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 :
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
Tags: #homotopy/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: 
Thom-Dold, Poincare duality, Chern classes 
Postnikov invariants: 
formal group laws and generalized cohomolology theory and complex oriented cohomology theory: 
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.