Higher homotopy categories, higher derivators, and K-theory. (15th July 2022) - Raptis, George
Abstract: For every ∞ -category C, there is a homotopy n -category h_n C and a canonical functor γ _n C → h_n 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 C to the K -theory of the associated n -derivator D_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 C, can) associated to h_n C . We prove that the canonical comparison map from the Waldhausen K -theory of C to K(h_n C, can) is n -connected.