Volume 22, issue 4 (2018)

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From operator categories to higher operads

Clark Barwick

Geometry & Topology 22 (2018) 1893–1959
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 $\Lambda \left(\Phi \right)$ 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 ${E}_{n}$ ($1\le n\le +\infty$) and also a collection of new examples.

Keywords
operator categories, $\infty$–operads, $E_n$–operads, wreath product, Boardman–Vogt tensor product, Leinster category, Segal spaces
Mathematical Subject Classification 2010
Primary: 18D50, 55U40