Matrads, Biassociahedra and A_{\infty}-Bialgebras
Algebraic Topology
2010-12-07 v9
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 H_{\infty} generated by a singleton in each bidegree (m,n) and realize H_{\infty} as the cellular chains on biassociahedra KK_{n,m} = KK_{m,n}, of which KK_{n,1} = KK_{1,n} 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 H_{\infty}.
Cite
@article{arxiv.math/0508017,
title = {Matrads, Biassociahedra and A_{\infty}-Bialgebras},
author = {Samson Saneblidze and Ronald Umble},
journal= {arXiv preprint arXiv:math/0508017},
year = {2010}
}
Comments
54 pages; 22 Figures. Version 9 reorganizes the content of version 7 (v.8 posted incorrectly) and includes proofs of the following facts: (1) Upsilon products act associatively on bisequence matrices and (2) balanced factorizations are unique