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Trialgebras and families of polytopes. (English) Zbl 1065.18007
Goerss, Paul (ed.) et al., Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic \(K\)-theory. Papers from the international conference on algebraic topology, Northwestern University, Evanston, IL, USA, March 24–28, 2002. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3285-9/pbk). Contemporary Mathematics 346, 369-398 (2004).
The authors construct operads associated to the chain modules of simplexes and of Stasheff polytopes. The corresponding algebras have three operations; they are called associative trialgebras and dendriform trialgebras. The authors show that the two operads are acyclic and are Koszul dual to one another. They also show that the similar operad associated to cubes is acyclic and self-dual.
For the entire collection see [Zbl 1052.55001].

18D50 Operads (MSC2010)
17A50 Free nonassociative algebras
17D99 Other nonassociative rings and algebras
18G60 Other (co)homology theories (MSC2010)
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