Stable Postnikov data of Picard 2–categories

Gurski, Nick; Johnson, Niles; Osorno, Angélica; Stephan, Marc

doi:10.2140/agt.2017.17.2763

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Stable Postnikov data of Picard $2$–categories

Nick Gurski, Niles Johnson, Angélica M Osorno and Marc Stephan

Algebraic & Geometric Topology 17 (2017) 2763–2806

DOI: 10.2140/agt.2017.17.2763

Abstract

Picard $2$ –categories are symmetric monoidal $2$ –categories with invertible $0$ –, $1$ – and $2$ –cells. The classifying space of a Picard $2$ –category $D$ is an infinite loop space, the zeroth space of the $K$ –theory spectrum $K D$ . This spectrum has stable homotopy groups concentrated in levels $0$ , $1$ and $2$ . We describe part of the Postnikov data of $K D$ in terms of categorical structure. We use this to show that there is no strict skeletal Picard $2$ –category whose $K$ –theory realizes the $2$ –truncation of the sphere spectrum. As part of the proof, we construct a categorical suspension, producing a Picard $2$ –category $Σ C$ from a Picard $1$ –category $C$ , and show that it commutes with $K$ –theory, in that $K Σ C$ is stably equivalent to $Σ K C$ .

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Keywords

Picard $2$–category, stable homotopy hypothesis, Postnikov system, $k$–invariant, symmetric monoidal $2$–category, K–theory spectrum, $2$–monad

Mathematical Subject Classification 2010

Primary: 55S45

Secondary: 18C20, 18D05, 19D23, 55P42

References

Publication

Received: 8 July 2016

Revised: 1 March 2017

Accepted: 27 March 2017

Published: 19 September 2017

Authors

Nick Gurski
Department of Mathematics, Applied Mathematics and Statistics Case Western Reserve University Cleveland, OH United States
http://casfaculty.case.edu/nick-gurski/
Niles Johnson
Department of Mathematics The Ohio State University Newark Newark, OH United States
http://nilesjohnson.net
Angélica M Osorno
Department of Mathematics Reed College Portland, OR United States
http://people.reed.edu/~aosorno/
Marc Stephan
Department of Mathematics University of British Columbia Vancouver, BC Canada
http://www.math.ubc.ca/~mstephan/

Volume 17, issue 5 (2017)