Geodesic
Matrads, biassociahedra, and $A_{\infty}$-bialgebras
Homology, homotopy, and applications, Tome 13 (2011) no. 1, pp. 1-57.

Voir la notice de l'article provenant de la source International Press of Boston

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 $\mathcal{H}_\infty$.
DOI : 10.4310/HHA.2011.v13.n1.a2
Classification : 55P35, 55Pxx, 52B05
Keywords: $A_{\infty}$-bialgebra, operad, matrad, permutahedron, biassociahedron 
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Samson Saneblidze; Ronald Umble. Matrads, biassociahedra, and $A_{\infty}$-bialgebras. Homology, homotopy, and applications, Tome 13 (2011) no. 1, pp. 1-57. doi : 10.4310/HHA.2011.v13.n1.a2. http://geodesic.mathdoc.fr/articles/10.4310/HHA.2011.v13.n1.a2/

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