There is an operad whose algebras are objects with a homotopy unital multiplication -- the $A_2$ operad.
There is an operad whose algebras are objects with a homotopy unital, homotopy associative objects -- the $A_3$ operad.
There is an operad $O$ whose algebras are homotopy associative, homotopy commutative objects. It may be constructed by generators and relations.
Question 1: Is there a more enlightening description of $O$? What are the connectivities / dimensionalities of its spaces, for instance?
Question 2: In particular, does $O$ coincide with the operad $A_2 \otimes A_2$, where $\otimes$ is the Boardman-Vogt tensor product?
Question 2 is suggested by the Eckmann-Hilton argument.
