Matrads, biassociahedra, and A∞-bialgebras

Abstract

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 $\mathcal{H}_\infty$ generated by a singleton $\theta^n_m$ in each bidegree $(m,n)$ and realize $\mathcal{H}_\infty$ as the cellular chains on a new family of polytopes $\{KK_{n,m}=KK_{m,n}\}$, called biassociahedra, of which $KK_{n,1}$ 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 $\mathcal{H}_\infty$.