$2$–associahedra

Abstract

For any $r\ge 1$ and $n\in {\mathbb{Z}}_{\ge 0}^r\setminus \left\{0\right\}$ we construct a poset $W_n$ called a $2$–associahedron. The $2$–associahedra arose in symplectic geometry, where they are expected to control maps between Fukaya categories of different symplectic manifolds. We prove that the completion ${\hat{W}}_n$ is an abstract polytope of dimension $\left|n\right|+r-3$. There are forgetful maps $W_n\to K_r$, where $K_r$ is the $\left(r-2\right)$–dimensional associahedron, and the $2$–associahedra specialize to the associahedra (in two ways) and to the multiplihedra. In an appendix, we work out the $2$– and $3$–dimensional $2$–associahedra in detail.