Title: 2-associahedra,
Abstract: For any $r\geq 1$ and $\mathbf{n} \in \mathbb{Z}_{\geq0}^r \setminus
\{\mathbf0\}$ we construct a poset $W_{\mathbf{n}}$ 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 $|\mathbf{n}|+r-3$. There are forgetful maps $W_{\mathbf{n}} \to
K_r$, where $K_r$ is the $(r-2)$-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.