire rather highly developed field of linear A A_\infty-categories and their application in homological mirror symmetry, while it will resonate the way it is intended only among a select few who are still working on developing a theory to go under that name.

We could go around and try to tell everybody in homological mirror symmetry that they are using wrong terminology, but more useful seems to be the attitude that we stick to keeping qualifiers “simplicial”, “topological” when we are talking about non-linear A A_\infty-categories.

But that shall be as far as my interest in terminology discussion goes. What I’d be more intersted in is the answer to the question I have at A A_\infty-category: what is a good abstract-nonsense way, if any, to define linear A A_\infty-category so that the usual component-wise definition with all its signs and differentials drops out by turning the crank?

Mike pointed out that saying “Segal Chain complex” is not likely to work smoothly.

Another possible partial answer is hinted at in that reference by Konstsevich-Soibelman, which says essentially that if the space of objects is itself locally linear in some suitable sense, then an A A_\infty-category is just an A A_\infty-algebroid, i.e. a not-necessrily commutative NQ-supermanifold.

Maybe that’s the best encapsulation one can get.

Posted by: Urs Schreiber on April 21, 2009 1:29 PM | Permalink | Reply to this

Re: Categorification and Topology

We could go around and try to tell everybody in homological mirror symmetry that they are using wrong terminology, but more useful seems to be the attitude that we stick to keeping qualifiers ‘simplicial,’ ‘topological’ when we are talking about non-linear A A_\infty-categories.

I don’t think this is an either-or. We don’t need to try to get them to change their terminology, but neither do we need to constantly apologize for using precise terminology. I think it’s fairly commonplace in different fields for all categories to be implicitly assumed to be enriched over whatever is relevant, whether it’s topological spaces, abelian groups, or chain complexes. They are just using an implicit enrichment because that’s the particular case they care about, while we are studying a general concept.

what is a good abstract-nonsense way, if any, to define linear A A_\infty-category so that the usual component-wise definition with all its signs and differentials drops out by turning the crank?

I believe there is a particular A A_\infty operad OO in chain complexes so that an OO-category can be unraveled into the usual component-wise definition. I don’t remember its exact description, but one could probably reconstruct what it has to be.

Posted by: Mike Shulman on April 21, 2009 2:34 PM | Permalink | Reply to this

Re: Categorification and Topology

Mike,
Are you thinking of the dg operad formed from the top dim chains on the associahedra?

Posted by: jim stasheff on April 22, 2009 1:16 PM | Permalink | Reply to this

Re: Categorification and Topology

Are you thinking of the dg operad formed from the top dim chains on the associahedra?

Perhaps, if that’s what works.

Posted by: Mike Shulman on April 22, 2009 3:06 PM | Permalink | Reply to this

Re: Categorification and Topology

I believe there is a particular A A_\infty operad OO in chain complexes so that an OO-category can be unraveled into the usual component-wise definition.

I added the definition as nnLab: A A_\infty-operad.

Also created nnLab: category over an operad and added the operadic definition to A A_\infty-category.

Posted by: Urs Schreiber on April 24, 2009 10:25 PM | Permalink | Reply to this

Re: Categorification and Topology

Mike Shulman writes:

We don’t need to try to get them to change their terminology, but neither do we need to constantly apologize for using precise terminology. I think it’s fairly commonplace…

I agree with Mike here. Let’s not forget that math survives quite happily with terminology that’s only locally defined. For example, there are plenty of algebraic geometry books where “ring” means “commutative ring”. I used to find this appalling, but then I grew up. A brief word of explanation is all we need to know what’s going on.

Similarly, people working on A A_\infty-categories should feel free to use topologically enriched ones, simplicially enriched ones or chain-complex enriched ones as long as they say (or hint) which one they mean. It would be terrible to demand that a structure so fundamental as the A A_\infty operad be limited to the chain-complex enriched case.

what is a good abstract-nonsense way, if any, to define linear A A_\infty-category so that the usual component-wise definition with all its signs and differentials drops out by turning the crank?

I would just take the linear operad for linear categories with a given set SS of objects and hit it with the bar construction, using the adjunction between ‘linear operads with set of types S×SS \times S’ and ‘linear signatures with set of types S×SS \times S and distinguished identity operations’. This gives a simplicial linear operad whose algebras are A A_\infty algebras.

The basic idea here is described nicely in Todd’s work, but he considers the case S=1S = 1 and enriches over SetSet instead of VectVect.

Posted by: John Baez on April 22, 2009 1:49 AM | Permalink | Reply to this

Re: Categorification and Topology

in the dg context, “the usual component-wise definition with all its signs and differentials drops out by turning the crank” by using the alternate definition in terms of the tensor coalgebra on the (de)suspension

but perhaps that’s not what you are after

the NQ-supermanifold alternative I find less than transparent and slightly misleading -
what’s a manifold go to do with it? surely you wouldn’t refer to an algebraic variety as a manifold

Posted by: jim stasheff on April 22, 2009 1:12 PM | Permalink | Reply to this

Re: Categorification and Topology

Not to belabor the terminology issue further, but I realized I have one more question:

what is your (whoever feels addressed) standard notation for the associative operad in some category? It’s not usually just “AA”, is it? Don’t people usually write AsAs or AssAss or the like for it? And then As(Top)As(Top), As(sSet)As(sSet) etc?

Let me see… here the first Google hit I get has “AsAs”.

