\(K\)-theory for 2-categories (Q1678146)

The main result of the article, Theorem 1.1, is that the homotopy theories of the following three categories (equipped with suitable notions of stable weak equivalences) are equivalent: (1) Gamma-objects in the category of 2-categories and 2-functors, (2) permutative 2-categories, (3) permutative Gray monoids. The ``homotopy theories'' under consideration are constructed using the machinery of complete Segal spaces, and the comparison is accomplished by constructing various ``K-theory functors'' and ``inverse K-theory functors'', and maps between various compositions of such. As is to be expected the authors have to overcome a host of technical obstacles, which are documented in meticulous and helpful detail. There is also an appendix containing a list of the categories and functors employed to help navigating of the paper. Category (1) provides a model for the homotopy theory of connective spectra, while (3) provides a model for the homotopy theory of symmetric monoidal bicategories. The relevance of the result, as explained in the introduction, stems from the observation that ``a number of interesting symmetric monoidal structures are naturally 2-categorical'', and the from the authors' desire to ``work directly with this 2-dimensional algebra''.