Higher operad structure for Fukaya categories

TL;DR

This paper introduces a higher operad framework for Fukaya categories using $ extbf{fc}$-multicategories, providing a unified operadic approach to various $A_ abla$-type structures in symplectic geometry.

Contribution

It establishes a natural $ extbf{fc}$-multicategory structure on moduli spaces of pseudo-holomorphic polygons and develops dg $ extbf{fc}$-multicategories to uniformly describe $A_ abla$-type structures.

Findings

$ extbf{fc}$-multicategory structure on moduli spaces of pseudo-holomorphic polygons

Development of dg $ extbf{fc}$-multicategories for $A_ abla$-structures

Unified operadic formulation of $A_ abla$-algebras, modules, and categories

Abstract

Operads often arise from geometry. The standard $A_\infty$ operad can be derived from the cellular chains on the Stasheff associahedra, and an $A_\infty$ algebra is an algebra over this operad. The notion of an $\mathbf{fc}$-multicategory, also called a virtual double category, is a two-dimensional generalization of operads and multicategories. Here $\mathbf{fc}$ stands for the free category monad. We establish a natural $\mathbf{fc}$-multicategory structure on the collection of moduli spaces of pseudo-holomorphic polygons with boundary on sequences of Lagrangian submanifolds in a symplectic manifold. These moduli spaces are known to underlie the construction of Fukaya categories. Based on this, we develop the theory of differential graded (dg) variants of $\mathbf{fc}$-multicategories and show that a broad range of $A_\infty$-type structures, such as $A_\infty$ algebras, $A_\infty$ (bi)modules, and $A_\infty$ categories (possibly curved), admit a uniform operadic formulation as algebras over dg $\mathbf{fc}$-multicategories.