Roughly speaking, a matron is a bigraded associative unital algebra M = M_{n,m} whose submodules M_{∗,1} and M_{1,∗} are non-\Sigma operads. We define the bialgebra matron \mathcal{H} and construct its minimal model \mathcal{H_\infty}, the A-infinity bialgebra matron, generated by a singleton in each bidegree (m,n)≠(1,1). We define an A-infinity bialgebra as an algebra over \mathcal{H_\infty} and realize \mathcal{H_\infty} as the cellular chains of polytopes KK_{n-1,m-1}, of which KK_{n-1,0} = KK_{0,n-1} is the Stasheff associahedron K_n.