Homotopy theory of algebras of substitudes and their localisation

Abstract

We study the category of algebras of substitudes (also known to be equivalent to the regular patterns of Getzler and operads coloured by a category) equipped with a (semi)model structure lifted from the model structure on the underlying presheaves. We are especially interested in the case when the model structure on presheaves is a Cisinski style localisation with respect to a proper Grothendieck fundamental localiser. For example, for W = W ∞ \mathtt {W}= \mathtt {W}_{\infty } the minimal fundamental localiser, the local objects in such a localisation are locally constant presheaves, and local algebras of substitudes are exactly algebras whose underlying presheaves are locally constant. We investigate when this localisation has nice properties. We single out a class of such substitudes which we call left localisable and show that the substitudes for n n -operads, symmetric, and braided operads are in this class. As an application we develop a homotopy theory of higher braided operads and prove a stabilisation theorem for their W k \mathtt {W}_k -localisations. This theorem implies, in particular, a generalisation of the Baez-Dolan stabilisation hypothesis for higher categories.