#### Volume 19, issue 2 (2019)

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### Philip Hackney, Marcy Robertson and Donald Yau

Algebraic & Geometric Topology 19 (2019) 863–940
##### Abstract

We introduce a convenient definition for weak cyclic operads, which is based on unrooted trees and Segal conditions. More specifically, we introduce a category $\Xi$ of trees, which carries a tight relationship to the Moerdijk–Weiss category of rooted trees $\Omega$. We prove a nerve theorem exhibiting colored cyclic operads as presheaves on $\Xi$ which satisfy a Segal condition. Finally, we produce a Quillen model category whose fibrant objects satisfy a weak Segal condition, and we consider these objects as an up-to-homotopy generalization of the concept of cyclic operad.

##### Keywords
cyclic operad, dendroidal set, Quillen model category, Reedy category
##### Mathematical Subject Classification 2010
Primary: 05C05, 18D50, 55P48, 55U35
Secondary: 37E25, 55U10, 18G30, 18G55
##### Publication
Received: 10 October 2017
Revised: 9 June 2018
Accepted: 2 August 2018
Published: 12 March 2019
##### Authors
 Philip Hackney Institut für Mathematik Universität Osnabrück Osnabrück Germany Max-Planck-Institut für Mathematik Bonn Germany Department of Mathematics University of Louisiana at Lafayette Lafayette, LA United States http://phck.net Marcy Robertson School of Mathematics and Statistics University of Melbourne Melbourne Victoria Australia Donald Yau Department of Mathematics The Ohio State University at Newark Newark, OH United States