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Talk:A¹ homotopy theory

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Article improvements

[edit]
  • 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

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  • 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

Wundzer (talk) 16:58, 4 February 2021 (UTC)[reply]

Adding material on sphere spectra

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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

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This is in ch 5

  • Eilenberg-Maclane spectra
  • Motivic cohomology spectra

Some other references/resources

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Other

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pg 403 of Morel notes gives a Postnikov tower for t-structures, this should be included in the Postnikov article

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