--- date: 2021-10-03 tags: [ web/quick-notes ] --- # 2021-10-03 - [perfect complex](perfect%20complexes.md) and perfect modules? ## Spectra Stuff Tags: #homotopy/stable-homotopy Producing a LES: - Take a map $A \mapsvia{f} B$ - Extract cofibers to get $A \to B \to \hocofib(f) \to \cdots$ - Apply $[\Suspendpinf(\wait), E]_{-n}$ Integration pairing: for $E \in \SHC(\Ring)$, \[ E^*X &\too E_* X \\ \omega \in [\Suspendpinf X, E] &\too \alpha \in [\SS, E\smashprod X] \\ \\ \SS \mapsvia{\alpha} E \smashprod X \cong E\smashprod \SS \smashprod X &\cong E \smashprod \Suspendpinf X \mapsvia{1\smashprod \omega } E\smashpower{2} \mapsvia{\mu} E .\] - [Cohomology operations](Cohomology%20operations.md) : natural transformations $E^n(\wait)\to F^m(\wait)$. - Classified by maps $E_n \to F_m$, i.e. $F^m(E_n)$. - E.g. [Steenrod squares](Steenrod%20squares) $\Sq^i \in [K(C_2, n), K(C_2, n+i)]$. - 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 $\sset = [\Delta\op, \Set] = \Fun(\Delta\op, \Set) = \Set^{\Delta\op}$. - For $x_0 \in \cat C$, 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: \begin{tikzcd} {x_1} && {F(x_1)} \\ &&&& {x_0} \\ {x_2} && {F(x_2)} \arrow["f"', from=1-1, to=3-1] \arrow[from=2-5, to=1-3] \arrow[from=2-5, to=3-3] \arrow["{F(f)}", from=1-3, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMiwwLCJGKHhfMSkiXSxbMCwwLCJ4XzEiXSxbMCwyLCJ4XzIiXSxbMiwyLCJGKHhfMikiXSxbNCwxLCJ4XzAiXSxbMSwyLCJmIiwyXSxbNCwwXSxbNCwzXSxbMCwzLCJGKGYpIl1d) - [Extranatural transformations](extranatural%20transformations) are given by a certain [string calculus](string%20calculus): ![](attachments/2021-10-03_01-51-54.png) - [free cocompletion](Unsorted/free%20cocompletion.md) of a category: $\cat C \mapsto [\cat C, \Set ]$. - Cauchy completeness for a category: closure under all [colimits](colimits) that are preserved by every functor. - [Subfunctor](Subfunctor) : $G\leq F$ iff $G(x) \subseteq F(x)$ and for all $x \mapsvia{f} y$, require $F(f)(G(x)) \subseteq G(y)$. ## Lie Algebras? > References: > > and Tags: #projects/notes/reading [Lie algebra](Unsorted/Lie%20algebra.md) [L_infty algebra](Unsorted/L%20infty%20algebra.md) Defining the [Chevalley-Eilenberg complex](Unsorted/Chevalley-Eilenberg%20complex.md): ![](attachments/2021-10-03_14-50-57.png) ## Differential forms for L_infty algebras Differential forms for an [L_infty algebra](Unsorted/L%20infty%20algebra.md) ![](attachments/2021-10-03_13-39-30.png) [Cartan-Ehresmann connections](Cartan-Ehresmann%20connections), [descent](Unsorted/descent.md) ![](attachments/2021-10-03_13-42-38.png) [String structure](String%20structure) on $X$: spin structures on $\Loop X$. Computing Lie group cohomology using the [Chevalley-Eilenberg complex](Unsorted/Chevalley-Eilenberg%20complex.md): ![](attachments/2021-10-03_13-59-30.png) Relation to [BG](Unsorted/classifying%20space.md): ![](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](Feynman%20diagram) ![](attachments/2021-10-03_15-31-46.png) ![](attachments/2021-10-03_15-31-59.png) See [BV quantization](Unsorted/BV%20quantization.md) ## 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](Unsorted/derived%20category.md) in [infty-category](Unsorted/infinity%20categories.md) world: [derived infinity category](derived%20infinity%20category) : [dg nerve](dg%20nerve) of subcategory of [fibrant](Unsorted/fibrant%20and%20cofibrant%20objects.md) objects. Always a [stable infinity category](stable%20infinity%20category.md), equivalent to original category localized at weak equivalences. - Alternatively: take subcategory of fibrant objects, observe [enrichment](Unsorted/enriched%20category.md) over chain complexes, apply [Dold-Kan](Unsorted/Dold-Kan%20correspondence.md) to get a simplicial enrichment, then take the [homotopy coherent nerve](Unsorted/homotopy%20coherence.md) or [simplicial nerve](simplicial%20nerve). - Getting a chain complex from a [simplicial set](simplicial%20set.md) : take free \(\ZZ\dash\)modules levelwise, then apply [Dold-Kan](Unsorted/Dold-Kan%20correspondence.md). ## Postnikov/Whitehead stuff - How (I think?) Postnikov and Whitehead towers are related: \begin{tikzcd} {\lim_n X_n \cong \pt} && X && {\lim_n X_n \cong X} \\ \vdots && \vdots && \vdots \\ {X^2 = \hofib(f_2)} && X && {X_2 = \tau_{\leq 2} X} \\ {X^1 = \hofib(f_1)} && X && {X_1 = \tau_{\leq 1} X} \\ {X^0 = \hofib(0)} && X && {X_0 = \pt} \arrow["{f_2}", from=3-3, to=3-5] \arrow["{f_1}", from=4-3, to=4-5] \arrow["0", from=5-3, to=5-5] \arrow[from=5-1, to=5-3] \arrow[Rightarrow, no head, from=5-3, to=4-3] \arrow[Rightarrow, from=4-3, to=3-3] \arrow[from=4-1, to=4-3] \arrow[from=3-1, to=3-3] \arrow["\cong", from=1-3, to=1-5] \arrow[from=1-1, to=1-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) - Defining [factorization algebras](factorization%20algebras) : ![](attachments/2021-10-03_21-11-38.png) ![](attachments/2021-10-03_21-13-39.png)