2-associahedra

For any $r\ge 1$ and $$\mathbf{n}$ \in $\mathbb{Z}${\geq0}^r \setminus {\mathbf0}$weconstructaposet$W{$\mathbf{n}$}$ 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 $\widehat{W_{\mathbf{n}}}$ is an abstract polytope of dimension $\mathrm{\mid }\mathbf{n}\mathrm{\mid }+r-3$. There are forgetful maps $W_{$\mathbf{n}$} \to K_r$,where$K_r$isthe$(r-2)$-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 associahedra in detail.