Homotopy theory for algebras over polynomial monads

TL;DR

This paper establishes conditions under which model structures can be transferred to algebras over polynomial monads, covering a wide range of algebraic structures including operads and properads, with a focus on their existence and properness.

Contribution

It introduces the concept of tame polynomial monads and provides a general combinatorial criterion ensuring transferred model structures for their algebras.

Findings

Many important monads are tame polynomial.

Conditions for existence of transferred model structures are characterized.

Left properness of these structures is established under certain conditions.

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.