From operator categories to higher operads

Abstract

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 $\mathrm{\Lambda }\left(\mathrm{\Phi }\right)$ attached to a perfect operator category $\mathrm{\Phi }$ 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 $E_n$ ($1\le n\le +\infty$) and also a collection of new examples.