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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.
Keywords: Picard $2$–category, stable homotopy hypothesis, Postnikov system, $k$–invariant, symmetric monoidal $2$–category, K–theory spectrum, $2$–monad
Gurski, Nick 1 ; Johnson, Niles 2 ; Osorno, Angélica 3 ; Stephan, Marc 4
@article{10_2140_agt_2017_17_2763,
author = {Gurski, Nick and Johnson, Niles and Osorno, Ang\'elica and Stephan, Marc},
title = {Stable {Postnikov} data of {Picard} 2{\textendash}categories},
journal = {Algebraic and Geometric Topology},
pages = {2763--2806},
publisher = {mathdoc},
volume = {17},
number = {5},
year = {2017},
doi = {10.2140/agt.2017.17.2763},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2017.17.2763/}
}
TY - JOUR AU - Gurski, Nick AU - Johnson, Niles AU - Osorno, Angélica AU - Stephan, Marc TI - Stable Postnikov data of Picard 2–categories JO - Algebraic and Geometric Topology PY - 2017 SP - 2763 EP - 2806 VL - 17 IS - 5 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2017.17.2763/ DO - 10.2140/agt.2017.17.2763 ID - 10_2140_agt_2017_17_2763 ER -
%0 Journal Article %A Gurski, Nick %A Johnson, Niles %A Osorno, Angélica %A Stephan, Marc %T Stable Postnikov data of Picard 2–categories %J Algebraic and Geometric Topology %D 2017 %P 2763-2806 %V 17 %N 5 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2017.17.2763/ %R 10.2140/agt.2017.17.2763 %F 10_2140_agt_2017_17_2763
Gurski, Nick; Johnson, Niles; Osorno, Angélica; Stephan, Marc. Stable Postnikov data of Picard 2–categories. Algebraic and Geometric Topology, Tome 17 (2017) no. 5, pp. 2763-2806. doi: 10.2140/agt.2017.17.2763
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