Higher homotopy categories, higher derivators, and K-theory

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.