Associahedra are polytopes first introduced by Stasheff to model loop spaces, and together form a topological operad whose algebras are \(A_\infty\)-spaces, which have homotopy associative multiplication. Taking cellular chains on this operad gives a differential graded operad whose algebras are \(A_\infty\)-algebras. However, in both the topological and differential graded settings, it is assumed that units are strict. The question of whether units can be taken to be homotopy coherent was taken up by Fukaya, Oh, Ohta, and Ono [K. Fukaya et al., Lagrangian intersection Floer theory. Anomaly and obstruction. I. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (2009; Zbl 1181.53002); Lagrangian intersection Floer theory. Anomaly and obstruction. II. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (2009; Zbl 1181.53003)] in the differential graded setting in their work on symplectic geometry, and they defined an operad whose algebras have homotopy coherent unit.
In this paper, the authors seek to find a topological analogue, or a topological operad whose algebras are \(A_\infty\)-spaces with homotopy coherent, rather than strict, unit. They generalize the Boardman-Vogt cubical subdivision approach to associahedra to give a definition of unital associahedra using planar rooted trees. Then they show that these spaces can be defined cellularly, although they can no longer be realized as polytopes. Finally, they prove that taking cellular chains on the resulting operad recovers the differential graded operad of Fukaya-Oh-Ohno-Ono.
The diagrams in the paper are particularly helpful with lots of illustrations of operations on trees, and then several color pictures of unital associahedra.