Volume 19, issue 2 (2019)

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Higher cyclic operads

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 Ξ of trees, which carries a tight relationship to the Moerdijk–Weiss category of rooted trees Ω. We prove a nerve theorem exhibiting colored cyclic operads as presheaves on Ξ 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
References
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