Operadic bar constructions, cylinder objects, and homotopy morphisms of algebras over operads

Details Operadic bar constructions, cylinder objects, and homotopy morphisms of algebras over operads Operadic bar constructions, cylinder objects, and homotopy morphisms of algebras over operads

2009-02-02

arxiv.org/abs/0902.0177v2

Mathematics

Algebraic Topology

64 pages. Bibliographical updates and minor corrections in the revised version. To appear in the "Alpine perspectives on algeb

Scientific paper

The purpose of this paper is twofold. First, we review applications of the bar duality of operads to the construction of explicit cofibrant replacements in categories of algebras over an operad. In view toward applications, we check that the constructions of the bar duality work properly for algebras over operads in unbounded differential graded modules over a ring. In a second part, we use the operadic cobar construction to define explicit cyclinder objects in the category of operads. Then we apply this construction to prove that certain homotopy morphisms of algebras over operads are equivalent to left homotopies in the model category of operads.

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