What about As(sSet) As(sSet)_\infty-category? Hm, is it the pronounciation…

Posted by: Urs Schreiber on April 22, 2009 2:19 PM | Permalink | Reply to this

Re: Categorification and Topology

as in `toxic’? ;-D

Posted by: jim stasheff on April 22, 2009 4:25 PM | Permalink | Reply to this

Re: Categorification and Topology

I’ve seen AssAss. Apparently AsAs is also used. I believe Peter May uses 𝒩\mathcal{N} (or is it \mathcal{M}? I can never remember). I’ve never seen anything like As(Top)As(Top) or As(sSet)As(sSet); my inclination would be to just write AsAs or 𝒜\mathcal{A} with the enriching category left implicit.

Posted by: Mike Shulman on April 22, 2009 7:45 PM | Permalink | Reply to this

Re: Categorification and Topology

One might specify before proceeding that throughout this paper
(or chapter or section) we work in the —–category

Posted by: jim stasheff on April 23, 2009 2:49 AM | Permalink | Reply to this

Re: Categorification and Topology

Hi Mike,

thanks for your reply. You write:

my inclination would be to just write AsAs or 𝒜\mathcal{A} with the enriching category left implicit.

Further above you said:

I would argue that for some other operad OO we should just say OO-categories

This sounds all very reasonable. Now I am wondering: taken together, doesn’t this imply that you would rather not want to say A A_\infty-category for a category where comoposition is an algebra over a resolution of the topological associatve operad but

As As_\infty-category

?

Hope I am not being a pain with my instsitence on this. I won’t further express hesitance about the use of “A A_\infty-category” for non-linear versions here and on the nnLab, but before closing this discussion, I’d just seriously be intersted in how you think about it.

Cause it seems to me that my concern was that the name should indicate roughly which operad one is actually thinking of, and now I am getting the impression that all in all you (and Jim) have argued for precisely that approach yourself.

No?

Posted by: Urs Schreiber on April 23, 2009 10:51 AM | Permalink | Reply to this

Re: Categorification and Topology

I haven’t thought deeply about making the terminology consistent, not having yet written much about operadic categories. Jim’s comment that O O_\infty should refer quite generally to a resolution of the operad OO, though quite natural, had not occurred to me before. I think that the meaning of A A_\infty is so well-known that it would be silly to try to change it, which suggests that perhaps for consistency the associativity operad should be called AA (as you’ll notice I suggested above, anticipating your comment). I’m wary of mandating a particular unadorned letter to always indicate a specific object anywhere in mathematics, though, so I would be just as happy to regard the AA in A A_\infty as an abbreviation of AsAs or AssAss or “associative.”

By the way, a question for Jim or whoever may know: the AA in A A_\infty presumably came originally from “associative,” likewise the LL in L L_\infty came from “Lie”—where did the EE in E E_\infty come from?

Posted by: Mike Shulman on April 23, 2009 4:48 PM | Permalink | Reply to this

Re: Categorification and Topology

Mike Shulman wrote:

I’ve seen Ass.

Since I’m a bit of a joker, I don’t like using ‘Ass’ as a name for the operad whose algebras in Vect are associative algebras. It’s just too easy to make fun of sentences like “I’ve seen Ass”.

In fact, I like the term ‘AssociativeAssociative’ for the operad whose algebras in Vect are associative algebras. Similarly for ‘LieLie’ and ‘CommutativeCommutative’. For any operad OO, I call its algebras ‘OO-algebras’ — this is quite standard. Then I can state these theorem:

Thm. An AssociativeAssociative-algebra is an associative algebra.

Thm. A CommutativeCommutative-algebra is a commutative algebra.

Thm. A LieLie-algebra is a Lie algebra.

But as I said, I’m a bit of a joker.

Posted by: John Baez on April 23, 2009 7:30 PM | Permalink | Reply to this

Re: Categorification and Topology

Yes, the urge to abbreviate at times is not to be followed.

Posted by: jim stasheff on April 24, 2009 1:43 PM | Permalink | Reply to this

Re: Categorification and Topology

Hi Jim,

you replied to John:

I don’t like using ‘Ass’ as a name for the operad whose algebras in Vect are associative algebras.

Yes, the urge to abbreviate at times is not to be followed.

However unfortunate it may be, it seems to me that writing “AssAss” for the associative operad is entirely standard and widely adopted.

After all, you use it throughout your book

Markl, Shnider, Stasheff, Operads in topology and physics,

Martin Markl uses it in his further articles, in particular in

Markl, Models for operads

which seems to be the origin and source of the theorem that the dgdg-A A_\infty-operad is the free resolution of the linear associative operad, if I understand correctly.

For whatever it’s wroth: Google finds 141 references for “associative operad”+”ass”.

I don’t really care about how these things are called, but I am hoping we can at least eventually recount on the nnLab accurately the usual practice of what they are actually called, possibly related to ourr own proposal for what things should be called.

If you have a minute, I’d like to ask you to have a look at

nnLab: A-infinity-operad

on which I keep working. I am trying to list the relevant references with page and verse for the relevant theorems.

Please have a look and please complain about missing references or wrong statements etc.

Posted by: Urs Schreiber on April 25, 2009 1:06 PM | Permalink | Reply to this

Re: Categorification and Topology

Thanks. Skimming it I’m well pleased.
You might want to add a link to

math.univ-lyon1.fr/~chapoton/stasheff.html

where there is a rotatable image of an associahedron

and also to

math.univ-lyon1.fr/~chapoton/petitespages.html

for other hedras

Posted by: jim stasheff on April 25, 2009 2:20 PM | Permalink | Reply to this

Re: Categorification and Topology

Thanks, Jim.

rotatable image of an associahedron

I have now split off nnLab. associahedron from [[A-infinity-operad]] and included those links there.

I am hoping to eventually fill that with more detailed information than it currently has.

