Norms in motivic homotopy theory

Chapter 1. Introduction
1.1. Norm functors
1.2. Normed motivic spectra
1.3. Examples of normed spectra
1.4. Norms in other contexts
1.5. Norms vs. framed transfers
1.6. Summary of the construction
1.7. Summary of results
1.8. Guide for the reader
1.9. Remarks on -categories
1.10. Standing assumptions
1.11. Acknowledgments
Chapter 2. Preliminaries
2.1. Nonabelian derived -categories
2.2. Unstable motivic homotopy theory
2.3. Weil restriction
Chapter 3. Norms of pointed motivic spaces
3.1. The unstable norm functors
3.2. Norms of quotients
Chapter 4. Norms of motivic spectra
4.1. Stable motivic homotopy theory
4.2. The stable norm functors
Chapter 5. Properties of norms
5.1. Composition and base change
5.2. The distributivity laws
5.3. The purity equivalence
5.4. The ambidexterity equivalence
5.5. Polynomial functors
Chapter 6. Coherence of norms
6.1. Functoriality on the category of spans
6.2. Normed -categories
Chapter 7. Normed motivic spectra
7.1. Categories of normed spectra
7.2. Cohomology theories represented by normed spectra
Chapter 8. The norm-pullback-pushforward adjunctions
8.1. The norm-pullback adjunction
8.2. The pullback-pushforward adjunction
Chapter 9. Spectra over profinite groupoids
9.1. Profinite groupoids
9.2. Norms in stable equivariant homotopy theory
Chapter 10. Norms and Grothendieck's Galois theory
10.1. The profinite étale fundamental groupoid
10.2. Galois-equivariant spectra and motivic spectra
10.3. The Rost norm on Grothendieck-Witt groups
Chapter 11. Norms and Betti realization
11.1. A topological model for equivariant homotopy theory
11.2. The real Betti realization functor
Chapter 12. Norms and localization
12.1. Inverting Picard-graded elements
12.2. Inverting elements in normed spectra
12.3. Completion of normed spectra
Chapter 13. Norms and the slice filtration
13.1. The zeroth slice of a normed spectrum
13.2. Applications to motivic cohomology
13.3. Graded normed spectra
13.4. The graded slices of a normed spectrum
Chapter 14. Norms of cycles
14.1. Norms of presheaves with transfers
14.2. The Fulton-MacPherson norm on Chow groups
14.3. Comparison of norms
Chapter 15. Norms of linear -categories
15.1. Linear -categories
15.2. Noncommutative motivic spectra and homotopy K-theory
15.3. Nonconnective K-theory
Chapter 16. Motivic Thom spectra
16.1. The motivic Thom spectrum functor
16.2. Algebraic cobordism and the motivic J-homomorphism
16.3. Multiplicative properties
16.4. Free normed spectra
16.5. Thom isomorphisms
Appendix A. The Nisnevich topology
Appendix B. Detecting effectivity
Appendix C. Categories of spans
C.1. Spans in extensive -categories
C.2. Spans and descent
C.3. Functoriality of spans
Appendix D. Relative adjunctions
Table of notation
Bibliography