From operator categories to topological operads

TL;DR

This paper introduces operator categories and develops two models for the homotopy theory of $ abla$-operads, extending Lurie's framework and presenting new constructions and examples in higher algebra.

Contribution

It defines perfect operator categories, constructs associated categories with Segal maps, and introduces a wreath product and tensor product for operator categories, expanding the theory of $ abla$-operads.

Findings

Provides models extending Lurie's $ abla$-operads theory.

Defines perfect operator categories and associated categories with Segal maps.

Offers new examples and universal properties for $A_n$ and $E_n$ 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.