One question:

do I understand correctly that the associahedron K 4K_4 – the pentagon – corresponds precisely to the the 4th oriental?

And that it accordingly controls the associativity coherence law for monoidal categories, as discussed and illustrated at monoidal category?

It seems this must be clearly true, but I am just checking.

If so, how are the higher associahedra related to the higher orientals, i.e. the higher simplices?

Posted by: Urs Schreiber on April 25, 2009 3:15 PM | Permalink | Reply to this

Re: Categorification and Topology

Now that the link to the rotatable K_5 is available to the orientalists, I’ll let them decide. Judging from my interactions with Gordon (of Gordon, Power and Street),
I expect the answer is yes. Perhaps it’s obvious from your pictures, but, just to remind, the relevant in and out of the pentagon are the two paths from all parentheses accumulated at one end versus at the other

at one time known as perestroika ;-)

Posted by: jim stasheff on April 25, 2009 3:54 PM | Permalink | Reply to this

Re: Categorification and Topology

Then I can state these theorem:

Thm. An AssociativeAssociative-algebra is an associative algebra.

Thm. A CommutativeCommutative-algebra is a commutative algebra.

Thm. A Lie-algebra is a Lie algebra.

We could play this game seriously to the end and get some nice results that would solve the discussion we are having here:

first rename these categories as follows

VectLinearVect \mapsto Linear

TopTopologicalTop \mapsto Topological

SSetSimplicialSSet \mapsto Simplicial

and maybe

SetOrdinarySet \mapsto Ordinary

etc.

This way the usual habit of saying “linear category” for VectVect(-enriched) category and “simplicial category” for SSetSSet(-enriched) category would be precise.

next, use your joker-sytle convention for the names of operads, but add the information about the enriching category in the above style by saying

[enriching category]-[operad]-category.

then we could have

Thm. An OrdinaryAssociativeOrdinary-Associative-category is an ordinary associative category.

Thm. A LinearAssociativeLinear-Associative-category is a linear associative category.

Etc.

Next, for any operad [enriching category]-OO we should say homotopyhomotopy-[enriching category]-OO for its free resolution.

Then we’d have

Thm. A TopologicalhomotopyAssociativeTopological-homotopy-Associative-category is a topological homotopy-associative category.

Thm. A LinearhomotopyAssociativeLinear-homotopy-Associative-category is a linear homotopy-associative category.

Posted by: Urs Schreiber on April 25, 2009 1:55 PM | Permalink | Reply to this

Re: Categorification and Topology

SetOrdinarySet \mapsto Ordinary

Actually, we should rename SetSet to be called 11. And then CatCat should be called 22, and so on. All the way up to ωCat\omega Cat being called ω\omega.

Posted by: Mike Shulman on April 28, 2009 4:25 PM | Permalink | Reply to this

Re: Categorification and Topology

Simplicial complete Segal spaces and simplicial A A_\infty-categories are definitely morally (though not definitionally) the same. Same with topological ones.

The linear case is actually a lot trickier because it’s hard to say what you mean by a complete Segal space. In a (complete) Segal space, the “composition” operation comes from the composite X 1×X 1X 2X 1X_1\times X_1 \leftarrow X_2 \to X_1 where we assume the the backwards-pointing arrow is an equivalence, so we can invert it. Note that the backwards-pointing arrow is induced by the two face maps X 2X 1X_2 \;\rightrightarrows\; X_1. Therefore, it depends essentially on X 1×X 1X_1\times X_1 being a cartesian product. But when you talk about kk-linear categories, you want your composition operations to map out of the tensor product X(b,c)X(a,b)X(a,c).X(b,c)\otimes X(a,b) \to X(a,c). It is not clear how to get this with a Segalic structure, which I think is one reason that the people working in the kk-linear case use A A_\infty-categories so much.

(You can put in the necessary maps, such as X 2X 1X 1X_2\to X_1\otimes X_1, by hand. But now you have maps into a tensor product, which are really hard to describe formally and get a handle on homotopically, not to mention produce in examples.)

Posted by: Mike Shulman on April 19, 2009 4:45 PM | Permalink | Reply to this

Re: Categorification and Topology

Looks like a great workshop. I left it too late to register, and I was told there wasn’t any space left — very disappointing!

Jacob Lurie came to Oxford to give his talks on the homotopy hypothesis, and I talked to him a little bit about (,n)(\infty,n)-categories. He said there is essentially no problem in letting nn go to infinity, which then yields a theory of fully weak nn-categories. Is this pretty much the first full definition of weak nn-category?

Posted by: Jamie Vicary on April 7, 2009 9:11 AM | Permalink | Reply to this

Re: Categorification and Topology

He said there is essentially no problem in letting nn go to infinity, which then yields a theory of fully weak nn-categories.

Well, any theory of (,n)(\infty,n)-categories already contains a theory of nn-categories by looking at the (,n)(\infty,n)-categories where any two parallel kk-morphisms for k>nk\gt n are equal. So I assume you meant a theory of \infty-categories, and to that I would say “well, maybe.” The CSS approach is fundamentally a recursive definition, so you would have to define the collection of \infty-categories as some sort of limit of (,n)(\infty,n)-categories. Morally, it should work, but I’ll wait until I see the details before I get behind a statement like “there’s no problem.”

Is this pretty much the first full definition of weak nn-category?

No, there are plenty (and many can “do \infty” more directly than an iterative approach can.

Posted by: Mike Shulman on April 7, 2009 10:10 AM | Permalink | Reply to this

Re: Categorification and Topology

and many can “do \infty” more directly than an iterative approach can.

And does one have an idea to which of these direct approaches, if any, the iterative approach is asymptoting to?

