Homotopy theory of algebras over polynomial monads

For many applications it is important to know if there exists a transferred model structures for ``monoid-like'' objects in monoidal model categories. These include genuine monoids, but also all kinds of operads, e.g. symmetric, cyclic, modular, n-operads, dioperads, properads and (wheeled) PROP's etc. All these structures can be realised as algebras over polynomial monads.

In my talk I will explain a general condition for a polynomial monad called tameness which ensures the existence and left properness of a transferred model structure for its algebras in an h-monoidal model category. This condition is of a combinatorial nature and singles out a special class of polynomial monads which we call tame. Many important polynomial monads are shown to be tame. On the other hand there are interesting polynomial monads which are not tame. For example, monads for modular operads or PROPs. We show that failure of tameness condition can be used to find obstructions for the existence of transferred model structure on algebras.