2-associahedra

TL;DR

This paper introduces 2-associahedra, a new class of posets inspired by symplectic geometry, which generalize associahedra and multiplihedra and have applications in Fukaya categories.

Contribution

The construction and proof that 2-associahedra form abstract polytopes, expanding the combinatorial tools for symplectic geometry and category theory.

Findings

2-associahedra are abstract polytopes of dimension |n|+r-3.

They generalize associahedra and multiplihedra.

Explicit descriptions of 2- and 3-dimensional cases.

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.