Higher homotopy categories, higher derivators, and K-theory. (15th July 2022) - Raptis, George
Abstract: For every $∞ $ -category $\mathscr {C}$, there is a homotopy n -category $h_n \mathscr {C}$ and a canonical functor $γ _n \colon \mathscr {C} → h_n \mathscr {C}$ . 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 $∞ $ -categories. Following the work of Maltsiniotis and Garkusha, we define K -theory for $∞ $ -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 $∞ $ -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 $K(h_n \mathscr {C}, can)$ associated to $h_n \mathscr {C}$ . We prove that the canonical comparison map from the Waldhausen K -theory of $\mathscr {C}$ to $K(h_n \mathscr {C}, can)$ is n -connected.