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

摘要

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 $heta^n_m$ 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 $mathcal{H}_infty$.