Matrads, biassociahedra, and $A_{\infty}$-bialgebras

We introduce the notion of a matrad M = {M_{n,m}} whose submodules M_{*,1} and M_{1,*} are non-Sigma operads. We define the free matrad H_{\infty} generated by a singleton in each bidegree (m,n) and realize H_{\infty} as the cellular chains on biassociahedra KK_{n,m} = KK_{m,n}, of which KK_{n,1} = KK_{1,n} is the associahedron K_{n}. We construct the universal enveloping functor from matrads to PROPs and define an A_{\infty}-bialgebra as an algebra over H_{\infty}.