Higher homotopy categories, higher derivators, and K-theory

For every $\mathrm{\infty }$-category $\mathcal{C}$, there is a homotopy $n$-category ${\mathrm{h}}_n\mathcal{C}$ 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 $\mathrm{\infty }$-categories. Following the work of Maltsiniotis and Garkusha, we define $K$-theory for $\mathrm{\infty }$-derivators and prove that the canonical comparison map from the Waldhausen $K$-theory of $\mathcal{C}$ to the $K$-theory of the associated $n$-derivator ${\mathbb{D}}_{\mathcal{C}}^{(n)}$ is $(n+1)$-connected. We also prove that this comparison map identifies derivator $K$-theory of $\mathrm{\infty }$-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}, $\mathrm{c}\mathrm{a}\mathrm{n}$)$associatedto$$\mathrm{h}$_n \mathscr{C}$. We prove that the canonical comparison map from the Waldhausen $K$-theory of $\mathcal{C}$ to $K({\mathrm{h}}_n\mathcal{C},\mathrm{c}\mathrm{a}\mathrm{n})$ is $n$-connected.