The K-theory of left pointed derivators

TL;DR

This paper proves that derivator K-theory satisfies additivity for a broad class of derivators, including all triangulated derivators, and shows it forms an infinite loop space, with implications for algebraic K-theory of stable infinity-categories.

Contribution

It establishes the additivity of derivator K-theory for left pointed derivators and demonstrates that it is an infinite loop space, extending previous work and providing new insights.

Findings

Additivity of derivator K-theory proven for left pointed derivators.

Derivator K-theory is shown to be an infinite loop space.

Implications for algebraic K-theory of stable infinity-categories discussed.

Abstract

We build on work of Muro-Raptis in [Ann. K-Theory 2 (2017), no. 2, 303-340] and Cisinski-Neeman in [Adv. Math. 217 (2008), no. 4, 1381-1475] to prove that the additivity of derivator K-theory holds for a large class of derivators that we call left pointed derivators, which includes all triangulated derivators. The proof methodology is an adaptation of the combinatorial methods of Grayson in [Doc. Math. 16 (2011), 457-464]. As a corollary, we prove that derivator K-theory is an infinite loop space. Finally, we speculate on the role of derivator K-theory as a trace from the algebraic K-theory of a stable $\infty$-category \`a la Blumberg-Gepner-Tabuada in [Geom. Topol. 17 (2013), no. 2, 733-838].