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Let $X$ be a curve over $\mathbf{C}$. As I understand from the 2008 Talbot notes, a chiral category on $X$ consists of a crystal of categories on the Ran space $\mathrm{Ran}(X)$ (see these notes of Gaitsgory) together with factorization data: that is, the category associated to $X^I$ should be the same as the $I$-fold tensor product of the category associated to $X$, when restricted to the locus where the coordinates are distinct. Factorization spaces over $X$, such as the affine Grassmannian, give one source of chiral categories.

As I understand, the notion of a chiral category is supposed to be the algebro-geometric analog of a factorization algebra (in categories) in topology, as defined in ch. 5 of Lurie's "Higher Algebra" (using an operad $\mathbf{E}_M$ associated to $M$, but equivalently cosheaves on the topological version of the Ran space). There, it is proved that locally constant factorization algebras over euclidean space $\mathbf{R}^n$ are equivalent to $\mathbf{E}_n$-algebras. So, chiral categories, especially over $\mathbf{A}^1$, are supposed to be a version of $\mathbf{E}_2$-algebras in the algebro-geometric setting.

I'm looking for a precise comparison statement (rather than an analogy) along these lines between the algebro-geometric and topological pictures. In particular, given a chiral category (e.g., over $\mathbf{A}^1$), I'd like to know:

1) Is there a natural way to extract an $\mathbf{E}_2$-category by taking sections over $\mathbf{A}^1$?

2) If so, what is the formula for the $\mathbf{E}_2$-tensor product?

The example I have in mind, which I would like to understand, is the tensor structure on the geometric Satake category constructed in Mirkovich-Vilonen.

Edit: It seems like the construction in this paper of Richarz is based upon the following. Given two objects in the category over $X$ (which here is equivariant perverse sheaves on the affine Grassmannian), one forms their external tensor product in the category over the complement of the diagonal in $X \times X$ (which makes sense) and then intermediate extends to obtain an object over $X \times X$, and then applies a vanishing cycles functor to obtain a new object over $X$. In what generality can something like this be done, and why should such a construction yield a (braided) monoidal structure?

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  • $\begingroup$ What does it mean for a crystal of categories to be quasi-coherent? $\endgroup$
    – S. Carnahan ♦
    Commented Oct 30, 2014 at 22:29
  • $\begingroup$ @S.Carnahan : You are right, perhaps this is part of the definition. $\endgroup$
    – Akhil Mathew
    Commented Oct 31, 2014 at 1:16
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    $\begingroup$ I think it's not quite right to think of a chiral algebra as an $E_2$ algebra. The latter should give examples of the former (up to some framing, perhaps), but chiral algebras should not, in general, be "locally constant". Think about a much less categorical setting of functions. Locally constant functions are very different from holomorphic functions. Or am I misunderstanding the notion of "chiral algebra" --- is there a "de Rham" that I missed somewhere? $\endgroup$
    – Theo Johnson-Freyd
    Commented Oct 31, 2014 at 1:24
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    $\begingroup$ I think much is said in the folklore but you won't find much written down. At least I know of three big groups working in Boston, Chicago and Paris trying to sort these things. In particular, the AG analog of Topological Factorization Algebra does exist (and predates the topological one) and is a Factorization algebra in Beilinson-Drinfeld. E_2 algebras are locally constant (or constructible) such algebras, their AG counterpart correspond to what are called Topological Vertex algebras. Presumably RH goes from the AG side to the top. side and this has been claimed by many to be done. $\endgroup$
    – Reimundo Heluani
    Commented Oct 31, 2014 at 9:20
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    $\begingroup$ Continuing: little is known for general vertex algebras or general factorization algebras in the Beilinson-Drinfeld case. Namely those which will not come from holonomic D-modules (which is the right finiteness and locally constant condition in the AG side). I've heard many times and from many that applying Chiral Homology and then RH one should obtain a factorization algebra. However, there are some subtleties involved since analitification does not preserve the chiral operations for general D-modules (quasi-coherent as O-mod). $\endgroup$
    – Reimundo Heluani
    Commented Oct 31, 2014 at 9:24

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