Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

November 12, 2007

Geometric Representation Theory (Lecture 10)

Posted by John Baez

In my last lecture, I explained that when we simultaneously wave the magic wands of qq-deformation and categorification over the humble binomial coefficient

(nk)=n!k!(nk)!\binom{n}{k} = \frac{n!}{k! (n-k)!}

it transforms into a marvelous thing: the Grassmannian of kk-dimensional subspaces of F q nF_q^n, where F qF_q is the field with qq elements.

This time in the Geometric Representation Theory seminar, I sketch what happens when we work the same magic on the binomial formula

(x+y) n= k=0 n(nk)y kx nk (x + y)^n = \sum_{k = 0}^n \binom{n}{k} y^k x^{n-k}

We’re soon led into deep waters: categorified quantum groups!

To qq-deform Pascal’s triangle amounts to putting it in a constant magnetic field, so a little electrically charged ball rolling down from the apex to a given point in the triangle picks up a phase depending on the path it takes. The qq-binomial coefficient

(nk) q=(n) q!(k) q!(nk) q!\binom{n}{k}_q = \frac{(n)_q!}{(k)_q! (n-k)_q!}

is then the sum of these phases over all paths from the apex to the kkth slot in the nnth row of the triangle. This is a baby version of a path integral.

If we use xx to denote the process of rolling one step down and to the left, and yy for rolling one step down and to the right, we have

xy=qyxx y = q y x

So, these variables satisfy the qq-deformed binomial formula:

(x+y) n= k=0 n(nk) qy kx nk (x + y)^n = \sum_{k = 0}^n \binom{n}{k}_q y^k x^{n-k}

We can think of xx and yy as coordinates on the ‘quantum plane’ — a mysterious object from the land of noncommutative geometry. The symmetries of the quantum plane are then the ‘quantum group’ GL q(2,k)GL_q(2,k). So, if we succeed in categorifying the qq-deformed binomial formula, we should be well on our way towards categorifying this quantum group!

But, to really explain all this, I needed to review the basics of noncommutative geometry. And then I needed to pose the question: what do the variables xx and yy really mean here? And — a closely connected question — how do we categorify them?

  • Lecture 10 (Oct. 30) - John Baez on the qq-deformed Pascal’s triangle and the quantum group GL q(2,k)GL_q(2,k). Putting Pascal’s triangle in a magnetic field, we obtain the qq-deformed Pascal’s triangle. Now the operation of moving down and to right (called xx) and the operation of moving down and to the left (called yy) no longer commute, but instead satisfy:

    xy=qyxx y = q y x

    This relation implies the qq-deformed binomial formula:

    (x+y) n= k=0 n(nk) qy kx nk (x + y)^n = \sum_{k = 0}^n \binom{n}{k}_q y^k x^{n-k}

    Picking a field kk, the ‘algebra of functions on the quantum plane’, k q[x,y]k_q[x,y], is the associative algebra over kk generated by variables xx and yy satisfying the relation xy=qyxx y = q y x. The symmetries of the quantum plane form the quantum group GL q(2,k)GL_q(2,k) The basic philosophy of algebraic geometry. The functor from geometry to algebra. Noncommutative geometry as a mutant version of algebraic geometry. Hopf algebras, and how they ‘coact’ on algebras.

    A sketch of how we’ll simultaneously qq-deform and categorify the following structures:

    • binomial coefficients (to obtain Grassmanians)
    • the variables xx and yy showing up in the binomial theorem (to obtain certain Hecke operators)
    • the group GL(2,k)GL(2,k) (to obtain a categorified version of the quantum group GL q(2,k)GL_q(2,k))





In case you’re wondering, I’m writing the binomial formula in this funny way:

(x+y) n= k=0 n(nk)y kx nk (x + y)^n = \sum_{k = 0}^n \binom{n}{k} y^k x^{n-k}

because of slightly suboptimal conventions I chose concerning the qq-deformed Pascal’s triangle. Actually, all the conventions I could think of seemed slightly suboptimal one way or another. But it’s no big deal.

