Stable Postnikov data of Picard 2-categories (Q2409806)

The Stable Homotopy Hypothesis claims that there is an equivalence of homotopy theories between Picard \(n\)-categories (i.e. \(n\)-categories with all \(i\)-morphisms invertible for \(0 \leq i \leq n\)) and \(Sp^n_0\), i.e. spectra \(X\) whose homotopy groups are trivial outside degrees \(0\) through to \(n\). In the case of \(n=1\) and \(2\), there is a \(K\)-theory functor for symmetric monoidal \(n\)-categories to spectra. This paper makes some of the related constructions and properties explicit, particularly focusing on the Postnikov tower of the \(K\)-theory spectrum of a Picard \(2\)-category and on the interaction of \(K\)-theory with suspension.