Higher homotopy categories, higher derivators, and 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.

Higher weak (co)limits, adjoint functor theorems, and higher Brown representability

“…Higher weak (co)limits are simultaneously a higher categorical generalization of ordinary weak (co)limits and a weakening of the notion of (co)limits in higher categories. We review the definition of higher weak (co)limits and some of their basic properties from [17,Section 3].…”

“…Proof. This is shown by induction on the dimension d of the simplicial set K using [17,Proposition 3.10]. We sketch the details for completeness.…”

“…Example 3.3.6. Let D be a complete ∞-category and consider the canonical functor γ n : D → h n D. By [17,Corollary 3.22], the functor γ n preserves small products and weak pullbacks of order (n − 1). However, γ n does not admit a left adjoint unless D is equivalent to an n-category and γ n is an equivalence.…”

“…The general construction and the properties of the homotopy n-category were studied in [12]. Homotopy n-categories were studied further in [17] in connection with a higher categorical notion of weak (co)limit that was introduced mainly for this purpose. Similarly to the usual homotopy category, the homotopy n-category h n C of a (co)complete ∞-category C does not admit all small (co)limits in gen-eral, but it does admit better (= higher) weak (co)limits as n grows.…”

“…Similarly to the usual homotopy category, the homotopy n-category h n C of a (co)complete ∞-category C does not admit all small (co)limits in gen-eral, but it does admit better (= higher) weak (co)limits as n grows. The general properties of the homotopy n-categories suggest the following class of n-categories as a convenient context for refined versions of the GAFTs (see [17,Section 3]).…”