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Dévissage for Algebraic K-theory of Small Stable $\infty$-categories

Chunhui Wei

TL;DR

This work extends Quillen's dévissage to small stable $\infty$-categories by introducing dévissage and fillability conditions for a stable pair $(\mathcal{C},\mathcal{A})$, and shows that the inclusion $\mathcal{A}\hookrightarrow\mathcal{C}$ induces isomorphisms on higher $K$-groups $K_n$ for $n\ge 1$ under a weak dévissage condition and on $K_0$ under strong dévissage. It develops a framework of Gap and Gap^w constructions to categorify fibers and loops, enabling a robust additivity and fiber/loop calculus for universal localizing invariants, including $K$-theory. Higher and stronger notions of fillability yield vanishing results for quotients $\mathrm{Q}(\mathcal{C},\mathcal{A})$ and, in many cases, force $K$-groups of the larger category to vanish or coincide with those of the subcategory, providing structural constraints on stable $\infty$-categories. The paper also connects these dévissage techniques with classical quasi-abelian and abelian settings, offering a new proof pathway for Quillen's theorem through derived categories and functor categories, and highlighting the versatility of the $\infty$-categorical approach in higher algebraic $K$-theory.

Abstract

In this article, we extend Quillen's Dévissage Theorem to small stable $\infty$-categories. To be precise, we establish sufficient conditions under which the non-negative $K$-groups of a small stable $\infty$-category coincide with those of a stable subcategory thereof.

Dévissage for Algebraic K-theory of Small Stable $\infty$-categories

TL;DR

This work extends Quillen's dévissage to small stable -categories by introducing dévissage and fillability conditions for a stable pair , and shows that the inclusion induces isomorphisms on higher -groups for under a weak dévissage condition and on under strong dévissage. It develops a framework of Gap and Gap^w constructions to categorify fibers and loops, enabling a robust additivity and fiber/loop calculus for universal localizing invariants, including -theory. Higher and stronger notions of fillability yield vanishing results for quotients and, in many cases, force -groups of the larger category to vanish or coincide with those of the subcategory, providing structural constraints on stable -categories. The paper also connects these dévissage techniques with classical quasi-abelian and abelian settings, offering a new proof pathway for Quillen's theorem through derived categories and functor categories, and highlighting the versatility of the -categorical approach in higher algebraic -theory.

Abstract

In this article, we extend Quillen's Dévissage Theorem to small stable -categories. To be precise, we establish sufficient conditions under which the non-negative -groups of a small stable -category coincide with those of a stable subcategory thereof.
Paper Structure (20 sections, 73 theorems, 160 equations)

This paper contains 20 sections, 73 theorems, 160 equations.

Table of Contents

  1. Preliminaries on Stable $\infty$-category
  2. Stable $\infty$-category
  3. Exactness and stable subcategories
  4. Localizing invariants
  5. Dévissage condition for stable subcategory
  6. Category of sequences
  7. Definition of dévissage condition
  8. Theorems for invariants
  9. Additivity theorem
  10. Categorfication of fiber with dévissage condition
  11. Categorfication of loop
  12. Fillability
  13. $1$-fillable subcategories
  14. Vanishing of $K_0$-group of quotients
  15. Higher fillability for higher $K$-groups
  16. ...and 5 more sections

Key Result

Theorem A

(Theorem mainresult1) Let $\mathcal{C}\in\mathrm{Cat}_\infty^{\mathrm{perf}}$ be a small stable idempotent-complete $\infty$-category and $\mathcal{A}$ be an idempotent $n$-fillable subcategory of $\mathcal{C}$. If $(\mathcal{C},\mathcal{A})$ satisfies the weak dévissage condition(resp. dévissage co

Theorems & Definitions (163)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.5
  • proof
  • Lemma 1.6
  • ...and 153 more