From operator categories to higher operads

Abstract

In this paper we introduce the notion of an operator category and two different models for homotopy theory of $\infty$-operads over an operator category -- one of which extends Lurie's theory of $\infty$-operads, the other of which is completely new, even in the commutative setting. We define perfect operator categories, and we describe a category $\Lambda(\Phi)$ attached to a perfect operator category $\Phi$ that provides Segal maps. We define a wreath product of operator categories and a form of the Boardman--Vogt tensor product that lies over it. We then give examples of operator categories that provide universal properties for the operads $A_n$ and $E_n$ ($1\leq n\leq +\infty$), as well as a collection of new examples.