Posted at November 12, 2007 8:52 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1495

23 Comments & 1 Trackback

Re: Geometric Representation Theory (Lecture 10)

Now the operation of moving down and to right (called xx) and the operation of moving down and to the left (called yy) no longer commute, but instead satisfy: xy=qyx x y = q y x

You’d think that the q=0q = 0 deformation would be quite simple. The contribution of a path in which there is left move followed by a right is 00. There’s only one path between the apex and any given slot, so all entries of the deformed Pascal’s triangle are 11.

Then concerning your earlier question about qq-deformed Gaussians, in the q=0q = 0 case we’d have the limit of a uniform distribution as the range increases.

One could easily be led to believe that q=0q = 0 is a bit boring. But then crystal bases and free probability seem rich enough. What, for instance, does the latter have to do with noncrossing partitions?

Posted by: David Corfield on November 13, 2007 9:23 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

Yeah, the qq-deformed Pascal’s triangle seems pretty dull at qq = 0 when you look at it this way!

Maybe there’s some interesting way to take the q0q \to 0 limit while rescaling???

Posted by: John Baez on November 13, 2007 8:17 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

With all the integers [n] 0[n]_0 equal to 1, and hence all the factorials, it’s hard to see where the excitement comes from.

If you’ve got quantum groups GL q(2,k)GL_q(2, k) acting on quantum planes, perhaps we could see there how something interesting happens as q0q \to 0. Or is it only the q=0q = 0 limit of deformations of universal enveloping algebras which are interesting?

Posted by: David Corfield on November 15, 2007 9:14 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

Can we take it that, eleven years on, the Seminar is expanding on this comment?

Ordinarily quantization is conceived of in terms of a deformation of a commutative algebra to give a noncommutative algebra (the deformation parameter being Planck’s constant). Quantum groups were a big suprise, and here one has a deformation of a symmetric monoidal category (the category of representations of your group) to a braided monoidal category (the category of representations of the corresponding quantum group). The same sort of thing, in other words, but one step up the n-categorical ladder. The crystal basis/canonical basis stuff suggests that there is also a kind of deformation of a symmetric monoidal 2-category to a braided monoidal 2-category going on! But this is not just a further prolongation of the same sort of pattern, since 1) if it were, we would be deforming a strongly involutory 2-category to a weakly involutory one, and 2) as far as I know, the canonical basis stuff only works when the deformation parameter qq is a root of unity, so the sense in which we have a “deformation” is subtler.

Posted by: David Corfield on November 13, 2007 10:11 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

David wrote:

Can we take it that, eleven years on, the Seminar is expanding on this comment?

Yes! I’m glad someone is paying such careful attention to what I say. I’ve been trying to understand quantum groups more deeply ever since they came out… and understanding them seems to involve understanding why they have an ‘extra categorical dimension’ beyond what one would naively expect, involving braided monoidal 2-categories instead of braided monoidal categories.

By now this is a big business, often called ‘Khovanov homology’. But, I think we still need to work harder to understand it! From a lot of stuff one reads, Khovanov homology still feels like a ‘trick’. But I’m claiming it’s not; you can begin to see its traces just from thinking hard about basic entities like Pascal’s triangle.

Interestingly, the passage you quote is my reply to this question from Phil Gibbs:

Does this suggest that the process of quantisation might take us one step further up the categorical ladder than previously thought?

By now it’s clear the answer is yes, and the phenomenon that explains this is called ‘groupoidification’. Urs has written eloquently on this ‘one step further up’… and the geometric representation theory is headed in this direction. But I still feel it’s mysterious.

Posted by: John Baez on November 13, 2007 8:35 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

Urs has written eloquently on this ‘one step further up’ […]. But I still feel it’s mysterious.

I also still feel it is mysterious – but to me it now feels much less mysterious than it did a while ago. In other words, I got the feeling that I am understanding at least some basics underlying it.

Let me maybe recall the latest version of how I think about what is going on:

That Shift in Dimension™ which we encounter all over the place, in different guises, is essentially the result of a kind of obstruction problem which is very fundamental:

The obstruction to some nn-thing being flat is itself an (n+1)(n+1)-thing: its curvature.

