  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. In this paper, 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 $\Sigma K C$.
