Higher homotopy categories, higher derivators, and K-theory

George Raptis

Forum of Mathematics, Sigma, Tome 10 (2022)

Résumé

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.

Détail

DOI : 10.1017/fms.2022.47