On the Q construction for exact quasicategories

Abstract

We prove that the K-theory of an exact quasicategory can be computed via a higher categorical variant of the Q construction. This construction yields a quasicategory whose weak homotopy type is a delooping of the K-theory space. We show that the direct sum endows this homotopy type with the structure of a infinite loop space, which agrees with the canonical one. Finally, we prove a proto-devissage result, which gives a necessary and sufficient condition for a "nilimmersion" of stable quasicategories to be a K-theory equivalence. In particular, we prove that a well-known conjecture of Ausoni and Rognes is equivalent to the weak contractibility of a particular quasicategory.

22 pages. Revised and expanded with the proto-devissage result. Comments always welcome

References in corpus (1)

• On exact infinity-categories and the Theorem of the Heart

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• Simplicial spaces, lax algebras and the 2-Segal condition
• Comparison of Waldhausen constructions
• Stable moduli spaces of hermitian forms
• Integral Models for Spaces via the Higher Frobenius
• Characters and transfer maps via categorified traces
• A variant of algebraic K-theory
• A dévissage theorem of non-connective $K$-theory