#### Volume 17, issue 5 (2017)

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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
##### Abstract

Picard $2$–categories are symmetric monoidal $2$–categories with invertible $0$–, $1$– and $2$–cells. The classifying space of a Picard $2$–category $\mathsc{D}$ is an infinite loop space, the zeroth space of the $K$–theory spectrum $K\mathsc{D}$. This spectrum has stable homotopy groups concentrated in levels $0$, $1$ and $2$. We describe part of the Postnikov data of $K\mathsc{D}$ 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 $\Sigma C$ from a Picard $1$–category $C$, and show that it commutes with $K$–theory, in that $K\Sigma C$ is stably equivalent to $\Sigma KC$.

##### 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