The n-Category Café

A veritable bombardment of free beautiful geometry. I am so lucky to be living in this time period; ten years ago if you couldn’t make it to a conference you would miss out big time. You still do of course, but now at least a life buoy is thrown out to you and the skipper shouts out valiantly No one left behind on my watch!

Yes, thanks Alex and Christoph. Urs, it Evan Jenkins’ notes also cover Kevin’s talk. Unfortunately, not enough for me to understand the defintion of what a “factorization algebra” is… though I suspect that would have been the case even if I had been there! Might be important though, a more elegant way to approach chiral algebras, conformal field theory, etc.

not enough for me to understand the defintion of what a “factorization algebra” is…

By the way, directly relevant is

section 4.1 “topological chiral homology” of Lurie’s TFT thing.

Unfortunately, not enough for me to understand the defintion of what a “factorization algebra” is…

But let me see, is there a secret here? It seems to be a pretty straightforward definition. I may be missing something, let me know:

It seems to me that a factorization algebra is just an algebra (in Vect) over a version of the little disk operad for which all these little disks come with an embedding into a fixed given manifold.

So, for each $n$-disk image $D \subset X$ in your manifold $X$ there is a vector space $V_D$, and for all collections of such disks $D_1, \cdots, D_k \subset D \subset X$ sitting inside a single big little disk $D \subset X$ there is a linear map

$$V_{D_1} \otimes \cdots \otimes V_{D_k} \to V_D$$

and these satisfy the obvious operadic compatibility condition that says that when you have disks inside disks inside … inside one big disk, it doesn’t matter in which order you map everything finally to the outermost $V_D$.

So, the whole things is like a $n$-fold commutative higher algebroid with one set of elements per disk in $X$.

The best way to think of this probably is as a TFT all by itself on “genus 0” /tree-levl cobordisms embedded in $X$.

In Lurie’s section 4.1 is actually a simpler special case of this, that where these algebroid elements don’t actually depend on the position of the disks inside $X$.

See the displayed equation in the middle of his page 90: he looks at such beasts obtained from just algebras of the little disk operad by taking a collection of disks in $X$, forgetting that they sit in $X$, and just assigning objects to them.

I am guessing (but will have to remind me) that the original Beilinson-Drinfeld definition looked slightly more complicated, as they used (unless I am misremebering) not open disks but points, so that everything is a bit more singular.