Stable Postnikov data of Picard 2-categories
| Autor: | Gurski, Nick, Johnson, Niles, Osorno, Angélica M., Stephan, Marc |
|---|---|
| Rok vydání: | 2016 |
| Předmět: | |
| Zdroj: | Algebraic & Geometric Topology, vol. 17 (2017), pp. 2763 -- 2806 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.2140/agt.2017.17.2763 |
| Popis: | 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 $\Sigma K C$. Comment: 31 pages. To appear in Algebraic and Geometric Topology |
| Databáze: | arXiv |
| Externí odkaz: |
|