The n-Category Café

Re: Categorification and Topological Invariants in Luminy

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

Is layer cake here to be read Postnikov tower ?

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:

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:

then we define the differential:

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.,

and

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.