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
elliptic curve and deformation theory :
presentable infinity category. deformation-obstruction theory :
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
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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*}
Volcano Stuff
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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.
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Minor aside: \({\mathbf{B}}\mathsf{C} \coloneqq{ {\left\lvert {{ \mathcal{N}({\mathsf{C}}) }} \right\rvert} }\).
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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.