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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

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 KD. This spectrum has stable homotopy groups concentrated in levels 0, 1 and 2. We describe part of the Postnikov data of KD 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 ΣC from a Picard 1–category C, and show that it commutes with K–theory, in that KΣC is stably equivalent to ΣKC.

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
Received: 8 July 2016
Revised: 1 March 2017
Accepted: 27 March 2017
Published: 19 September 2017
Nick Gurski
Department of Mathematics, Applied Mathematics and Statistics
Case Western Reserve University
Cleveland, OH
United States
Niles Johnson
Department of Mathematics
The Ohio State University Newark
Newark, OH
United States
Angélica M Osorno
Department of Mathematics
Reed College
Portland, OR
United States
Marc Stephan
Department of Mathematics
University of British Columbia
Vancouver, BC