## June 29, 2010

### Categorification and Topological Invariants in Luminy

#### Posted by Alexander Hoffnung

Yesterday began the petit groupe de travail with the title “Categorification and Topological Invariants” at the Centre International des Rencontres Mathématiques (CIRM) in Luminy. This week-long meeting is meant to be both introductory as well as an opportunity to explore interesting questions in the fields of interest of the various participants. So far this seems to be going quite well.

I will give a little diary below the fold, and I will try to fill in some of what I hear or say in the talks as comments.

I arrived in Marseille on Sunday with my girlfriend Vera Dao and we fumbled our way through the airport, bus station, metro station, the house and car of a kind (although odd) stranger and the campus at Luminy using a mix of English, broken Spanish, and mutilated French until we happily found ourselves at the door to the institute. (We even saw a small brawl on the metro!)

The grounds here are very nice from what I have seen so far, and the weather is quite pleasant. I am told that I have been inside during the “hot” hours though. The sun seems as though it will never set, although around 9:30 or so it finally dips out of site. Vera is doing all of the site-seeing, along with her new friend Carola, and I am site-seeing vicariously though them, or rather her camera. With some luck I will be able to figure out how to get those picture up here. In the meantime, here is a picture courtesy taken by George Gollin.

There is another meeting here this week on non-linear dispersive evolution equations, so we are quite outnumbered by the analysts at mealtime. However, it was nice to see a friendly face from Riverside among this crowd. I had never talked to Ben Dodson while in Riverside, but he recognized my shirt (possibly a hint that I should change more often). We each leave Riverside for new cities this summer, so this was probably a first and last chance to speak with him a bit, at least for some time. It is proving difficult, however, to get the PDE people to explain their work to me beyond a few sentences!

Now a little about our meeting. This is actually a brand new experience for me, as well as the others I believe. The first day of the meeting consisted of four people:Yael Fregier (Luxembourg), Raphael Zentner (Muenster), Andrew Lobb (Stony Brook), and myself. There was not a lot of organization prior to the meeting as we were not sure exactly who would be in attendance. Nonetheless, it seems to be working out quite nicely.

At this first meeting, Yael began to speak about $A_\infty$-modules, although most of the day was spent on various aspects of $A_\infty$-algebras, in particular the correspondence with Maurer-Cartan elements in a graded Lie algebra. I will try to recap what he explained soon, mostly to see if I have understood him correctly.

Today, Adrian Lim (Luxembourg) joined us and we heard from Raphael in the morning on introductory knot theory. This was very nice since I very often find myself in knot theory talks (I am not completely sure why) without the requisite definitions on hand. Even better someone suggested yesterday that the audience be obliged to step up to the board and recap some parts of the lecture at various points throughout the day. Raphael had us work through some nice exercises in calculating the Alexander polynomial of various knots.

After lunch, Andrew took over. He spoke about a conjecture of Milnor on the slice genus of a knot, and more briefly about categorifications of various famous knot polynomials. This got us into some low-dimensional topology, which I just do not know. If I can manage, I will try to recap what Andrew has discussed after his next talk.

Tomorrow we will quit early and I will go to the beach and undoubtedly return looking like a lobster. Then Thursday, I will speak on categorification of various sorts, but will likely focus on some of the more basic category theory which tends to show up.

Friday, Adrian will speak on Chern-Simons theory, and then Vera and I will head off to Paris.

Posted at June 29, 2010 8:58 PM UTC

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### Re: Categorification and Topological Invariants in Luminy

Alex wrote:

Yael began to speak about A-modules, although most of the day was spent on various aspects of A-algebras, in particular the correspondence with Mauer-Cartan elements in a graded Lie algebra. I will try to recap what he explained soon, mostly to see if I have understood him correctly.

Please do - Being deeply engrossed with the infty version of MauRer-Cartan elements, I can hardly wait.

Also delving into Yael’s and Zambon’s pre-pre-print.

Posted by: jim stasheff on June 30, 2010 12:44 PM | Permalink | Reply to this

### Re: Categorification and Topological Invariants in Luminy

Fixed, thanks! I think Yael will speak one more time, so that we may actually get to $A_\infty$-modules. Then I will certainly make some remarks since there is interest. However, this will have to wait a little bit, since this afternoon is beach time!
Posted by: Alex Hoffnung on June 30, 2010 12:58 PM | Permalink | Reply to this

### Re: Categorification and Topological Invariants in Luminy

Basically, in my talk there was nothing new to experts: Iwas just recalling the interpretation of L algebras in terms of homological coderivations, which can be seen as a solution of [Q,Q]=0 for the graded commutator bracket of coderivations. This was the Maurer-Cartan equation I was talking about, so nothing to do with infinity version of Maurer-Cartan - sorry to not satisfy your thirst of knowledge.

