The relative 2-operad of 2-associahedra in symplectic geometry

Abstract: The Fukaya A-infinity category $\mathrm{Fuk}(M)$ is a rich invariant of a symplectic manifold $M$, and its manipulation and computation is a core focus of current symplectic geometry. Building on work of Wehrheim and Woodward, I have proposed that the correct way to encode the functoriality properties of $\mathrm{Fuk}$ is by defining an "$(A_\infty,2)$-category" called Symp, in which the objects are symplectic manifolds and hom($M,N$) is defined to be $\mathrm{Fuk}(M^-\times N)$. Underlying the new notion of an $(A_\infty,2)$-category is a family of abstract polytopes called 2-associahedra, which form a "relative 2-operad" (another new notion, which is related to Batanin's theory of higher operads). I will describe all of these constructions from scratch, without assuming any knowledge of symplectic geometry. This talk is based partly on joint work with Shachar Carmeli, and I will mention related joint work with Alexei Oblomkov.

algebraic topologycategory theoryquantum algebra

Audience: learners

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