Higher homotopy categories, higher derivators, and K-theory. (15th July 2022)

• Record Type: Journal Article
• Authors: Raptis, George
• Is Part Of: Forum of mathematics. Volume 10(2022)
• Publication Date: 2022-07-15
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1017/fms.2022.47
• Languages: English
• ISSNs: 2050-5094

Abstract

Abstract: For every $\infty $ -category $\mathscr {C}$, there is a homotopy n -category $\mathrm {h}_n \mathscr {C}$ and a canonical functor $\gamma _n \colon \mathscr {C} \to \mathrm {h}_n \mathscr {C}$ . We study these higher homotopy categories, especially in connection with the existence and preservation of (co)limits, by introducing a higher categorical notion of weak colimit. Using homotopy n -categories, we introduce the notion of an n -derivator and study the main examples arising from $\infty $ -categories. Following the work of Maltsiniotis and Garkusha, we define K -theory for $\infty $ -derivators and prove that the canonical comparison map from the Waldhausen K -theory of $\mathscr {C}$ to the K -theory of the associated n -derivator $\mathbb {D}_{\mathscr {C}}^{(n)}$ is $(n+1)$ -connected. We also prove that this comparison map identifies derivator K -theory of $\infty $ -derivators in terms of a universal property. Moreover, using the canonical structure of higher weak pushouts in the homotopy n -category, we also define a K -theory space $K(\mathrm {h}_n \mathscr {C}, \mathrm {can})$ associated to $\mathrm {h}_n \mathscr {C}$ . We prove that the canonical comparison map from the Waldhausen K -theory of $\mathscr {C}$ to $K(\mathrm {h}_n \mathscr {C}, \mathrm {can})$ is n -connected.