For any r≥ 1 and n∈Z_≥0^r ∖{0} 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 W_n is an abstract polytope of dimension |n|+r-3. There are forgetful maps W_n→ 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

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... ndix, we work out the 2- and 3-dimensional associahedra in detail.