# 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

• 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:

• Extranatural transformations are given by a certain string calculus:

• 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?

Defining the Chevalley-Eilenberg complex:

## Differential forms for L_infty algebras

Differential forms for an L_infty algebra

Cartan-Ehresmann connections, descent

String structure on $$X$$: spin structures on $${\Omega}X$$. Computing Lie group cohomology using the Chevalley-Eilenberg complex:

Relation to BG:

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

Feynman diagram

## 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.