Higher operad structure for Fukaya 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.

Figures (6)

• Figure 1: Moduli space of pseudo-holomorphic (stable) disks bounded by a single embedded Lagrangian
• Figure 2: In Theorem \ref{['fc_moduli_introduction_thm']}, a pseudo-holomorphic polygon with boundary paths $\gamma_i$ in the Lagrangian component $L_{v_i}$ (blue) is viewed as a $2$-cell diagram in the corresponding $\mathbf{fc}$-multicategory (gray), and an intersection point in $L_{v_i}\cap L_{v_{i+1}}$ is viewed as a horizontal 1-cell.
• Figure 3: a 2-cell in a vertically discrete $\mathbf {fc}$-multicategory
• Figure 4: The partial composition of 2-cells $\mathbf u$ and $\mathbf u'$
• Figure 5: The partial composition with $\mathbf u'$ of empty input resolves the horizontal 1-cell input and removes that slot.

Theorems & Definitions (55)

• Proposition 1.1
• Theorem 1.2: Theorem \ref{['fc_moduli_unlabeled_thm']}
• Remark 1.3
• Remark 1.4
• Theorem 1.5
• Example 2.1: Cartesian categories
• Definition 2.3
• Example 2.4: Recovery of category
• Proposition 2.5
• Definition 2.7