W-locally constant functors and localisation of operads

Michael Batanin – 20 June 2018

We study the category of algebras of a polynomial monad equipped with a (semi)model structure lifted from the model category of W-locally constant functors on its underlying category. Here W is a proper Grothendieck fundamental localiser and operads and algebras have values in a combinatorial monoidal model category.

We investigate when this localisation has some good properties. In particular, we are interested when the forgetful functor from the category of algebras reflects local weak equivalences. We single out a class of such polynomial monads which we call locally left derivable and show that the monads for n-operads, symmetric and braided operads are in this class.

As an application we develop a homotopy theory of higher braided operads and prove some stabilization theorems for them from which we obtain yet another proof of generalised Baez-Dolan stabilization hypothesis.