From operator categories to higher operads

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 $Λ (Φ)$ 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 $E_{n}$ ( $1 \leq n \leq + \infty$ ) and also a collection of new examples.

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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

References

Publication

Received: 26 April 2013

Revised: 5 June 2017

Accepted: 20 July 2017

Published: 5 April 2018

Proposed: Ralph Cohen

Seconded: Stefan Schwede, Mark Behrens

Authors

Clark Barwick
School of Mathematics University of Edinburgh Edinburgh United Kingdom

Volume 22, issue 4 (2018)

Clark Barwick

Abstract