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Controlled objects in left-exact $\infty$-categories and the Novikov conjecture

Ulrich Bunke, Denis-Charles Cisinski, Daniel Kasprowski, Christoph Winges

Abstract

We associate to every $G$-bornological coarse space $X$ and every left-exact $\infty$-category with $G$-action a left-exact infinity-category of equivariant $X$-controlled objects. Postcomposing with algebraic K-theory leads to {new} equivariant coarse homology theories. This allows us to apply the injectivity results for assembly maps by Bunke, Engel, Kasprowski and Winges to the algebraic K-theory of left-exact $\infty$-categories.

Controlled objects in left-exact $\infty$-categories and the Novikov conjecture

Abstract

We associate to every -bornological coarse space and every left-exact -category with -action a left-exact infinity-category of equivariant -controlled objects. Postcomposing with algebraic K-theory leads to {new} equivariant coarse homology theories. This allows us to apply the injectivity results for assembly maps by Bunke, Engel, Kasprowski and Winges to the algebraic K-theory of left-exact -categories.

Paper Structure

This paper contains 37 sections, 147 theorems, 545 equations.

Table of Contents

  1. Introduction
  2. Functors on the orbit category and split injectivity of assembly maps
  3. Controlled objects and coarse homology theories
  4. infinity-category background
  5. Left-exact infinity-categories
  6. The calculus of fractions formula
  7. Localisations of left-exact infinity-categories
  8. Stabilisation and cofibres
  9. Excisive squares in CatLex and homological functors
  10. Sheaves on bornological coarse spaces
  11. Presheaves
  12. Sheaves
  13. Sheaves on GCoarse
  14. Equivariantly small sheaves on $G$-bornological coarse spaces
  15. Localisation
  16. ...and 22 more sections

Key Result

Theorem 1.1.5

The functor $K\mathbf{D}_{G}$ is a hereditary CP-functor.

Theorems & Definitions (426)

  • Theorem 1.1.5: \ref{['rgiuhreiguhgwergergrwegwergreg']}
  • Theorem 1.1.6
  • Example 1.1.7
  • Theorem 1.1.10: desc,desc
  • Example 1.1.11
  • Theorem 1.1.12: Bunke:aa
  • Example 1.1.13
  • Example 1.1.14
  • Example 1.1.15
  • Remark 1.1.19
  • ...and 416 more