2021-10-03

  • perfect complex and perfect modules?

Spectra Stuff

Tags: #stable-homotopy

Producing a LES:

  • Take a map \(A \xrightarrow{f} B\)
  • Extract cofibers to get \(A \to B \to {\operatorname{hocofib}}(f) \to \cdots\)
  • Apply \([\mathop{\mathrm{{\Sigma_+^\infty}}}({-}), E]_{-n}\)

Integration pairing: for \(E \in {\mathsf{SHC}}(\mathsf{Ring})\), \begin{align*} E^*X &\longrightarrow E_* X \\ \omega \in [\mathop{\mathrm{{\Sigma_+^\infty}}}X, E] &\longrightarrow\alpha \in [{\mathbb{S}}, E\wedge X] \\ \\ {\mathbb{S}}\xrightarrow{\alpha} E \wedge X \cong E\wedge{\mathbb{S}}\wedge X &\cong E \wedge\mathop{\mathrm{{\Sigma_+^\infty}}}X \xrightarrow{1\wedge\omega } E{ {}^{ \scriptscriptstyle\wedge^{2} } } \xrightarrow{\mu} E .\end{align*}

  • Cohomology operations : natural transformations \(E^n({-})\to F^m({-})\).
  • Classified by maps \(E_n \to F_m\), i.e. \(F^m(E_n)\).
  • E.g. Steenrod squares \(\operatorname{Sq}^i \in [K(C_2, n), K(C_2, n+i)]\).
  • They’re in fact stable, so live in \(HC_2^*(HC_2)\).
  • In general, algebras of stable operations for a cohomology theory \(E\) are exactly \(E^*(E)\).

Categories

Tags: #higher-algebra/category-theory #higher-algebra/simplicial #higher-algebra/infty-cats

  • Recall \({\mathsf{sSet}}= [\Delta^{\operatorname{op}}, {\mathsf{Set}}] = {\mathsf{Fun}}(\Delta^{\operatorname{op}}, {\mathsf{Set}}) = {\mathsf{Set}}^{\Delta^{\operatorname{op}}}\).

  • For \(x_0 \in \mathsf{C}\), a cone from \(x_0\) to \(F\in [J, C]\) for \(J\) any diagram category is a family \(\psi_x: x_0 \to F(x)\) making diagrams commute:

Link to Diagram

  • Extranatural transformations are given by a certain string calculus:

attachments/2021-10-03_01-51-54.png

  • free cocompletion of a category: \(\mathsf{C} \mapsto [\mathsf{C}, {\mathsf{Set}}]\).

  • Cauchy completeness for a category: closure under all colimits that are preserved by every functor.

  • Subfunctor : \(G\leq F\) iff \(G(x) \subseteq F(x)\) and for all \(x \xrightarrow{f} y\), require \(F(f)(G(x)) \subseteq G(y)\).

Lie Algebras?

References: https://arxiv.org/pdf/0801.3480.pdf and https://people.math.umass.edu/~gwilliam/thesis.pdf

Tags: #projects/notes/reading Lie algebra L_infty algebra

Defining the Chevalley-Eilenberg complex: attachments/2021-10-03_14-50-57.png

Differential forms for L_infty algebras

Differential forms for an L_infty algebra attachments/2021-10-03_13-39-30.png

Cartan-Ehresmann connections, descent attachments/2021-10-03_13-42-38.png

String structure on \(X\): spin structures on \({\Omega}X\). Computing Lie group cohomology using the Chevalley-Eilenberg complex: attachments/2021-10-03_13-59-30.png

Relation to BG: attachments/2021-10-03_14-02-16.png

Defining algebra-valued forms when curvature doesn’t vanish.

attachments/2021-10-03_14-05-07.png

attachments/2021-10-03_14-04-31.png

attachments/2021-10-03_14-09-46.png

Feynman diagrams

Feynman diagram attachments/2021-10-03_15-31-46.png

attachments/2021-10-03_15-31-59.png

See BV quantization

Differentiable vector spaces, connections

attachments/2021-10-03_19-23-07.png

attachments/2021-10-03_19-23-30.png

Some derived cats

  • A version of derived categories in infty-category world: derived infinity category : dg nerve of subcategory of fibrant objects. Always a stable infinity category, equivalent to original category localized at weak equivalences.

  • Alternatively: take subcategory of fibrant objects, observe enrichment over chain complexes, apply Dold-Kan to get a simplicial enrichment, then take the homotopy coherent nerve or simplicial nerve.

  • Getting a chain complex from a simplicial set : take free \({\mathbb{Z}}{\hbox{-}}\)modules levelwise, then apply Dold-Kan.

Postnikov/Whitehead stuff

  • How (I think?) Postnikov and Whitehead towers are related:

Link to Diagram

  • Defining factorization algebras :

attachments/2021-10-03_21-11-38.png

attachments/2021-10-03_21-13-39.png

Links to this page
#web/quick-notes #stable-homotopy #higher-algebra/category-theory #higher-algebra/simplicial #higher-algebra/infty-cats #projects/notes/reading