## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 13, Number 1 (2011), 1-57.

### Matrads, biassociahedra, and A∞-bialgebras

Samson Saneblidze and Ronald Umble

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

#### Article information

**Source**

Homology Homotopy Appl., Volume 13, Number 1 (2011), 1-57.

**Dates**

First available in Project Euclid: 29 July 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1311953345

**Subjects**

Primary: 55P35: Loop spaces 55P99: None of the above, but in this section

Secondary: 52B05: Combinatorial properties (number of faces, shortest paths, etc.) [See also 05Cxx]

**Keywords**

$A_\infty$--bialgebra operad matrad permutahedron biassociahedron