Homotopy theory for algebras over polynomial monads

Abstract

We study the existence and left properness of transferred model structures for "monoid-like" objects in monoidal model categories. These include genuine monoids, but also all kinds of operads as for instance symmetric, cyclic, modular, higher operads, properads and PROP's. All these structures can be realised as algebras over polynomial monads. We give a general condition for a polynomial monad which ensures the existence and (relative) left properness of a transferred model structure for its algebras. This condition is of a combinatorial nature and singles out a special class of polynomial monads which we call tame polynomial. Many important monads are shown to be tame polynomial.

Final version. Remark 5.16 extended. Bibliography completed

References in corpus (5)

• Dendroidal sets and simplicial operads
• On the homotopy theory of enriched categories
• Theories of analytic monads
• Operads as polynomial 2-monads
• Moduli spaces of algebras over non-symmetric operads

Cited by in corpus (33)

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• A criterion for existence of right-induced model structures
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• Deformation theory of Cohomological Field Theories
• Faà di Bruno for operads and internal algebras
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• Operads in Unstable Global Homotopy Theory
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• Hyper-commutative algebras and cyclic cohomology
• Admissible replacements for simplicial monoidal model categories
• Familial Monads as Higher Category Theories