- [[perfect complexes|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: #category_theory #simplicial #infinity_cats
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Recall \({\mathsf{sSet}}= [\Delta^{\operatorname{op}}, {\mathsf{Set}}] = {\mathsf{Fun}}(\Delta^{\operatorname{op}}, {\mathsf{Set}}) = {\mathsf{Set}}^{\Delta^{\operatorname{op}}}\).
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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|Extranatural transformations]] are given by a certain string calculus:
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Free cocompletion of a category: \(\mathsf{C} \mapsto [\mathsf{C}, {\mathsf{Set}}]\).
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Cauchy completeness for a category: closure under all colimits that are preserved by every functor.
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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: #reading_notes #lie_algebras
- Some spin stuff and algebra valued differential forms
String structures on \(X\): spin structures on \({\Omega}X\).
Defining algebra-valued forms when curvature doesn’t vanish:
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derived infinity category : differential graded nerve of subcategory of fibrant objects. Always a stable infinity category, and localizes at weak equivalences.
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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.
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Getting a chain complex from a simplicial set : take free \({\mathbb{Z}}{\hbox{-}}\)modules levelwise, then apply Dold-Kan.
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How (I think?) Postnikov and Whitehead towers are related:
- Defining factorization algebras :