In fact I am, in my talk, leading towards the exposition of the work in progress with John about the extension of the layer cake construction to L algebras, not to the exposition of the work with Marco!!! By the way, I have received your mail concerning our work with Marco, thanks a lot :-)

Posted by: yael fregier on June 30, 2010 3:04 PM | Permalink | Reply to this

### Re: Categorification and Topological Invariants in Luminy

the layer cake construction to L-$\infty$ algebras.

Is layer cake here to be read Postnikov tower ?

Posted by: Urs Schreiber on June 30, 2010 5:43 PM | Permalink | Reply to this

### Re: Categorification and Topological Invariants in Luminy

Yes, ‘layer cake’ is my slang to help people understand Postnikov towers and cohomology. The idea is that the homotopy groups are the ‘layers’ of the layer cake, while the Postnikov invariants are the ‘frosting’ that helps glue the layers together. This metaphor is designed to help people understand how mouth-wateringly delicious Postnikov towers actually are!

Anyone who wants a taste could start with cohomology: the layer-cake philosophy, where I wrote:

In a minute we’ll see that from the viewpoint of homotopy theory, a space is a kind of ‘layer cake’ with one layer for each dimension. I claim that cohomology is fundamentally the study of classifying ‘layer cakes’ like this. There are many other kinds of layer cakes, like chain complexes (which are watered-down versions of spaces), $L_\infty$-algebras and $A_\infty$-algebras (which are chain complexes with extra bells and whistles), and so on. But let’s start with spaces.

Posted by: John Baez on June 30, 2010 8:52 PM | Permalink | Reply to this

### Re: Categorification and Topological Invariants in Luminy

So homotopy group functors slice the other way as in a loaf of bread, except I can’t think of a metaphor to glue the slices together. Maybe marquetry would be a better image.

Posted by: jim stasheff on July 1, 2010 12:59 PM | Permalink | Reply to this

### re: cohomology: the layer-cake philosophy,

cohomology: the layer-cake philosophy

weird switch from very high brow categorical jazz to such a down home image

also that chain complexes slice it the other way!

Posted by: jim stasheff on July 1, 2010 1:03 PM | Permalink | Reply to this

### Re: Categorification and Topological Invariants in Luminy

I claim that cohomology is fundamentally the study of classifying ‘layer cakes’ like this.

I’d say this is just one aspect of the notion of cohomology. Fundamentally, I think cohomology is this. But probably you can argue that one perspective is fully reflected in the other.

Posted by: Urs Schreiber on June 30, 2010 11:27 PM | Permalink | Reply to this

### Re: Categorification and Topological Invariants in Luminy

I have heard that you would probably give a minicourse on “this” at the “easy” center in Vienna in September, do you have more information about it so that I can book my flight ticket?

Posted by: yael fregier on July 1, 2010 3:28 PM | Permalink | Reply to this

### cohomology etc.

I have heard that you would probably give a minicourse on “this” …

Right, it’s a mini-course on “differential this”, actually.

…at the “easy” center in Vienna…

Yup, at the ESI the Erwin-Schrödinger Institute for mathematical physics .

…in September, do you have more information about it so that I can book my flight ticket?

Dates haven’t been completely fixed yet, it seems, but I am being told to hang around “second and third week of September”. The minicourse web-site is here, I suppse this will be updated with more information at some point.

Posted by: Urs Schreiber on July 1, 2010 4:27 PM | Permalink | Reply to this

### Re: Categorification and Topological Invariants in Luminy

Since I promised some recap of the local events of this week, I will not be deterred by Yael’s brief explanation for experts. Instead I will try to recall the basics of Yael’s talk on $A_\infty$-algebras. Unfortunately, he did not ever get the chance to cover $A_\infty$-modules, but I may keep bugging him until he explains them to me, or even better to all of us here. Ah, I also see that Keller describes these in his notes.

Of course, giving the definition to the small but diverse audience here took some time. Therefore, a good deal of what I will say can be seen here at the $n$Lab. In particular, the $n$Lab gives the compact definition: an $A_\infty$-algebra is an algebra for the cofibrant resolution of the operad $Ass$ of associative algebras. Since “cofibrant resolution” doesn’t make me happy unless I am having a particularly good day, then it will be good to begin by delving into the more concrete definition following the precedent of both the $n$Lab article and Yael’s talk.

