Homology, Homotopy and Applications

Matrads, biassociahedra, and A∞-bialgebras

Samson Saneblidze and Ronald Umble

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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

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

First available in Project Euclid: 29 July 2011

Permanent link to this document

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]

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