Higher homotopy categories, higher derivators, and K-theory

TL;DR

This paper explores higher homotopy categories, introduces $n$-derivators, and establishes new connections between $ ext{K}$-theory of $ ext{infinity}$-categories and their homotopy $n$-categories, revealing new universal properties and connectivity results.

Contribution

It defines $n$-derivators and higher weak pushouts, and proves new connectivity and universal property results relating $ ext{K}$-theory of $ ext{infinity}$-categories to their homotopy $n$-categories.

Findings

The comparison map from Waldhausen $ ext{K}$-theory to $n$-derivator $ ext{K}$-theory is $(n+1)$-connected.

The comparison map from Waldhausen $ ext{K}$-theory to $K( ext{h}_n ext{C}, ext{can})$ is $n$-connected.

Introduction of a universal property characterizing derivator $ ext{K}$-theory.

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.