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Big pictures of motivic and classical homotopy theories

Ahmad Rouintan

Abstract

Motivic homotopy theory is meant to play the role of algebraic topology, in particular homotopy theory, in the context of algebraic geometry. As proved by Oliver Rondigs and Paul Arne Ostvaer, this theory is closely connected to Voevodsky's triangulated category of motives. A connection that is the motivic analogue of the connection between algebraic topology and homological algebra. In this paper, we try to understand the big picture of motivic homotopy theory and its connection to Voevodsky's motives by comparison to the classical counterpart.

Big pictures of motivic and classical homotopy theories

Abstract

Motivic homotopy theory is meant to play the role of algebraic topology, in particular homotopy theory, in the context of algebraic geometry. As proved by Oliver Rondigs and Paul Arne Ostvaer, this theory is closely connected to Voevodsky's triangulated category of motives. A connection that is the motivic analogue of the connection between algebraic topology and homological algebra. In this paper, we try to understand the big picture of motivic homotopy theory and its connection to Voevodsky's motives by comparison to the classical counterpart.
Paper Structure (20 sections, 16 theorems, 37 equations)

This paper contains 20 sections, 16 theorems, 37 equations.

Table of Contents

  1. The picture in classical homotopy theory
  2. Reduced cohomology theories
  3. Homotopy category of spaces
  4. Stable homotopy category of spaces
  5. Reduced (co)homology theories as spectra
  6. The link between spaces and chain complexes of abelian groups
  7. The picture in motivic homotopy theory: Part 1
  8. Homotopy category of sites with an interval
  9. Motivic unstable homotopy category
  10. Spheres, homotopy sheaves, and suspensions
  11. Motivic stable homotopy category
  12. Cohomology theories
  13. The picture in motivic homotopy theory: Part 2
  14. Motivic spaces with transfers
  15. Unstable homotopy category of motivic spaces with transfers
  16. ...and 5 more sections

Key Result

Theorem 1.2

The category $\mathbf{Top}$ with the class of usual weak equivalences and the class of Serre fibrations becomes a model category. Also, the category $\mathbf{Top}_\bullet$ of pointed spaces is a model category with its weak equivalences and fibrations being the usual weak equivalences and Serre fibr

Theorems & Definitions (51)

  • Definition 1.1
  • Theorem 1.2: Hirschhorn2002, Theorem 7.10.10 and 7.10.11
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5: Freudenthal suspension theorem, Freudenthal1938
  • Definition 1.6
  • Example 1.7
  • Definition 1.8
  • Definition 1.9
  • Example 1.10: Singular (Co)homology
  • ...and 41 more