And the point is: here I am using the word “obstruction” in a very technical, very systematic sense.

This is what I think obstruction theory is about, and the way it ought to be conceived:

it is about having an exact sequence of nn-things in some context ABC A \to B \to C and having a map into CC A B C P \array{ A &\to& B &\to& C \\ &&&& \uparrow \\ &&&& P } and asking whether this map can be lifted to a map to BB A B C dl P. \array{ A &\to& B &\to& C \\ &&&{}_{dl}\nwarrow& \uparrow \\ &&&& P } \,.

Here “dl” is for “desired lift”.

What is the systematic way to ask for the obstruction for this lift to exist? The systematic way uses weak cokernels – and that’s where the shift in dimension appears.

And there is a nice story to be told as the formalism proceeds: it’s a story of weakness and of equivalence, and of things holding almost, up to something else. So therefore it is a story in the world of nn-categories.

The thing is this: nn-categories offer a nice way to say

“We can try to lift the map, but we might make a mistake thereby. However, the kinds of mistakes we can possibly make are rather well behaved.”

All this is encoded in the following technical statement:

regard the above diagram of nn-things as a diagram of (n+1)(n+1)-things. Then the desired lift always exists in a weak sense. And then the obstruction problem amounts to constructing the weak lift and checking if it does restrict to a strict lift.

So it’s really a big tautology, in a sense, but formalized nicely enough to be actually useful.

So this is how it works: we form the cokernel of ABA \to B in the context of (n+1)(n+1)-things. That’s the weak cokernel of ABA \to B (or “homotopy quotient”, if you like). Since its definition turns out to be somewhat tautologous, also its denotation is somewhat tautologous: I call this weak cokernel itself (AB)(A \to B). So

AB(AB) A \to B \to (A \to B)

is an exact sequence now. And since the previous sequence was exact, too, we get a universal morphism from (AB)(A \to B) onto CC. And in nice cases this is actually an equivalence, in a suitable way. So we get (AB) al A B C dl P. \array{ &&&& (A\to B) \\ &&&\nearrow & \uparrow^{al} \\ A &\to& B &\to& C \\ &&&{}_{dl}\nwarrow& \uparrow \\ &&&& P } \,.

Here “al” is for “attempted lift”. This attempt always exists, but it need not be successful!

But we can systematically measure the failure of it being succesful: that’s simply what the map to the cokernel of B(AB)B \hookrightarrow (A \to B) projects out. In simple situations this is simply A[1]A[1] (AA shifted in degree by one, what I used to write as ΣA\Sigma A) (AB) fl A[1] i al A B C dl P. \array{ &&&& (A\to B) &\stackrel{fl}{\to}& A[1] \\ &&&{}^i\nearrow & \uparrow^{al} \\ A &\to& B &\to& C \\ &&&{}_{dl}\nwarrow& \uparrow \\ &&&& P } \,.

Here “fl” is for “failed lift”.

Then if we are in a situation where each map is the kernel of its cokernel, by the universal property of cokernels we find that the desired lift dl exists if and only if the composite denoted “obs” in (AB) fl A[1] i al A B C obs dl P \array{ &&&& (A\to B) &\stackrel{fl}{\to}& A[1] \\ &&&{}^i\nearrow & \uparrow^{al} \\ A &\to& B &\to& C && \nearrow_{obs} \\ &&&{}_{dl}\nwarrow& \uparrow & \nearrow \\ &&&& P } is trivializable (equivalent to the trivial such map).

Here “obs” is for, guess what, “obstruction”.

It is this kind of reasoning that makes an additional categorical dimension appear out of nowhere:

since we are bound to be making mistakes in the nn dimensions that we started with, we device an ingenious method to export all these mistakes into an additional dimension, to get rid of them.

It’s some very elementary principle of category theory at work here, really, which says something like: don’t take quotients, take weak quotients if possible.

While I think it provides a huge reduction of my internal entropy, you might still feel the above diagrammatics is too baroque to serve as a good explanation for That Shift in Dimension™.

