Operads are combinatorial gadgets encoding categories of algebras. In modern homotopy theory, where categories are replaced by infinity-categories, strict operads are replaced by their homotopy-coherent version, infinity-operads. A natural way infinity-operads arise is through the process of localization, the process of freely inverting a class of morphisms in a homotopy-coherent way.

After recalling the above notions, in this talk I will prove that all infinity-operads arise as a localization of a strict one, and moreover that the localization functor establishes an equivalence of homotopy theories. These results generalize analogue ones for infinity-categories proven by Joyal and Stevenson, resp. Barwick and Kan. We will see that essential to prove such results is the dendroidal formalism for infinity-operads, based on a certain category of trees and its combinatorics. If time permits, I will also highlight some open questions. Part of this work is joint with K. Arakawa and V. Carmona.

Abstract: TBA