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Dualité de Koszul des PROPs

Abstract : We generalize the Koszul duality theory for algebras and operads to PROPs. Whereas the operads are algebraic objects that represent the operations with multiple inputs but only one output on any type of algebras, the PROPs model the operations with multiple inputs and multiple outputs acting on algebraic structures like bialgebras and Lie bialgebras. To do this, we introduce a new monoidal product describing the compositions between the operations. We consider the connected part of a PROP, which we call a properad, by analogy with the operads. A properad is a monoid in this monoidal category. We generalize to the properads the homological objects associated to algebras and operads like the bar and cobar construction, the quasi-free modules and the quasi-free properads. To any properad, we associate a Koszul dual coproperad and a Koszul complex whose acyclicity is a criterion that determines whether the cobar construction is a resolution, called the minimal model, of the properad. To prove this theorem, we have used an additional graduation coming from the analytic functors generated by the monoidal product. This theory gives the notion of homotopy type "bialgebra" over a Koszul PROP. This notion corresponds to the one of homotopy type algebra which is a every important notion in algebraic topology.
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Contributor : Bruno Vallette <>
Submitted on : Thursday, January 8, 2004 - 2:52:14 PM
Last modification on : Friday, June 19, 2020 - 9:10:04 AM
Long-term archiving on: : Wednesday, September 12, 2012 - 12:20:28 PM


  • HAL Id : tel-00004118, version 1



Bruno Vallette. Dualité de Koszul des PROPs. Mathématiques [math]. Université Louis Pasteur - Strasbourg I, 2003. Français. ⟨tel-00004118⟩



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