I promise that feeling will disappear after some time, but then there is still room to wonder why this kind of shift appears to be so crucial throughout all of quantum theory.

And I think this also has a good answer:

Quantum theory is, I claim, about colimits of transport nn-functors. But actually not of the transport nn-functors themselves, but of the (n+1)(n+1)-functors which obstruct flat nn-functors to exist.

There is really a pretty vast generalization of

- the fundamental theorem of calculus

- Stokes’ law

hiding here, encompassing most of the occurences of the word “curvature” as it appears in the physical literature.

So I am saying: if you are willing to accept that the notion of “gauge theory” is at all relevant, then from that it follows that you are bound to run into the mysterious Shift in Dimension™ from time to time:

the extra dimension is the home for the obstructions to flatness.

Posted by: Urs Schreiber on November 13, 2007 10:18 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

May I say, Urs, that this was a very helpful distillation indeed; I think you’ve been trying to pound these messages home for some time now, but only now do I think I get a coherent sense of what you’ve been talking about (you’re not to blame for that; I am).

I don’t have anything much more substantive to say at the moment, so I’ll just say for now: thanks very much!

Posted by: Todd Trimble on November 14, 2007 4:47 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

May I say, Urs, that this was a very helpful distillation indeed;

Okay, thanks for letting me know!

When we met in Yale a few weeks ago and Hisham Sati was pressing me to use all the available time to think about our project, I had that crazy idea that we could actually think together, the three of us, about Chern-Simons 3-bundles, because on the one hand they appear in “M-theory”, and Hisham is maybe the single person on the planet most expert in the huge and tantalizing collection of hints that goes under this name (and its such a pity that these hints haven’t propagated further into the math community, there is so much applied nn-category theory to be identified there, it’s a shame), while on the other hand Chern-Simons 3-bundles are nothing but a special case of precisely this obstruction theory I just summarized, so a genuinely nn-categorical problem, and you being the local nn-category guru.

I was rather upset with myself when I realized that I completely failed to achieve the desired fusion of worlds when I tried to talk about that big diagram to you. I noticed that something I said pretty much scared you away, or something.

Whatever: if, as you said, the above summary was helpful, maybe you might want to take a deep breath and return to the big diagram again? (It’s a 6mb download, I am hoping you have a fast internet connection available!)

Because as you can probably see very clearly now, this big diagram we were looking at in Yale in Hisham’s office is nothing but three copies of the diagrams which i have just drawn in the above comment!

Or in other words (this will be very clear to you): it is just one such diagram in a wolrd of 3-term sequences.

While the diagram may look intimidating, it is really just two things combined:

a) the general idea of obstruction theory as presented above

b) its internalization in the world of Lie nn-algebras, where all the steps can actually nicely be realized for all nn explicitly. The details for this are described in On weak cokernels (html). Possibly there is a hurdle here when the dg-algebras enter the picture, but after some getting used to that the nice thing is that everything is totally explicit.

Then there is some theory involved that says why these 3-term sequences depicted in this diagram have anything to do with nn-bundles with connection, but maybe it’s good to ignore that for the moment. I am in the process of writing up a more coherent account of that anyway. The only other review of the big picture underlying the big diagram is the section called “Obstruction theory” in my slides String- and Chern-Simons nn-Transport.

Whatever you think of all this, I would like to emphasize: I am and always will be very interested in your comments and crititical questions. I know there are vasts amounts of things I can and still should learn from you.

Posted by: Urs Schreiber on November 14, 2007 6:22 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

Khovanov homology still feels like a ‘trick’

Now this I don’t understand. Maybe I’ve just grown up too close to it, but Khovanov homology never seemed like a trick to me, and especially not once it was broken down into the combinatorics for links.

The bracket skein relation shoves one way of splitting the crossing up in degree and the other one down, providing an algebraic separation. What Khovanov homology does is turn this algebraic degree separation into a “firmer” homological degree separation, and reads the crossing as a “path” from one splitting to the other rather than just a relation between them. Equations turn into processes, just as they should.

