Talk:A¹ homotopy theory

Article improvements

 * Discuss fiber functors in the relevant article as it relates here
 * Clean up denoting between unstable and A^1 local unstable theories
 * Add cohomology theories such as K-theory, (pg 381 of Morel notes http://users.ictp.it/~pub_off/lectures/lns015/Morel/Morel.pdf and pg 313 of Thomason's higher K-theory of schemes paper https://www.gwern.net/docs/math/1990-thomason.pdf)
 * Mention how K-theory works in Zariski, but not etale topology here
 * Example 3.1.6 in morel notes
 * Example 3.1.7 gives references to infinite grassmanian and BGL, giving algebraic k-theory the expected representatives
 * Example 3.1.10
 * Theorem 3.3.4 on Thom spaces
 * The two stable categories

T-spectra
Wundzer (talk) 16:58, 4 February 2021 (UTC)
 * Define the homotopy spheres (remark 5.1.5) + pg 417-418
 * Rost motives (Rost cycles)
 * Theorems 6.3.3 and onwards on Milnor K-theory

Adding material on sphere spectra
There should be a discussion for the stable motivic homotopy spheres. This should include the definition of the spheres and the related vanishing theorems. Moreover, there should be ample motivation behind their definition/construction so it's clear why they are defined the way they are.

Using https://web.archive.org/web/20201128162813/http://users.ictp.it/~pub_off/lectures/lns015/Morel/Morel.pdf
 * Extract symmetric monoidal structure from chapter 4 and state it for the spectra in chapter 5
 * This way the spheres can be defined in chapter 5 using T-spectra
 * Discuss A^1 localization
 * Give diagram showing the isomorphism P^1/A^1 \cong T
 * pg 417 gives the definition, note the negative smash products come from the symmetric monoidal structure
 * Also note the unit is the zero sphere

Additional spectra
This is in ch 5
 * Eilenberg-Maclane spectra
 * Motivic cohomology spectra

Some other references/resources

 * Stable connectivity over a base - https://arxiv.org/abs/1911.05014
 * Vanishing in stable motivic homotopy sheaves - https://arxiv.org/abs/1704.04744
 * A1-Algebraic Topology over a Field - 6.43 on page 167 gives vanishing result in the unstable situation - 978-3-642-29513-3

Other
pg 403 of Morel notes gives a Postnikov tower for t-structures, this should be included in the Postnikov article
 * https://arxiv.org/abs/1002.5007 contains a collapsing spectral sequence
 * Voevodsky's Seattle Lectures: K-theory and Motivic Cohomology - https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/Seatle_lectures_notes_by_Weibel_published.pdf

Wundzer (talk) 16:23, 19 February 2021 (UTC)