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Stable Postnikov data
of Picard $2$–categories
Nick Gurski, Niles Johnson, Angélica M Osorno and Marc Stephan |
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Algebraic & Geometric Topology 17 (2017) 2763–2806
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Abstract |
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Picard –categories are symmetric monoidal –categories with invertible –, – and –cells. The classifying space of a Picard –category is an infinite loop space, the zeroth space of the –theory spectrum . This spectrum has stable homotopy groups concentrated in levels , and . We describe part of the Postnikov data of in terms of categorical structure. We use this to show that there is no strict skeletal Picard –category whose –theory realizes the –truncation of the sphere spectrum. As part of the proof, we construct a categorical suspension, producing a Picard –category from a Picard –category , and show that it commutes with –theory, in that is stably equivalent to . |
Keywords
Picard $2$–category, stable homotopy hypothesis, Postnikov
system, $k$–invariant, symmetric monoidal $2$–category,
K–theory spectrum, $2$–monad
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Mathematical Subject Classification 2010
Primary: 55S45
Secondary: 18C20, 18D05, 19D23, 55P42
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References |
Publication
Received: 8 July 2016
Revised: 1 March 2017
Accepted: 27 March 2017
Published: 19 September 2017
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Authors |
| Nick Gurski | |
| Department of Mathematics, Applied
Mathematics and Statistics Case Western Reserve University Cleveland, OH United States |
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| http://casfaculty.case.edu/nick-gurski/ | |
| Niles Johnson | |
| Department of Mathematics The Ohio State University Newark Newark, OH United States |
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| http://nilesjohnson.net | |
| Angélica M Osorno | |
| Department of Mathematics Reed College Portland, OR United States |
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| http://people.reed.edu/~aosorno/ | |
| Marc Stephan | |
| Department of Mathematics University of British Columbia Vancouver, BC Canada |
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| http://www.math.ubc.ca/~mstephan/ | |
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