Tags: #quick_notes

2021-10-05

DAG-X

Tags: #reading_notes #derived #infinity_cats

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

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

Elliptic Cohomology Paper

Tags: #stable_homotopy #physics #summaries

Refs: Elliptic cohomology, Thom-Dold, [[Orientability of spectra|orientability]], formal group law, ring spectra, Bousfield localization, [[Topological modular forms|tmf]],

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

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|coslice 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.