The relative 2-operad of 2-associahedra

Abstract: I will explain my construction of the 2-associahedra, which are abstract polytopes indexed by sequences of nonnegative integers. The elements of a 2-associahedron correspond to the degenerations in the configuration space of marked points on vertical lines in R^2, up to translations and positive dilations. There is a forgetful map from each 2-associahedron to an associahedron, and these maps enable us to equip the 2-associahedra with an operad-like structure: they form a "relative 2-operad" over the associahedra. This 2-operadic structure allows us to define the notion of an $(A_\infty,2)$-space. I will produce an example of such a space, which specializes to a double loop space.