Posted by: Urs Schreiber on April 7, 2009 10:33 AM | Permalink | Reply to this

Re: Categorification and Topology

And does one have an idea to which of these direct approaches, if any, the iterative approach is asymptoting to?

Morally, of course they are all asymptoting to the same thing, namely weak ω\omega-categories, which is the same thing that all the direct approaches are trying to define. On a technical level it doesn’t look to me as though the nonalgebraic (Segal nn-categories, complete nn-Segal spaces) iterative approaches are asymptoting to any direct approach. On the other hand one can make precise sense out of the Trimble-May operadic approach asymptoting to the Batanin theory of higher operads. And if you call Rezk’s new definition iterative (which it sort of is), then it is evidently asymptoting to its own ω\omega-version.

Posted by: Mike Shulman on April 7, 2009 10:09 PM | Permalink | Reply to this

Re: Categorification and Topology

Thanks for that, Mike! The last two nns should indeed have been \inftys, I’m afraid those sort of mistakes invariably creep in when I post in my pyjamas.

Is this pretty much the first full definition of weak nn-category?

No, there are plenty (and many can “do \infty” more directly than an iterative approach can.

Right, thanks, I should have known about Tom Leinster’s survey article. Is the status of the homotopy hypothesis known for any of these definitions of weak nn-category? (I can’t find a mention of it in Tom’s article.) Presumably it goes through for Lurie’s weak nn-categories, since it forms the foundation for the entire approach.

Posted by: Jamie Vicary on April 7, 2009 10:40 AM | Permalink | Reply to this

Re: Categorification and Topology

There is an appoach of weak (,n)(\infty,n)-categories which uses (simplicial) presheaves on the category Θ n\Theta_n (where Θ n op\Theta^{op}_n is Joyal’s category of nn-disks): this is done by Charles Rezk in this paper. Rezk’s approach is a “complete Segal space”-like one and works directly with 1n1\leq n\leq\infty. The fact we still have the homotopy hypothesis in this setting is essentially the contents of Clemens Berger’s paper “A Cellular nerve for higher categories”, Adv. Math. 169 (2002), 118-175; an elementary proof of this comes from the fact that the categories Θ n\Theta_n are test categories in the sense of Grothendieck (see Section 1.8 there). There is also a quasi-category-like version of this (which leads to an equivalent theory), and this theory is also (Quillen) equivalent to the theory of nn-Segal categories (but this has still to be written down).

Posted by: Denis-Charles Cisinski on April 7, 2009 1:50 PM | Permalink | Reply to this

Re: Categorification and Topology

The fact we still have the homotopy hypothesis in this setting is essentially the contents of Clemens Berger’s paper “A Cellular nerve for higher categories”… an elementary proof of this comes from the fact that the categories Θ n\Theta_n are test categories in the sense of Grothendieck…

Thanks! I didn’t realize it was established in this case. For me, the crucial thing to say that “the homotopy hypothesis holds” is section 11.25 of Rezk’s paper, where he gives an explicit localization of his model category of (n+k,n)(n+k,n)-categories, which models (n+k)(n+k)-types, and whose fibrant objects are the groupoidal ones. The homotopy hypothesis doesn’t just say that there is some localization of nn-categories that models nn-types (which I think is all that the Berger and Grothendieckian model structures say), but that it is specifically the nn-groupoids. But please correct me if I am wrong.

Posted by: Mike Shulman on April 7, 2009 10:20 PM | Permalink | Reply to this

Re: Categorification and Topology

You are right: we want precisely that nn-groupoids classify nn-homotopy types. But in the Grothendieckian point of view, to see that the “complete Segal space”-like model structure (or the “quasi-category”-like model structure) satisfies the homotopy hypothesis, one can check that the Bousfield localization which forces the fibrant objects to be groupoid-like is obtained by inverting the maps between representable presheaves: this latter property is essentially the description of the Grothendieckian model structures (which model homotopy types). Hence, to be very precise, the homotopy hypothesis in its precise form follows here from the conjunction of the fact Θ n\Theta_n is a (regular) test category and from (characterization (3) of) Proposition 11.26 in Rezk’s paper.

Posted by: Denis-Charles Cisinski on April 8, 2009 11:46 PM | Permalink | Reply to this

Re: Categorification and Topology

Is the status of the homotopy hypothesis known for any of these definitions of weak n-category?

Well…

  • It’s true for Street’s definition, because his ω\omega-groupoids are precisely Kan complexes. This includes quasicategories as (,1)(\infty,1)-categories.

  • We had a conversation (somewhere on the cafe which I’m too lazy to find) about how Trimble’s definition ought to work fairly easily since it is designed with that in mind, but I don’t know if anyone has written it down.

  • Michael Batanin has done some work on fundamental ω\omega-groupoids for the operadic versions, but I don’t remember whether he has a full proof yet.

  • For Joyal’s definition, there is some progress, cf Clemens Berger “A cellular nerve for higher categories,” which also has bearing on the higher-operadic approach.

  • I think that something is known for the Simpson/Tamsamani approach, but I’m too lazy to look it up right now. In the (,1)(\infty,1)-case Julie Bergner has a proof.

  • As usual, I think no one knows anything about opetopes.

So, something is known in some cases, but as far as I know it hasn’t been completely verified in any nontrivial case.

Presumably it goes through for Lurie’s weak n-categories, since it forms the foundation for the entire approach.

I don’t think that’s obvious at all. It’s certainly true that Lurie’s definition of (,n)(\infty,n)-category, when specialized to n=0n=0, does give spaces by definition. But the full homotopy hypothesis would relate also to (,n)(\infty,n)-categories in which all kk-morphisms are invertible for all kk; we are accustomed to regard these as the same as (,0)(\infty,0)-categories, but in Lurie’s iterative approach they are literally different and must be proven to be equivalent.

