Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Every spectrum is the K-theory of a stable $\infty$-category

Maxime Ramzi, Vladimir Sosnilo, Christoph Winges

Abstract

We prove that any spectrum is equivalent to the nonconnective K-theory of a stable $\infty$-category. We use these results to construct a stable $\infty$-category $\mathcal{C}$ with a bounded t-structure such that $\operatorname{K}(\mathcal{C})$ is not equivalent to $\operatorname{K}(\mathcal{C}^\heartsuit)$, disproving a conjecture of Antieau, Gepner, and Heller.

Every spectrum is the K-theory of a stable $\infty$-category

Abstract

We prove that any spectrum is equivalent to the nonconnective K-theory of a stable -category. We use these results to construct a stable -category with a bounded t-structure such that is not equivalent to , disproving a conjecture of Antieau, Gepner, and Heller.
Paper Structure (13 sections, 20 theorems, 76 equations)

This paper contains 13 sections, 20 theorems, 76 equations.

Table of Contents

  1. Notation
  2. Acknowledgements
  3. Reminder on localizing invariants
  4. Reminder on the Dundas-McCarthy theorem
  5. Categorification of spectra
  6. Reminder on the Calkin construction
  7. Categorifying suspensions and loops
  8. Bounded t-structures and the theorem of the heart
  9. Basic properties
  10. K-theoretic properties of t-structures
  11. Chromatic obstructions to bounded t-structures
  12. Counterexample to the nonconnective theorem of the heart
  13. Construction of the counterexample

Key Result

Theorem 1

For every spectrum $M$ there exists a small idempotent complete stable $\infty$-category $\mathcal{C}\xspace_M$ such that Here $\operatorname{K}$ denotes the nonconnective K-theory spectrum. Moreover, the construction of $\mathcal{C}\xspace_M$ is functorial in $M$.

Theorems & Definitions (58)

  • Theorem 1: Theorem \ref{['thm:cat_spectrum']}
  • Theorem 2: \ref{['sssec:counterexample']}
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Example 1.4
  • Theorem 1.5: Blumberg, Gepner & Tabuada blumberg2013universal
  • Theorem 1.6: Blumberg, Gepner & Tabuada blumberg2013universal
  • Corollary 1.7
  • proof
  • ...and 48 more