# 2021-10-05 quick_notes

Tags: #web/quick-notes

# 2021-10-05

## DAG-X

dg Lie algebras :

elliptic curve and deformation theory :

presentable infinity category. deformation-obstruction theory :

k-linear category :

## 10:49

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

## 20:02

partition function:

## Elliptic Cohomology Paper

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:

Thom-Dold, Poincare duality, Chern classes

Postnikov invariants:

Brown-Peterson spectra and Johnson-Wilson spectrum:

tmf:

## 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 :

# 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.