Posted by: Mike Shulman on April 7, 2009 2:00 PM | Permalink | Reply to this

Re: Categorification and Topology

Jamie wrote:

Looks like a great workshop. I left it too late to register, and I was told there wasn’t any space left — very disappointing!

That’s too bad! Can’t you sneak in the back door?

Is this pretty much the first full definition of weak n-category?

Far from it — and worse, it won’t be the last, either.

The first definition of weak nn-category is actually Street’s definition of a simplicial weak \infty-category, which dates back to 1987! But he was so modest about it that few people took it very seriously until much later — though those few included Dominic Verity, who took it very seriously.

The second definition of weak nn-category was the opetopic one, in 1997. This seems to have opened the floodgates: apparently all the experts thought “if those two upstarts can dream up a definition, so can I!” By 2001 there were more definitions than you can shake a stick at. In particular, Makkai has a definition of opetopic weak \infty-category, which he is proposing as a new foundations for mathematics.

At this point I advocate a government subsidy that pays mathematicians to not invent new definitions of weak nn-category — sort of like how the US government paid farmers to not grow wheat.

Of course the analogy is flawed: wheat, at least, you can eat.

Posted by: John Baez on April 7, 2009 5:07 PM | Permalink | Reply to this

Re: Categorification and Topology

Thanks Denis-Charles and Mike! Sorry for the slow reply, there’s been a workshop on here. From this great little survey of Mike’s, it seems that despite all the effort that’s been made, the homotopy hypothesis has still only been proven in those cases — Street’s ω\omega-groupoids, and (,1)(\infty,1)-categories — where it’s relatively easy to demonstrate. Is this fair?

Mike wrote:

Presumably it goes through for Lurie’s weak nn-categories, since it forms the foundation for the entire approach.

I don’t think that’s obvious at all … the full homotopy hypothesis would relate also to (,n)(\infty,n)-categories in which all kk-morphisms are invertible for all kk

Ah, good point, thanks!

John suggested:

That’s too bad! Can’t you sneak in the back door?

Crikey! I’m not sure if I’d be capable of that.

John also wrote:

The second definition of weak nn-category was the opetopic one, in 1997.

Right! So, do you still defend these opetopic nn-categories tooth-and-nail? Or has this formulation been superseded by something more modern?

Finally: let’s say somebody wanted to have a first stab at building an ω\omega-category of their very own, and playing around to see if they can do anything interesting with it. What flavour of ω\omega-category would the crowd recommend as the best to get stuck into?

Posted by: Jamie Vicary on April 9, 2009 11:05 PM | Permalink | Reply to this

Re: Categorification and Topology

Jamie wrote:

Right! So, do you still defend these opetopic nn-categories tooth-and-nail? Or has this formulation been superseded by something more modern?

I never defended them tooth-and-nail. That conjures up the image of a mother bear. I was more like a snake that lays its eggs and slithers away, leaving the poor baby snakies to fend for themselves. Jim did a lot more work with them, but he rarely writes anything up.

Anyway, almost everyone who fell in love with opetopic nn-categories felt the need to rework the foundations of the subject in their own way. The same basic structures, the opetopes, arise from many different viewpoints:

A few people understand how everyone’s work fits together, but it’s a bit of a wriggly mess. Eugenia Cheng has done the best job of straightening things out:

But perhaps the grandest vision of the whole subject — apart from Jim’s — comes from Makkai:

Actually, I urge you to start with this paper:

or at least the first four sections, which explain the philosophy behind opetopic (or multitopic) nn-categories, and the simple but fascinating notion of virtual functor.

Posted by: John Baez on April 9, 2009 11:31 PM | Permalink | Reply to this

Re: Categorification and Topology

Any thoughts about what’s being done to your periodic table over here?

Posted by: David Corfield on April 7, 2009 4:37 PM | Permalink | Reply to this

Re: Categorification and Topology

I prefer to think they’re doing something with the periodic table. Doing something to it sounds like they’re poking the poor thing with a sharp stick.

They’re digging deeper into Michael Müger’s idea that it’s really interesting to iterate the ‘generalized center’ construction that pushes you one notch down the periodic table. Check out page 2 of his paper for a simple intro to this notion.

Posted by: John Baez on April 7, 2009 5:24 PM | Permalink | Reply to this

Re: Categorification and Topology

Is it time to admit that the k=0k = 0 row of the periodic table really consists of pointed sets, categories, etc, as described in the appendix to your paper with Mike? Or are you not convinced of that?

Posted by: Toby Bartels on April 7, 2009 11:01 PM | Permalink | Reply to this

Re: Categorification and Topology

They concluded:

So we have two different periodic tables, and it isn’t that one is right and one is wrong, but rather that one is talking about monoidal structures (or equivalently, by the delooping hypothesis, pointed and connected things) and the other is talking about connectivity. Note that unlike the monoidal periodic table, the connectivity periodic table does not stabilize. (p. 54)

Posted by: David Corfield on April 8, 2009 8:54 AM | Permalink | Reply to this

Re: Categorification and Topology

But John's not talking about the connectivity table; he's talking about a table that stabilises. For that matter, even the connectivity table has a condition in the k=0k = 0 row (that the nn-category be inhabited, that is non-empty).

Posted by: Toby Bartels on April 9, 2009 9:34 AM | Permalink | Reply to this

literal table that stabilises; Re: Categorification and Topology

My brain, in an oddly literal mode, read: “talking about a table that stabilises” and thought about How to stabilize a wobbly table.