Or maybe there’s something else – something more general – that you’re referring to as Khovanov homology?

Posted by: John Armstrong on November 14, 2007 12:23 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

John Armstrong wrote:

John Baez wrote:

Khovanov homology still feels like a ‘trick’.

Now this I don’t understand. Maybe I’ve just grown up too close to it, but Khovanov homology never seemed like a trick to me, and especially not once it was broken down into the combinatorics for links.

As you clearly know, it’s often handy to step back from something you’re familiar with, view it from a different angle, and ask yourself if you really understand it — while applying very demanding standards on what counts as ‘really understanding’.

For me, I don’t even need to step back from Khovanov homology, since I was never all that close to it. So I can skip that step.

For me the story goes like this: people discovered knot invariants like the Jones polynomial, which were mysterious at first, but then they ‘understood’ them in terms of quantum groups. Every simple Lie group has a god-given 1-parameter deformation in the world of quantum groups. The category of representations of a Lie group is symmetric monoidal, but the category of representations of a quantum group is only braided monoidal. So, we get a nice invariant of braids — and ultimately knots — from any representation of a simple Lie group (which gives a rep of the corresponding quantum group). This invariant depends on the parameter qq that shows up in the deformation game. It becomes dull as q1q \to 1, which is the limit where our quantum group reduces to a plain old group.

Fine and dandy. But then people start realizing that this whole story should categorify! Maybe for any representation of a simple Lie group we should get a braided monoidal 2-category and an invariant of 2-tangles!

The sensible way to study this is to go back to the basics and categorify the whole theory of simple Lie groups — or for technical reasons, maybe simple Lie algebras — and their deformations.

At its best, categorifying something means understanding it better, by seeing where the equations come from. So, we should take pretty much all the basic equations in the theory of simple Lie algebras and see where they come from!

This sounds really cool, since simple Lie algebras are a venerable subject that people have thought about for over a century. Digging beneath the surface of simple Lie theory and getting a whole deeper level of understanding… that should be really interesting!

And in fact it is. But, most of what I read about Khovanov homology and its relatives does not give me the feeling that I’m digging beneath the surface of simple Lie theory and understanding it all better. Instead, I’m just getting recipes for computing invariants.

So, from that viewpoint, what I see so far seems more like a ‘trick’.

Posted by: John Baez on November 19, 2007 12:41 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

Now that I understand. No, what he did doesn’t really seem to lead to any deeper insight into Lie theory. In fact, I might go so far as to say that it only gets halfway to deeper insight into the structure of the Kauffman bracket because (as far as I can tell) we still can’t see any topology there.

I guess I’d say that I tend to see Khovanov (and related) homologies as natural next steps, but they’re still all dancing around one central idea that hasn’t spoken its name as yet. This idea has these tantalizing connections to Lie theory, category theory, knot theory, and mathematical physics, which makes it very juicy for those of us who look at more than one of those fields. Still, we’re just walking the borders here, just starting to get the idea that there might be something inside that makes these casual similarities feel like more than “tricks”.

But Khovanov homology makes one important step forwards on that front at least. And the original formulation does so in a way that might lead someone to finally close the circuit if they really grok all the different fields involved along the way.

Posted by: John Armstrong on November 19, 2007 5:26 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

John A. wrote:

I guess I’d say that I tend to see Khovanov (and related) homologies as natural next steps, but they’re still all dancing around one central idea that hasn’t spoken its name as yet.

Right. Immodestly, I’ll suggest that its name is groupoidification. The long-term goal of the geometric representation theory seminar is to make good on this claim.

But Khovanov homology makes one important step forwards on that front at least. And the original formulation does so in a way that might lead someone to finally close the circuit if they really grok all the different fields involved along the way.

I agree.

Posted by: John Baez on November 21, 2007 8:38 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

Actually, I’m thinking more along the lines of spans.

Khovanov homology for links is a straightforward generalization of the combinatorics of the Kauffman bracket. But for tangles he has to pass to a certain derived category. IIRC, one construction of such involves taking spans to add new morphisms. On the other hand, more topological invariants of tangles seem to require spans straight away. Each side seems to need spans sooner or later.

