Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

K-theory and localizing invariants of large categories

Alexander I. Efimov

Abstract

In this paper we introduce and study the so-called continuous $K$-theory for a certain class of "large" stable $\infty$-categories, more precisely, for dualizable presentable categories. For compactly generated categories, the continuous $K$-theory is simply the usual (non-connective) $K$-theory of the full subcategory of compact objects. More generally, we show that any localizing invariant of small stable $\infty$-categories can be uniquely extended to a localizing invariant of dualizable categories. We compute the continuous $K$-theory for categories of sheaves on locally compact Hausdorff spaces. Using the special case for sheaves on the real line, we give an alternative proof of the theorem of Kasprowski and Winges \cite{KW19} on the commutation of $K$-theory with infinite products for small stable $\infty$-categories. We also study the general theory of dualizable categories. In particular, we give an "explicit" proof of Ramzi's theorem \cite{Ram24a} on the $ω_1$-presentability of the category of dualizable categories. Among other things, we prove that dualizability is equivalent to "flatness" in the category of presentable stable categories.

K-theory and localizing invariants of large categories

Abstract

In this paper we introduce and study the so-called continuous -theory for a certain class of "large" stable -categories, more precisely, for dualizable presentable categories. For compactly generated categories, the continuous -theory is simply the usual (non-connective) -theory of the full subcategory of compact objects. More generally, we show that any localizing invariant of small stable -categories can be uniquely extended to a localizing invariant of dualizable categories. We compute the continuous -theory for categories of sheaves on locally compact Hausdorff spaces. Using the special case for sheaves on the real line, we give an alternative proof of the theorem of Kasprowski and Winges \cite{KW19} on the commutation of -theory with infinite products for small stable -categories. We also study the general theory of dualizable categories. In particular, we give an "explicit" proof of Ramzi's theorem \cite{Ram24a} on the -presentability of the category of dualizable categories. Among other things, we prove that dualizability is equivalent to "flatness" in the category of presentable stable categories.
Paper Structure (47 sections, 155 theorems, 310 equations)

This paper contains 47 sections, 155 theorems, 310 equations.

Table of Contents

  1. Introduction
  2. General theory of dualizable categories
  3. Infinity-categories, limits and colimits
  4. Presentable and accessible stable $\infty$-categories
  5. Well generated triangulated categories
  6. Dualizable categories
  7. Examples of compactly assembled categories, stable and non-stable
  8. Continuous stabilization
  9. Duality as a covariant involution on dualizable categories
  10. Dualizability via compact morphisms
  11. The axiom (AB6)
  12. Automatic dualizability and images of functors
  13. Calkin categories
  14. Colimits of dualizable categories
  15. Semi-orthogonal decompositions
  16. ...and 32 more sections

Key Result

Theorem 1

Let ${\mathcal{E}}$ be a stable $\infty$-category. The precomposition functor induces an equivalence between the full subcategories of localizing invariants. The inverse equivalence is given by $F\mapsto F^{\operatorname{cont}}.$

Theorems & Definitions (382)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Definition 1.1
  • Proposition 1.2
  • proof
  • Remark 1.3
  • Proposition 1.4
  • proof
  • ...and 372 more