From operator categories to topological operads

In this paper we introduce the notion of an operator category and two different models for homotopy theory of $\mathrm{\infty }$-operads over an operator category – one of which extends Lurie’s theory of $\mathrm{\infty }$-operads, the other of which is completely new, even in the commutative setting. We define perfect operator categories, and we describe a category $\mathrm{\Lambda }(\mathrm{\Phi })$ 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 $E_n$ ($1\le n\le +\mathrm{\infty }$), as well as a collection of new examples.