Hopefully, we will see why an $A_\infty$-algebra is a degree $1$ coderivation $D$ on the reduced tensor coalgebra over a graded vector space with $D^2=0$. To begin with an $A_\infty$-algebra is a graded vector space $V$ over $\mathbb{C}$, for instance, with a family of graded maps $\mu_n : T^n V \rightarrow V$ satisfying a particular property. First, the grading is given by $deg(\mu_n) = 2-n$. Then the first map $\mu_1: V\rightarrow V$ is of degree $1$. The fact that the degree is odd will be important when we talk about Maurer-Cartan elements. For each $m\geq 1$, this family of maps should satisfy the property:

(1)$\sum_{i+j=m+1}\sum_{k=0}^{m-j+1} sgn(i)\mu_i(Id^k\otimes \mu_j\otimes Id^{m-j-k}) = 0.$

Let’s hope I got those indices right! Here I write $sgn(i)$ to signify the fact that I do not know the correct sign convention.

Now there is a quite general fact that linear maps between chain complexes themselves form a chain complex. Given the linear map:

(2)$L : (W,d_W)\rightarrow (W',d_{W'})$

then we define the differential:

(3)$dL = d_{W'}L - (-1)^{|L|}Ld_W.$

One can check by computation that $d^2=0$. We can then rewrite the axiom of the multiplication maps in an $A_\infty$-algebra as a commutator $[\mu_1,\mu_m]$, for any $m$. In particular, $\mu_m$ is a linear map of chain complexes and $\mu_1$ is the differential.

Now the focus of this talk was on “formal geometry” and the relationship between Mauer-Cartan elements in a graded Lie algebra and $A_\infty$-algebras.

We have the definition: Let $(V,[\;,\;])$ be a graded Lie algebra, i.e.,

(4)$[a,b] = (-1)^{|a||b|}[b,a] \;(graded\; anti-symmetry)$

and

(5)$[a,[b,c]] = [[a,b],c] + (-1)^{|a||b|}[b,[a,c]]\;(gr. Jacobi),$

then $m$ is Maurer-Cartan if it satisfies the master equation, i.e., $[m,m] = 0$ and $|m|=1$.

If I am not using the master equation terminology correctly, please let me know.

A nice fact about these Maurer-Cartan elements is that they give chain complexes of the form $(V,ad_m)$. In fact, $\mu_1$ is a Maurer-Cartan element and $ad_{\mu_1}$ is an endomorphism on linear maps from $T^{m}V$ to $V$. This brings us to another useful point of terminology. That is, when we say $A_\infty$-algebras are “homotopy algebras”, we mean that $[\mu_1,\mu_m]$ is a homotopy operator.

I need some sleep, but if I seem to be getting this more or less correct so far, then I will try to explain the motivation via formal geometry and the correspondence between Maurer-Cartan elements and $A_\infty$-algebras. With any luck someone will jump in here and say a bit about $A_\infty$-modules.

Posted by: Alex Hoffnung on July 1, 2010 10:04 PM | Permalink | Reply to this

### A-infinity algebra

the nLab gives the compact definition: an A ∞-algebra is an algebra for the cofibrant resolution of the operad Ass of associative algebras. Since “cofibrant resolution” doesn’t make me happy unless I am having a particularly good day, then it will be good to begin by delving into the more concrete definition

On $A_\infty$-matters the $n$Lab is generally still underdeveloped. Luckily you can hit the “edit”-button until the entry makes you happy also on a bad day.

But notice that the concrete definition of $A_\infty$-algebras that you mention is precisely the same as the analogous concrete definition of $L_\infty$-algebras, the only difference being that in the latter case the tensor coalgebra is also skew-symmetrized.

And the $L_\infty$-story is spelled out in fair detail at $n$Lab: $L_\infty$-algebra.

Maybe that can serve as a template for you for developig the page on $A_\infty$-algebras a bit further.

Posted by: Urs on July 1, 2010 10:45 PM | Permalink | Reply to this

### Re: A-infinity algebra

What I’d really like to see at the $A_\infty$-algebra page is the connection between the algebraic and coalgebraic formulations fleshed out in some detail.

Posted by: Todd Trimble on July 2, 2010 12:08 AM | Permalink | Reply to this

### Re: A-infinity algebra

Alex wrote:
We can then rewrite the axiom of the multiplication maps in an A ∞-algebra as a commutator [μ 1,μ m], for any m.

