The Diagonal of the Stasheff Polytope

Part of the Progress in Mathematics book series (PM, volume 287)


We construct an A-infinity structure on the tensor product of two A-infinity algebras by using the simplicial decomposition of the Stasheff polytope. The key point is the construction of an operad AA-infinity based on the simplicial Stasheff polytope. The operad AA-infinity admits a coassociative diagonal and the operad A-infinity is a retract by deformation of it. We compare these constructions with analogous constructions due to Saneblidze–Umble and Markl–Shnider based on the Boardman–Vogt cubical decomposition of the Stasheff polytope.

Key words

Stasheff polytope Associahedron Operad Bar–cobar construction Cobarconstruction A-infinity algebra AA-infinity algebra Diagonal 



I thank Bruno Vallette for illuminating discussions on the algebras up to homotopy and Samson Saneblidze for sharing his drawings with me some years ago. Thanks to Emily Burgunder, Martin Markl, Samson Saneblidze, Jim Stasheff and Ron Umble for their comments on the previous versions of this paper. I warmly thank the referee for his careful reading and his precious comments which helped me to improve this text.This work is partially supported by the French agency ANR.


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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institut de Recherche Mathématique AvancéeCNRS et Université de StrasbourgStrasbourg CedexFrance

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