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# Homology, Homotopy and Applications

## Volume 13 (2011)

### Number 1

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

Pages: 1 – 57

DOI: http://dx.doi.org/10.4310/HHA.2011.v13.n1.a2

#### Authors

#### 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 $\mathcal{H}_\infty$.

#### Keywords

$A_{\infty}$-bialgebra; operad; matrad; permutahedron; biassociahedron

#### 2010 Mathematics Subject Classification

Primary 55P35, 55Pxx. Secondary 52B05.