Posted by: Jonathan Vos Post on April 9, 2009 7:17 PM | Permalink | Reply to this

Re: Categorification and Topology

Toby wrote:

Is it time to admit that the k=0k=0 row of the periodic table really consists of pointed sets, categories, etc, as described in the appendix to your paper with Mike? Or are you not convinced of that?

There are many alternatives besides not being convinced of something and being time to admit that it’s true… especially when it comes to 1-hour talks.

Posted by: John Baez on April 10, 2009 1:15 AM | Permalink | Reply to this

Re: Categorification and Topology

There are many alternatives besides not being convinced of something and being time to admit that it’s true… especially when it comes to 1-hour talks

Maybe I should ask then, is it time to stop spreading around a version of the table that you know is wrong … if you are convinced that it's wrong. A 1-hour talk doesn't have to explain the reasons for the correction, or even point out the correction, just include it.

Posted by: Toby Bartels on April 11, 2009 1:48 AM | Permalink | Reply to this

Re: Categorification and Topology

Something which bothers me about Lurie’s formulation of the Cobordism Hypothesis, which is also the one which David Ben-Zvi employs, is that he really combines the Extended TQFT Hypothesis and the Cobordism Hypothesis into one.

I think it’s best to keep them separate. Do you agree John (or Jim)? I’m taking my cue from you here, both on these recent slides as well as on previous comments on the café I’ve read of yours.

Let’s have a discussion about this!

To a certain degree, John and Jim are to blame on this, because they originally gave these hypotheses awkward names, calling them the Extended TQFT Hypotheses I and II:

Extended TQFT Hypothesis, Part I. The nn-category of which nn-dimensional extended TQFTs are representations is the free stable weak nn-category with duals on one object.

Extended TQFT Hypothesis, Part II. An nn-dimensional unitary extended TQFT is a weak nn-functor, preserving all levels of duality, from the free stable weak nn-category with duals on one object to nnHilb.

The way I would like these hypotheses to be renamed, taking my cue from what John has written, is as follows:

Cobordism Hypothesis. The nn-category of cobordisms nnCob is the free stable weak nn-category with duals on one object.

Extended TQFT Hypothesis. An nn-dimensional unitary extended TQFT is a weak nn-functor, preserving all levels of duality, from the nn-category of cobordisms nnCob to nnHilb, the nn-category of nn-Hilbert spaces.

Here is my reasoning (if John or Jim disagree, I stand corrected!). We should disentangle the Cobordism Hypothesis from any concept having to do with “TQFT”. It should be viewed simply as a statement about manifolds, namely that “n-categories with duals are the perfect language to describe manifolds and their cobordisms”.

The Extended TQFT Hypothesis (as I wrote it above) should be regarded as hypothesis about what an “extended TQFT” really is, in a mathematical sense. I think this underscores an important point about TQFT, which I have noticed sort of makes a dividing line between two views about what a TQFT is, in a sense.

In the one camp, are those that believe that an extended TQFT is a pre-existing gismo, coming from physics and path integrals and sigma-models and quantum gravity etc., and our task is to describe it mathematically. So when we say “An extended TQFT is…”, it’s not so much of a definition of a concept which was not defined beforehand, but rather an attempt to formalize that concept.

I think there might be another camp, where the definition “An extended TQFT is…” is taken basically at face value, and that concept is not regarded as having any apriori meaning (eg. from physics). In that camp, the Extended TQFT Hypothesis as I have outlined it above is in a sense meaningless… it would be thought of as a definition.

I am in the first camp, and the reason is it is a huge thing to claim that this “idea” of an extended TQFT, which basically means “a topological quantum field theory which satisfies the best possible locality properites”, can be formalized in terms of this weird mumbo-jumbo about higher categories and so on. Many physicists certainly are not convinced of that notion (I’ve heard Witten scoffs at it, for example!). So it should be viewed as a genuine hypothesis. The only way we could prove it would be to show that it is useful to think of extended TQFT’s in that way.

If we combine the Cobordism Hypothesis and the Extended TQFT Hypothesis above together, we get something which I like to call “the primacy of the point”… and this is (almost) the thing which Lurie calls “The Cobordism Hypothesis”:

The ‘primacy of the point’. An nn-dimensional unitary extended TQFT is completely described by the nn-Hilbert space it assigns to a point.

I say “almost” above because the feature I loved the most about the original Baez-Dolan version was that word “unitary”, and hence also the idea of an “nn-Hilbert space”. Lurie’s version doesn’t seem to pay as much homage to these concepts as I would like, which bothers me, but I guess I must just read harder, which I am.

The Cobordism Hypothesis. (Lurie). Let CC be a stable (,n)(\infty, n)-category. Then there is a bijection between equivalence classes of stable (,n)(\infty, n)-functors

(1)Z:nCobC Z : n {Cob} \rightarrow C

and equivalence classes of fully dualizable objects aCa \in C.

Anyhow, let’s discuss this thing.

Posted by: Bruce Bartlett on April 22, 2009 5:33 PM | Permalink | Reply to this

Re: Categorification and Topology

If an extended TQFT is not defined as a functor, then what is it? Are there any other competing precise definitions? Can you or anyone give any intuition for what the physicists think it “should” be?

Posted by: Mike Shulman on April 22, 2009 7:39 PM | Permalink | Reply to this

Re: Categorification and Topology

Physics intuition is all about the path integral, but that is of course a horribly ill-defined object (modulo the fact that many TQFTs localize onto something one can deal with).

