2021-10-05 quick_notes

Tags: #web/quick-notes

2021-10-05

DAG-X

Tags: #projects/notes/reading #higher-algebra/derived #higher-algebra/infty-cats

Derived AG: https://people.math.harvard.edu/~lurie/papers/DAG-X.pdf

dg Lie algebras :

attachments/2021-10-05_00-03-49.png

elliptic curve and deformation theory :

attachments/2021-10-05_00-05-28.png

presentable infinity category. deformation-obstruction theory :

attachments/2021-10-05_00-08-54.png

k-linear category :

attachments/2021-10-05_00-19-40.png

attachments/2021-10-05_00-21-36.png

attachments/2021-10-05_00-28-30.png

attachments/2021-10-05_00-30-48.png

attachments/2021-10-05_00-33-46.png

attachments/2021-10-05_00-34-14.png

10:49

Weak weak approximation would imply a positive answer to the inverse Galois problem.

20:02

partition function: attachments/2021-10-05_20-02-50.png

Elliptic Cohomology Paper

Tags: #stable-homotopy #physics #projects/notes/summaries

Refs: Elliptic cohomology, Thom-Dold, orientability, formal group law, ring spectra, Bousfield localization, tmf,

Reference: M-theory, type IIA superstrings, and elliptic cohomology https://arxiv.org/pdf/hep-th/0404013.pdf

orientability of spectra: attachments/2021-10-05_20-39-39.png

attachments/2021-10-05_20-40-20.png

Thom-Dold, Poincare duality, Chern classes attachments/2021-10-05_20-41-16.png

Gysin: attachments/2021-10-05_20-41-33.png

Postnikov invariants: attachments/2021-10-05_20-41-56.png

formal group laws and generalized cohomolology theory and complex oriented cohomology theory: attachments/2021-10-05_20-42-42.png

attachments/2021-10-05_20-43-37.png

Brown-Peterson spectra and Johnson-Wilson spectrum: attachments/2021-10-05_20-44-09.png

Morava K theory: attachments/2021-10-05_20-44-36.png

heights of formal group laws: attachments/2021-10-05_20-45-25.png

Elliptic cohomology: attachments/2021-10-05_20-46-47.png

Bousfield localization: attachments/2021-10-05_20-48-43.png

attachments/2021-10-05_20-51-54.png

tmf: attachments/2021-10-05_20-51-38.png

22:49

  • Hom is a continuous functor, i.e. it preserves limits in both variables. Just remember that the first argument is contravariant, so \begin{align*} \cocolim_i \cocolim_j \mathsf{C}(A_i, B_j) = \mathsf{C}(\colim_i A_i, \cocolim_j B_j) .\end{align*}

  • tannaka duality and tannaka reconstruction :

attachments/2021-10-05_23-01-03.png

attachments/2021-10-05_23-04-52.png

Volcano Stuff

  • How K-theory goes:

    • Form a symmetric monoidal category \(\mathsf{C}\), which is a commutative monoid object in infinity categories.
    • Apply \({ \mathsf{core} }: \mathsf{Cat}\to{\mathsf{Grpd}}\) to replace \(\mathsf{C}\) with \({ \mathsf{core} }\mathsf{C}\), which separates isomorphism classes into separate connected components. It turns out this lands in \({\mathbb{E}}_\infty{\hbox{-}}\)spaces, i.e. commutative monoid objects in infinity-groupoids.
    • Apply group completion of \((\infty, 1){\hbox{-}}\)categories to get an abelian group object in infinity-groupoids.
    • Identify these with connective spectra.
    • Include into the category of all spectra.
  • Minor aside: \({\mathbf{B}}\mathsf{C} \coloneqq{ {\left\lvert {{ \mathcal{N}({\mathsf{C}}) }} \right\rvert} }\).

  • Start with the category of elliptic curves : should be pointed algebraic group, so a slice category over a terminal object..?

    • Then take covering category: objects are based surjections \(E_1 \twoheadrightarrow E_2\), morphisms
    • Restrict to “covering spaces”: fibers are finite and discrete.
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