Unless I’m misreading, that’s incorrect. Perhaps what he had in mind is to assemble the maps m_i into a single map TV to V, then lift it to a coderivation of TV called m then [m,m]=0 is the desired condition.
Since there is no graded commutativity involved, the use of V[1] aka sV versus V is primarily to make the assemble m to be homogeneous of degree 1 (or -1 in the originalchain level setting).

Posted by: jim stasheff on July 2, 2010 1:17 PM | Permalink | Reply to this

### Re: A-infinity algebra

what Alex is allusing to is that in the relation defining A_\infty algebras, if one singles out the terms involving \mu_1, one gets [\mu_1,\mu_m] which can be thought as a homotopy, hence the remaining terms satisfy a relation “up to a homotopy”. This was just in order to motivate the name… I also explained how ad_\mu_1:=[\mu_1,-] forms a differential on maps of type Hochschild cochains.
For people interested in this I would recommand the lecture notes by Markl and Doubek and Zima http://arxiv.org/pdf/0705.3719

Posted by: yael fregier on July 2, 2010 4:39 PM | Permalink | Reply to this

### Re: A-infinity algebra

I forgot to add : of course, as James mentioned, the goal was to

“assemble the maps m_i into a single map TV to V, then lift it to a coderivation of TV called m then [m,m]=0” would be “the desired condition.”

and this is what we did, but I was obviously not enough pedagogical ;-).

Posted by: yael fregier on July 2, 2010 4:47 PM | Permalink | Reply to this

### Re: A-infinity algebra

Hi Yael,

Sorry for being thick-headed, but I would like to continue this conversation until I understand this well enough to move over to the nLab page and contribute something there.

I know that this commutator issue was only meant as motivation, but can you explain for me again which chain maps I should view $[\mu_1,\mu_m]$ as a homotopy between? Even more, what are the chain complexes I should have in mind?

Posted by: Alex Hoffnung on July 8, 2010 4:58 PM | Permalink | Reply to this

### Re: A-infinity algebra

The formalism works for both the A_\infty and the L_\infty cases. The A_\infty case is easier to describe in detail. Let me write m_n rather than mu_n.
That m_n is a higher’ homotopy for n>3. For n=3, it is an associating homotopy:
[m_1,m_3] = m_2(m_2 otimes 1) - m_2(1 otimes m_2).
With appropriate signs,
[m_1,m_n] =
\sum m_p((1 otimes)^i otimes m_q ( otimes 1)^j)
where the sum is over p+q = n+1 with 1 and all positions of m_q

Posted by: jim stasheff on July 9, 2010 2:13 PM | Permalink | Reply to this

### Re: A-infinity algebra

Curious that Ainfty is now described as an alternate to Linfty
rather than vice versa.

Physics rules?

Posted by: jim stasheff on July 2, 2010 1:19 PM | Permalink | Reply to this

### Re: A-infinity algebra

Jim queried:

Physics rules?

in response to the fact the $A_\infty$-algebras might not be the more more natural starting point over $L_\infty$-algebras.

I am just curious what this alludes to? I know a bit about $L_\infty$-algebras from work with Baez and Rogers, but have little more than a bit of classroom knowledge on $A_\infty$-algebras.

Posted by: Alex Hoffnung on July 8, 2010 5:04 PM | Permalink | Reply to this

### Re: A-infinity algebra

That does make me feel aged!
A_infty arose in the study of loop spaces and related H-spaces. In the classrooom’, one learns that addition of based loops is only homotopy associative.
A choice of associating homotopy satisfies a pentagon relation UP TO HOMOTOPY and so it goes.

A couple of decades past before L_infty appeared on the seen.

Posted by: jim stasheff on July 9, 2010 2:17 PM | Permalink | Reply to this

### Re: A-infinity algebra

Thanks for the history lesson! I hope you don’t begin to let what I don’t know make you feel aged. That could be a slippery slope as I am quite new to the scene. :)

Also, thanks for the previous comment, I am writing down the calculations to make sure I understand your last comment now.

Posted by: Alex Hoffnung on July 9, 2010 2:35 PM | Permalink | Reply to this

### Re: Categorification and Topological Invariants in Luminy

First big thing that is missing (and it is normal since I have hidden it under the carpet), is that there is a suspension hanging around, in particular one should consider TV[1] and not TV. I am sorry of not being to say more, but the talks are starting again ;-)

Posted by: yael fregier on July 2, 2010 10:04 AM | Permalink | Reply to this

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