I think we clearly have enough understanding to say that the n-categorical formulation of TQFTs capture a substantial part of the information in the physics formulation with its action functionals and symmetry, but the physics at the very least motivates techniques which don’t seem obvious from the n-category point of view and may contain things which are not there in the n-categories. Dan Freed emphasizes this in the final section of 0808.2507 for example.

Posted by: Aaron Bergman on April 22, 2009 10:02 PM | Permalink | Reply to this

Re: Categorification and Topology

the physics at the very least motivates techniques which don’t seem obvious from the n-category point of view and may contain things which are not there in the n-categories. Dan Freed emphasizes this in the final section of 0808.2507 for example.

My impression is that the required concept here is that of an extended QFT which is a σ\sigma-model.

There should be large classes of examples of (,n)(\infty,n)-functors on nCobn Cob which are represented (in the sense of representable functors) by the data given by an nn-bundle with connection on some \infty-stack (target space).

For Chern-Simons theory the target space is BGB G and the nn-bundle with conneciton on it is the Chern-Simons 2-gerbe with connection which obstructs the lift of the universal GG-bundle with its universal connection to a String GString_G-2-bundle with connection.

That formula (6.5) in Freed’s article should be seen, I think, as an aspect of this: the assignment of (,n)(\infty,n)-QFT functors that arise as σ\sigma-models are determined by the differential cohomological data encoded in that background nn-bundle with connection.

Posted by: Urs Schreiber on April 23, 2009 9:45 AM | Permalink | Reply to this

Re: Categorification and Topology

If an extended TQFT is not defined as a functor, then what is it? Are there any other competing precise definitions? Can you or anyone give any intuition for what the physicists think it “should” be?

Physicists don’t usually think about what something “should” be in the sense of formalizing the object under study by its stuff, structure and properties.

It’s a different way of thinking there. Physicists usually proceed mostly “operationally”. By many years of experience they have learned that starting from writing down an “action functional” one can construct lots of interesting expressions. Few to none thoughts are usually spent on which structure all these expressions form, one usually treats them by following tradition, intuition and the way formulas tend to lead to formulas as one writes them on paper.

I think that it is important, for both physicists and for pure mathematicians, to realize that an important task in beween the study of math and physics is very different from what many people want to understand under “Mathematical Phyisics”, which they just take to be the task of rigorously proving computations seen in physics: this important task is mathemtical modelling.

For many of the most interesting subjects in theoretical physics, it’s not that the formalism at issue is clear and that physicists just need a hand in helping them make a well-defined computation rigorous. For the most interesting issues it is the formalization of what’s going on in the first place that is still needed. The question is: which mathematical structures do naturally describe what they are talking about? What is the right natural languge?

An example of this is what Atiyah and Segal did back then when they came up with the idea that QFTs should be functors on cobordisms: they listened to what physicists were talking about, and thought about what that might actually mean when formalized. They essentially noticed that what physicsts call “locality”, “sewing law of the path integral” and the like is nothing but “functoriality”. This is not an insight of the kind one should have expected the physicists at that time (or today) to have come up with or even cared about in the first place. The question to them: “If not a functor, then what?” would just lead to shrugguing and at best the answer: “Well, quantum field theory as described in our textbooks.”

(In closing, one final remark: there is one other mathematical formalization of what QFT is, that’s called AQFT. )

I notice that, as far as I know, Lawvere was led to much of what he was led to by thinking about what the true natural mathematical formulation of classical continuum mathematics should be.

What we need today is the Lawvere-ification of quantum field theory.

Posted by: Urs Schreiber on April 23, 2009 9:26 AM | Permalink | Reply to this

Re: Categorification and Topology

Bruce, perhaps not surprisingly, I disagree, or maybe just don’t understand your point - after all Jacob’s result is precisely the cobordism hypothesis (and tangle hypothesis, and their generalizations with singularities) that you state. You can then look at functors with values in n-Hilb and emphasize unitarity, or look at functors with other kinds of values that arise also naturally from the physics. There’s certainly a lot to be gained from studying non-unitary field theories and theories with values other than (n-)vector spaces, even from a pure physics POV.. I also don’t see how functors to n-Hilb are the right objects to capture the most typical examples of TFTs, which come by topologically twisting supersymmetric QFTs - for example the topological B- and A-models in 2d, Rozansky-Witten theory in 3d, or twisted N=2 or N=4 SYM in 4d (Donaldson theory and geometric Langlands) – but I also don’t understand n-Hilbert spaces (you explain in your thesis that Costello’s work can be thought of as working with a derived version of 2-Hilbert spaces, but I’m not sure I see the unitary structure?) For these theories Jacob’s formalism appears to fit beautifully, and flexibility in what we take as the target category seems essential. Of course if you had a functor from a cobordism category to something totally nonlinear (eg to itself) that’s a TFT in a very abstract and (probably?) nonphysical sense, but as Mike Shulman and Jim Stasheff were arguing with the A A_{\infty} category terminology, I don’t see what’s wrong with keeping that name as long as the context makes it clear what we’re talking about..

(I also don’t at all get the feeling that Witten scoffs at this formulation - I think he’s not so interested in abstract formulations but rather in what one can learn about physics and math.. Kapustin OTOH does exactly talk about topological field theories from exactly this kind of POV. If we want to talk about scoffing, TOPOLOGICAL field theory - one with no dynamics! - is already itself a ridiculous toy model for most physicists, so we might as well just enjoy it without too much worry..)

Posted by: David Ben-Zvi on April 22, 2009 7:48 PM | Permalink | Reply to this

Re: Categorification and Topology

David wrote:

…or maybe just don’t understand your point - after all Jacob’s result is precisely the cobordism hypothesis…

Gosh, you’re right, sorry. Bruce slaps his head against his forehead. I had confused the actual result of Jacob Lurie with some weird amalgam that had formed in my mind; it’s a long story. I see it now, Jacob Lurie’s statement is precisely the cobordism hypothesis.

