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. Based on the idea of the homotopy $n$-category, 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 define also 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.
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