### 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.

Original language | English |
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Pages (from-to) | 148-253 |

Number of pages | 106 |

Journal | Theory and Applications of Categories |

Volume | 32 |

Issue number | 6 |

Publication status | Published - 6 Feb 2017 |

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### Keywords

- quillen model category
- polynomial monad
- coloured operad
- graph

### Cite this

*Theory and Applications of Categories*,

*32*(6), 148-253.