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Axiomatic homotopy theory for operads. (English) Zbl 1041.18011

The article concerns the existence of a model structure on operads in a symmetric monoidal model category. Sufficient conditions for the existence of such a model structure are given, concerning the existence of an interval with comultiplication. The weak equivalences and fibrations of the model structure are defined on the level of the underlying collections. The conditions are easy to check and they hold for topological, simplicial and chain operads. In the case of chain operads this recovers a result of V. Hinich [Commun. Algebra 25, No.10, 3291–3323 (1997; Zbl 0894.18008)].
With a model structure in place, homotopy algebras over an operad may be defined as the algebras over a cofibrant replacement of the operad. Existence of model structures for algebras over operads is discussed and a homotopy invariance property for algebras over cofibrant operads is proved. Various comparison theorems are proved, establishing that, under mild conditions, monoidal Quillen equivalences of monoidal model categories induce Quillen equivalences between categories of homotopy algebras.
The approach adopted here is conceptually simple and powerful. This paper provides a very useful reference for the homotopy theory of operads.

MSC:

18D50 Operads (MSC2010)
18G55 Nonabelian homotopical algebra (MSC2010)
55U35 Abstract and axiomatic homotopy theory in algebraic topology
55P48 Loop space machines and operads in algebraic topology

Citations:

Zbl 0894.18008
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