Now, spans are important in your Tale of Groupoidification, sure. Still, I’m not entirely sure that the groupoids are essential, or if it’s the spans that are the really important thing.

Posted by: John Armstrong on November 21, 2007 8:54 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

John A. wrote:

Now, spans are important in your Tale of Groupoidification, sure. Still, I’m not entirely sure that the groupoids are essential, or if it’s the spans that are the really important thing.

In the next quarter of the geometric representation theory, I’ll try to convince you (and the rest of the world) that groupoidification — the process of replacing linear operators by spans of groupoids — is the royal road to categorifying quantum groups and their representations. Jim and Todd and I have done a lot of work on this subject that we haven’t told anyone about yet. But, there’s a lot left to do. So, I’m not sure how the seminar will go. It should at least be entertaining.

Posted by: John Baez on November 22, 2007 12:09 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

Again, I think we’re talking at just slightly cross-purposes. Groupoidification may well be the royal road to categorifying quantum groups. Remember, though, that I’m looking out for the topology in the Jones polynomial as my primary interest here.

I can tell that there’s something really interesting here because so many people are interested in these things for so many different reasons. Scott Carter told me a couple weeks ago, “I keep forgetting that people are interested in braided monoidal 2-categories with duals for reasons other than 2-knots”.

Posted by: John Armstrong on November 22, 2007 12:58 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

John A. wrote:

Groupoidification may well be the royal road to categorifying quantum groups. Remember, though, that I’m looking out for the topology in the Jones polynomial as my primary interest here.

Okay. That’s a big mystery that I don’t feel I have any special insight into. I’m mainly hoping to make progress on a different question: “what sort of algebra really underlies the quantum invariants of 2-tangles?”

Posted by: John Baez on November 22, 2007 1:34 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

Sounds like a lot of fun! Eager to see how you guys understand quantum groups. (Is there a paper or series thereof in the works? or maybe you could give us an “executive summary”?)

There is an interesting area of activity now of people constructing braid group actions on categories using correspondences of stacks (which is the same as what you refer to as spans of groupoids?) In particular the work of Bezrukavnikov gives some very powerful tools to construct (affine!) braid group actions geometrically, which should have interesting implications for Khovanov-type questions.

This is all part of a four-dimensional TFT as we learn from Witten, Kapustin and Gukov, and so should live in the same
framework as Khovanov at the very least. (I guess the whole geometric Langlands program is in some sense about “groupoidification”, though we only learned recently that is was part of TFT and how it should relate to knot and tangle invariants and the like).

The q-deformation is also starting to appear more prominently, in particular Gaitsgory and Lurie have a big program to incorporate quantum groups into the same big picture - in fact Gaitsgory recently showed how to recover quantum groups (at rational parameters) from a Hecke-theoretic point of view.. it would be surprising
if the Khovanovian world wasn’t hiding in there somewhere! I wonder if your point of view relates to this stuff - the logic stuff is very neat but over my head a bit..

Happy Thanksgiving!
David

Posted by: David Ben-Zvi on November 23, 2007 7:20 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

Some aspects of what David mentions are discussed over at Secret Blogging Seminar.

Posted by: David Corfield on November 24, 2007 10:27 AM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

Forgive my laziness, but, since I’m (sort of) an algebraic geometer, and this terminology isn’t my native language: is it right, as BZ says, that what people refer to in the Cafe as a “span of groupoids” is the same notion as a diagram of stacks ZXYZ \leftarrow X \rightarrow Y? [I’m avoiding the word “correspondence” only because people often use that to mean that XX has to be a cycle or cycle class on Z×YZ \times Y, not just any old XX mapping to Z×YZ\times Y—although I prefer BZ’s use of the term as just meaning a diagram as above.] Thanks for your help!

Posted by: Thomas Nevins on November 26, 2007 5:06 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

I can answer part of that. In category theory, a span between objects ZZ and YY is a diagram ZXYZ \leftarrow X \rightarrow Y. (In their homological algebra book, Gelfand and Manin call this a roof.)

