Stable Postnikov data of Picard $2$–categories

Abstract

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