I think.

The only thing I’m still a bit worried about is that word “unitary”. If I read through Jacob Lurie’s proof of the cobordism hypothesis, in Section 2.3 he deals with the idea of duality for objects and morphisms in higher categories. It seems to boil down to this simple formulation: an object or morphism has a dual if it has both a left and a right dual. These left and right duals are not required to be related in any way. We require our (,n)(\infty, n)-category to have duals at all levels.

I find that very surprising; I thought the cobordism hypothesis had to do with duality in a deep and profound way; that to prove it one would need to formulate strong relationships between the duals at all levels. For instance, that they should somehow satisfy coherence laws like the swallowtail law which couples the left and right duals and the “interchanger”, and so on.

On the other hand, perhaps that is the power of the (,n)(\infty, n)-way of doing things, and I’ve just got to deal with it. Somehow the machinery does all that for you. If so, it’s incredible… but then somehow I would still feel a bit cheated, as if all my favourite parts were now relegated to stuff that happened “under the hood”.

Do I have a subtle (or not so subtle, which is the more common with me) misconception here?

I also don’t see how functors to n-Hilb are the right objects to capture the most typical examples of TFTs,

You’re right, nn-Hilb is not the right notion in general, and basically anywhere other than Chern-Simons theory you need to use other target higher categories. I would feel happier if the duality properties of those target higher categories were stressed more, but I guess that is already implicit in the statement of the cobordism hypothesis (the target higher category has to be capable of housing a “fully dualizable object”).

Aaron Bergman wrote:

Dan Freed emphasizes this in the final section of 0808.2507 for example.

Yes I was thinking about those remarks he made when I made my remark about physicists not always warming to the higher-categorical extended field theory point of view (of course he is not a physicist, but anyhow). I think Freed is very right when he says that our inability so far to prove the most basic fact about Chern-Simons TQFT, namely that the invariants it calculates take the form

(1)Z k(X)12e 3πi/4 𝒜 x 0e 2πiS X(A)(k+2)e 2πI A/4τ X(A) Z_k (X) \sim \frac{1}{2} e^{-3\pi i/4} \sum_{\mathcal{A} \in \mathcal{M}^0_x} e^{2 \pi i S_X (A) (k+2)} e^{-2 \pi I_A / 4} \sqrt{\tau_X(A)}

is an indictment on all our parading of higher-categorical treatments of extended TQFT. (By the way, to understand what each of those terms is above, look at the paper by Freed, I certainly can’t explain them all!)

I guess a higher-categorical proof of the above equation would go as follows: present your manifold XX as a complicated gluing of lower dimensional manifolds, right down to a point. Then “read off” what the invariant should be, from higher-categorical mumbo-jumbo. (i.e. “the invariant is the unit of the left adjunction coupled to the right dual of the weak transformation of the …”) and see if you reproduce the formula! If higher-categorical technology will never lead to such a proof, I also find it a bit disappointing.

Posted by: Bruce Bartlett on April 22, 2009 11:18 PM | Permalink | Reply to this

Re: Categorification and Topology

I just want to add that I’ve read a bit more of “On the classification of topological field theories (draft)” now and it’s really impressive, formiddable. Sometimes just a paragraph there corresponds to a major insight, a paper all on its own. I look forward to the expository articles which will help with understanding it! Did anyone at the n-category cafe perhaps take notes of Ben Hummon’s talk?

Posted by: Bruce Bartlett on April 23, 2009 1:14 AM | Permalink | Reply to this

Re: Categorification and Topology

(I’ve heard Witten scoffs at it, for example!)

I have heard of somebody who asked him: “So, if not higher categories, what do you think will in the end formalize quantum field theory?”. The reply was apparently: “Quantum Field Theory”.

I have also seen slides somewhere, where Witten tried to convince a mathematical audience that one shouldn’t think so much of groupoids, but better of Quantum Field Theory. :-)

(Well, I should better dig out my sources for this, so take this as anecdotal evidence, likely inaccurate, for the moment.)

Posted by: Urs Schreiber on April 23, 2009 9:01 AM | Permalink | Reply to this

Re: Categorification and Topology

Witten’s latest (from the Bott conference) not only does not scoff at extended TFT, it begins with a review of it!

Posted by: David Ben-Zvi on May 19, 2009 3:45 AM | Permalink | Reply to this

Re: Categorification and Topology

Posted by: David Corfield on May 19, 2009 9:13 AM | Permalink | Reply to this

Re: Categorification and Topology

Thanks, David and David!

It’s interesting to see the return to the expected nonabelian gerbe on the 5-brane worldvolume (section 4). That had originally been the main motivation for nonabelian gerbes applied to string theory.

(I mean, before it became clear that an orientifold gerbe is already an example of a nonabelian gerbe, as well as the String-gerbe and its cousins in the global description of the Green-Schwarz mechanism.)

Posted by: Urs Schreiber on May 19, 2009 9:45 AM | Permalink | Reply to this

Re: Categorification and Topology

Witten’s latest (from the Bott conference) not only does not scoff at extended TFT, it begins with a review of it!

Exciting news! The game’s up.

Posted by: Bruce Bartlett on May 19, 2009 4:22 PM | Permalink | Reply to this

Re: Categorification and Topology

Kapustin, Setter and Vyas use higher category theory in Surface operators in four-dimensional topological gauge theory and Langlands duality.

Posted by: David Corfield on February 5, 2010 12:45 PM | Permalink | Reply to this

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