Homotopy theory of algebras of substitudes and their localisation

Batanin, Michael; White, David

doi:10.1090/tran/8600

Homotopy theory of algebras of substitudes and their localisation
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by Michael Batanin and David White PDF

Trans. Amer. Math. Soc. 375 (2022), 3569-3640 Request permission

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 $\mathtt {W}= \mathtt {W}_{\infty }$ 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 $n$-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 $\mathtt {W}_k$-localisations. This theorem implies, in particular, a generalisation of the Baez-Dolan stabilisation hypothesis for higher categories.

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