---
date: 2021-10-05
tags: [ web/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](https://people.math.harvard.edu/~lurie/papers/DAG-X.pdf)

[dg Lie algebras](dg%20Lie%20algebras.md) : 

![attachments/2021-10-05_00-03-49.png](attachments/2021-10-05_00-03-49.png)

[elliptic curve](elliptic%20curve.md) and [deformation theory](deformation%20theory.md) : 

![attachments/2021-10-05_00-05-28.png](attachments/2021-10-05_00-05-28.png)

[presentable infinity category](presentable%20infinity%20category). [deformation-obstruction theory](deformation-obstruction%20theory) : 

![attachments/2021-10-05_00-08-54.png](attachments/2021-10-05_00-08-54.png)

[k-linear category](k-linear%20category) :

![attachments/2021-10-05_00-19-40.png](attachments/2021-10-05_00-19-40.png)

![attachments/2021-10-05_00-21-36.png](attachments/2021-10-05_00-21-36.png)

![attachments/2021-10-05_00-28-30.png](attachments/2021-10-05_00-28-30.png)

![attachments/2021-10-05_00-30-48.png](attachments/2021-10-05_00-30-48.png)

![attachments/2021-10-05_00-33-46.png](attachments/2021-10-05_00-33-46.png)

![attachments/2021-10-05_00-34-14.png](attachments/2021-10-05_00-34-14.png)

## 10:49

Weak [weak approximation](weak%20approximation.md) would imply a positive answer to the [inverse Galois problem](inverse%20Galois%20problem.md).

## 20:02

[partition function](partition%20function):
![attachments/2021-10-05_20-02-50.png](attachments/2021-10-05_20-02-50.png)

## Elliptic Cohomology Paper

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

Refs:
[Elliptic cohomology](Elliptic%20cohomology.md),
[Thom-Dold](Thom-Dold),
[orientability](orientability%20of%20spectra),
[formal group law](formal%20group%20law),
[ring spectra](ring%20spectra.md),
[Bousfield localization](Bousfield%20localization.md),
[tmf](Topological%20modular%20forms),


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

[orientability of spectra](orientability%20of%20spectra):
![attachments/2021-10-05_20-39-39.png](attachments/2021-10-05_20-39-39.png)

![attachments/2021-10-05_20-40-20.png](attachments/2021-10-05_20-40-20.png)

[Thom-Dold](Thom-Dold), [Poincare duality](Poincare%20duality), [Chern classes](Unsorted/Chern%20class.md)
![attachments/2021-10-05_20-41-16.png](attachments/2021-10-05_20-41-16.png)

[Gysin](Unsorted/Gysin%20sequence.md):
![attachments/2021-10-05_20-41-33.png](attachments/2021-10-05_20-41-33.png)

[Postnikov](Unsorted/Obstruction%20theory%20in%20homotopy.md) invariants:
![attachments/2021-10-05_20-41-56.png](attachments/2021-10-05_20-41-56.png)

[formal group laws](Unsorted/Formal%20group.md) and [generalized cohomolology theory](Unsorted/cohomolology%20theory.md) and [complex oriented cohomology theory](Unsorted/complex%20oriented%20cohomology%20theory.md):
![attachments/2021-10-05_20-42-42.png](attachments/2021-10-05_20-42-42.png)

![attachments/2021-10-05_20-43-37.png](attachments/2021-10-05_20-43-37.png)

[Brown-Peterson spectra](Brown-Peterson%20spectra) and [Johnson-Wilson spectrum](Johnson-Wilson%20spectrum):
![attachments/2021-10-05_20-44-09.png](attachments/2021-10-05_20-44-09.png)

[Morava K theory](Unsorted/Morava%20K%20theory.md):
![attachments/2021-10-05_20-44-36.png](attachments/2021-10-05_20-44-36.png)

[heights of formal group laws](Unsorted/Formal%20group.md):
![attachments/2021-10-05_20-45-25.png](attachments/2021-10-05_20-45-25.png)

[Elliptic cohomology](Unsorted/Elliptic%20cohomology.md):
![attachments/2021-10-05_20-46-47.png](attachments/2021-10-05_20-46-47.png)

[Bousfield localization](Unsorted/Bousfield%20localization.md):
![attachments/2021-10-05_20-48-43.png](attachments/2021-10-05_20-48-43.png)

![attachments/2021-10-05_20-51-54.png](attachments/2021-10-05_20-51-54.png)

[tmf](tmf):
![attachments/2021-10-05_20-51-38.png](attachments/2021-10-05_20-51-38.png)

## 22:49

- Hom is a [continuous functor](continuous%20functor), i.e. it preserves limits in both variables.
  Just remember that the first argument is contravariant, so
  \[
  \cocolim_i \cocolim_j \cat{C}(A_i, B_j) = \cat{C}(\colim_i A_i, \cocolim_j B_j)
  .\]

- [tannaka duality](Unsorted/Tannakian.md) and [tannaka reconstruction](tannaka%20reconstruction) : 

![attachments/2021-10-05_23-01-03.png](attachments/2021-10-05_23-01-03.png)

![attachments/2021-10-05_23-04-52.png](attachments/2021-10-05_23-04-52.png)

# Volcano Stuff

- How [K-theory](K-theory.md) goes:
  - Form a [symmetric monoidal category](Unsorted/monoidal%20category.md) $\cat{C}$, which is a commutative monoid object in [infinity categories](infinity%20categories.md).
  - Apply $\core: \Cat\to\Grpd$ to replace $\cat C$ with $\core \cat C$, which separates isomorphism classes into separate connected components.
  It turns out this lands in $\EE_\infty\dash$spaces, i.e. commutative monoid objects in infinity-groupoids.
  - Apply [group completion](group%20completion) of $(\infty, 1)\dash$categories to get an abelian group object in infinity-groupoids.
  - Identify these with [connective](connective) [spectra](spectra.md).
  - Include into the category of all spectra.

- Minor aside: $\B \cat{C} \da \realize{\nerve{\cat C}}$.


- Start with the category of [elliptic curves](elliptic%20curve.md) : should be pointed [algebraic group](algebraic%20group.md), so a [slice category](slice%20category) over a terminal object..?
  - Then take covering category: objects are based surjections $E_1 \surjects E_2$, morphisms 
  - Restrict to "[covering spaces](Unsorted/covering%20space.md)": fibers are finite and discrete.