Volume 22, issue 4 (2018)

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From operator categories to higher operads

Clark Barwick

Geometry & Topology 22 (2018) 1893–1959

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 An and En (1 n +) and also a collection of new examples.

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operator categories, $\infty$–operads, $E_n$–operads, wreath product, Boardman–Vogt tensor product, Leinster category, Segal spaces
Mathematical Subject Classification 2010
Primary: 18D50, 55U40
Received: 26 April 2013
Revised: 5 June 2017
Accepted: 20 July 2017
Published: 5 April 2018
Proposed: Ralph Cohen
Seconded: Stefan Schwede, Mark Behrens
Clark Barwick
School of Mathematics
University of Edinburgh
United Kingdom