Stable Postnikov data of Picard 2-categories

TL;DR

This paper studies the stable homotopy groups of Picard 2-categories' K-theory spectra, describing their Postnikov data and showing limitations on strict skeletal models, using categorical suspension techniques.

Contribution

It provides a categorical description of the Postnikov data of K-theory spectra of Picard 2-categories and demonstrates the non-existence of strict skeletal models realizing certain spectra.

Findings

Postnikov data of K-theory spectra described categorically

No strict skeletal Picard 2-category models the 2-truncation of the sphere spectrum

Categorical suspension commutes with K-theory, preserving stable equivalences

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. 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 $\Sigma K C$.