The associahedron is a well-known poset with connections to many different areas of combinatorics, algebra, geometry, topology, and physics. The associahedron has several different realizations as the face poset of a convex polytope and one realization, due to Loday, is a generalized permutahedron, i.e. a polymatroid. From the perspective of symplectic geometry, the associahedron encodes the combinatorics of morphisms in the Fukaya category of a symplectic manifold. In 2017, Bottman introduced a family of posets called 2-associahedra which encode the combinatorics of functors between Fukaya categories, and he conjectured that they can be realized as the face posets of convex polytopes. In this talk we will introduce categorical n-associahedra as a natural extension of associahedra and 2-associahedra, and we will produce a family of complete polyhedral fans called velocity fans whose face posets are the categorical n-associahedra. Categorical n-associahedra cannot be realized by generalized permutahedra or any of their known extensions. On the other hand, our velocity fan specializes to the normal fan of Loday’s associahedron suggesting a new extension of generalized permutahedra. This is joint work with Nathaniel Bottman and Daria Poliakova.