From operator categories to higher operads

In this paper we introduce the notion of an operator category and two different models for homotopy theory of ∞-operads over an operator category - one of which extends Lurie's theory of ∞-operads, the other of which is completely new, even in the commutative setting. We define perfect operator categories, and we describe a category Λ(Φ) attached to a perfect operator category Φ 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≤ n ≤ +∞), as well as a collection of new examples.