So the part of the question left is: how do groupoids relate to stacks?

Posted by: Tom Leinster on November 26, 2007 5:35 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

Great, thanks! I hadn’t realized that “span” was a
standard categorical term.

Posted by: Thomas Nevins on November 27, 2007 4:39 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

Thomas Nevins wrote:

… it it right, as BZ says, that what people refer to in the Cafe as a “span of groupoids” is the same notion as a diagram of stacks ZXYZ \leftarrow X \rightarrow Y?

Close!

A ‘span’ is a diagram of the form ZXYZ \leftarrow X \rightarrow Y, though we should really draw it as a little bridge to see why people call it a span.

When speaking of ‘stacks’ we have to be a bit careful, since there are stacks on an arbitrary site, topological and differentiable stacks, Deligne–Mumford stacks, Artin stacks and so on. A ‘groupoid’ is just a category where all the morphisms are invertible.

However, stacks and groupoids are closely related, since we can often specify (or ‘present’) some sort of stack by giving a groupoid with the same sort of structure. For example, you can present a stack over the site TopTop by giving a ‘topological groupoid’ — roughly, a groupoid where the set of objects and the set of morphisms are topological, and the groupoid operations are continuous.

From this viewpoint, you can almost think of stacks as groupoids with more flexible morphisms between them: not just functors, but specially nice spans (called ‘anafunctors’).

I’m deliberately being very sketchy here, trying to outline the overall picture instead of stating a precise result. We’ve had mammoth discussions about this, most of which sailed right over my head. However, here’s a very nice introduction to relation between groupoids and stacks:

If you want some precise theorems, this is a good place to start.

Everything in our seminar is far less subtle: we’re considering plain old groupoids — often finite ones — and spans of these, which are functors like

ZXYZ \leftarrow X \rightarrow Y

We’re trying to see how much we can do with very little.

Posted by: John Baez on November 26, 2007 6:17 PM | Permalink | Reply to this

Re: Geometric Representation Theory (Lecture 10)

David B-Z wrote:

Eager to see how you guys understand quantum groups.

So are we! There’s a lot we’ve worked out, but a bunch of details will only get ironed out in the course of this seminar.

As I explain at the beginning of lecture 16, we’re following the lead of the Japanese auto manufacturers and pioneering ‘just-in-time delivery’ of mathematics. Instead of proving a theorem, letting it sit in a warehouse for weeks, and then lecturing about it, we’re proving it right before we talk about it. This inevitably leads to screwups, but that’s part of the fun.

Is there a paper or series thereof in the works?

Most of this stuff will turn into papers and theses by my students — especially Christopher Walker, but also perhaps Chris Rogers. That may take a few years.

or maybe you could give us an “executive summary”?

I’ll try to outline the plan of next quarter’s lectures in my last lecture this quarter. I may also do a This Week’s Finds or two about this stuff.

I can guess what you’re secretly asking for: a nice lump of stuff written in LaTeX, instead of doled out slowly in videos. However, I’m so sick of writing math papers that I’m really trying to cut back — which is hard, because I have a bunch of papers I’m already committed to writing (which is the reason I’m sick of writing them).

So, I’m enjoying the ability to simply talk about math and have it recorded on videotape. It’s somehow more ‘human’ than writing papers.

There is an interesting area of activity now of people constructing braid group actions on categories using correspondences of stacks (which is the same as what you refer to as spans of groupoids?) In particular the work of Bezrukavnikov gives some very powerful tools to construct (affine!) braid group actions geometrically, which should have interesting implications for Khovanov-type questions.

That sounds very interesting and probably related — somehow — to what we’re doing. I’ll try to look at some of the papers you’re hinting at.

Posted by: John Baez on November 26, 2007 6:45 PM | Permalink | Reply to this
Read the post Lie oo-Connections and their Application to String- and Chern-Simons n-Transport
Weblog: The n-Category Café
Excerpt: A discussion of connections for general L-infinity algebras and their application to String- and Chern-Simons n-transport.
Tracked: January 9, 2008 10:20 AM

Post a New Comment