--- date: 2023-01-09 13:47 aliases: ["Misc"] flashcard: "Reading::Misc" --- Last modified: `=this.file.mday` --- - Tags: - #todo/untagged - Refs: - #todo/add-references - Links: - #todo/create-links --- # Misc - [x] What is a symmetric domain? ✅ 2023-01-23 - [ ] Let $G$ be a semisimple group over $\mathbb{Q}$. A subgroup $\Gamma \subseteq G(\mathbb{Q})$ is called congruence if there exists an embedding $G \hookrightarrow \mathrm{GL}_n$ such that $\Gamma$ contains$$\operatorname{ker}\left(\mathrm{GL}_n(\mathbb{Z}) \rightarrow \mathrm{GL}_n(\mathbb{Z} / N \mathbb{Z})\right) \cap G(\mathbb{Q})$$for some $N \geqslant 1$. Set $X_G^{+}=X^{+}:=G(\mathbb{R})^{+} / C$. Then, a space of the form $\Gamma^{+} \backslash X^{+}$is called a symmetric space for $G$ (here $\Gamma^{+}=\Gamma \cap G(\mathbb{R})^{+}$). - [ ] This is generally an orbifold since $\Gamma$ may contain torsion. - [x] What is a locally symmetric space? ✅ 2023-01-23 - [ ] Let $G$ be a reductive group over $\mathbb{Q}$. Let us fix a maximal compact subgroup $K_{\infty}^{\max }$ of $G(\mathbb{R})$ and let $C:=\left(K_{\infty}^{\max }\right)^{\circ}$. Let us then fix/denote: - $K_{\infty}$ to be a compact subgroup of $G(\mathbb{R})$ such that $C \subseteq K_{\infty} \subseteq K_{\infty}^{\max }$ - $A_G$ is the maximal $\mathbb{Q}$-split subtorus of $Z(G)$. Then, for any compact open subgroup $K_f \subseteq G\left(\mathbb{A}_f\right)$ we set $$S_G\left(K_f\right):=G(\mathbb{Q}) \backslash G(\mathbb{A}) /\left(K_{\infty} K_f A_G(\mathbb{R})^{+}\right)$$ A space of the form $S_G\left(K_f\right)$ is called a locally symmetric space for $G$. Here we are denoting by $\mathbb{A}=\left(\widehat{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q}\right) \times \mathbb{R}$ the adele ring for $\mathbb{Q}$ and by $\mathbb{A}_f$ its subring of finite adeles $\widehat{\mathbb{Z}} \otimes \mathbb{\mathbb { Z }} \mathbb{Q}$. We topologize this by the restricted direct topology and topologize $G(\mathbb{A})$ and $G\left(\mathbb{A}_f\right)$ in the usual way (e.g. see [Con12]). - [x] What is a Hermitian symmetric domain? ✅ 2023-01-23 - [ ] A hermitian space is a smooth manifold $M$ endowed with a riemannian structure $g$ and a complex structure $J$ that are compatible in the sense that $g(J x, J y)=g(x, y)$ for all $m \in M$ and all $x, y \in T_m M$. $^6$ A hermitian space is symmetric if every point is an isolated fixed point of an involution. A hermitian symmetric domain is a hermitian symmetric space with negative (sectional) curvature. - [ ] Every hermitian symmetric domain decomposes as a product of simple (irreducible) hermitian symmetric domains $D$ whose automorphism groups $\operatorname{Hol}(D)$ are simple Lie groups. - [x] What is a Shimura variety? ✅ 2023-01-23 - [ ] Let $D$ be a hermitian symmetric domain, and let $\operatorname{Hol}(D)^{+}$be the identity component of $\operatorname{Hol}(D)$. Then $\operatorname{Hol}(D)^{+}$is a connected semisimple Lie group with trivial centre. Consider a simply connected semisimple algebraic group $G$ over $\mathbb{Q}$ and a surjective homomorphism$$G(\mathbb{R}) \rightarrow \operatorname{Hol}(D)^{+}$$with compact kernel (such pairs always exist, and are classified). Let $\Gamma$ be a congruence subgroup in $G(\mathbb{Q})$ whose image $\bar{\Gamma}$ in $\operatorname{Aut}(D)^{+}$is torsion-free. Then $\bar{\Gamma}$ acts freely $D$, and so the quotient $\bar{\Gamma} \backslash D$ is a complex manifold. We'll see shortly that it is, in fact, an algebraic variety. - [ ] The Shimura varieties are the algebraic varieties that arise in this way. In other words, an algebraic variety over $\mathbb{C}$ is a Shimura variety if its universal covering space (in the topological sense) is a hermitian symmetric domain, and its fundamental group is the image of a congruence group as above. - [x] What are full and faithful functors? ✅ 2023-01-23 - [ ] Explicitly, let $C$ and $D$ be (locally small) categories and let $F: C \rightarrow D$ be a functor from $C$ to $D$. The functor $F$ induces a function $$F_{X, Y}: \operatorname{Hom}_{\mathcal{C}}(X, Y) \rightarrow \operatorname{Hom}_{\mathcal{D}}(F(X), F(Y))$$for every pair of objects $X$ and $Y$ in $C$. The functor $F$ is said to be - faithful if $F_{X, Y}$ is injective ${ }^{[1][2]}$ - full if $F_{X, Y}$ is surjective ${ }^{[2][3]}$ - fully faithful (= full and faithful) if $F_{X, Y}$ is bijective for each $X$ and $Y$ in $C$. A mnemonic for remembering the term "full" is that the image of the function fills the codomain; a mnemonic for remembering the term "faithful" is that you can trust (have faith) that $F(X)=F(Y)$ implies $X=Y$. - [ ] Warning: faithful need not imply injective on objects or morphisms! - [x] What is Chow's theorem for complex projective varieties? ✅ 2023-01-23 - [ ] Chow 1949: The functor $X \rightsquigarrow X(\mathbb{C})$ from nonsingular projective algebraic varieties to projective complex manifolds is an equivalence of categories. This remains true when singularities are allowed. - [ ] # Homotopy - [x] How is $\THoH(A)$ defined? ✅ 2023-01-23 - [ ] $\THoH(A) \da A \smashprod_{A^e} A$ where $A^e \da A\smashprod A$, defined for $A\in \algs{\EE_1}$. - [x] How is $\HoH^*_{\EE_n}(A)$ defined for $A\in \algs{\EE_n}$? ✅ 2023-01-23 - [ ] Essentially $\int_{S^n} A$. - [x] Describe factorization homology. ✅ 2023-01-23 - [ ] A homology theory for framed $n$-manifolds with coefficients given by $\mathcal{E}_n$-algebras, constructed as a topological analogue of Beilinson-Drinfeld's chiral homology (for factorization coalgebras, motivated by conformal field theory). - [x] Describe an $\EE_n\dash$algebra. ✅ 2023-01-23 - [ ] Algebras with multiplication maps parametrized by configuration spaces of $n$-dimensional disks inside a standard $n$-disk. - [x] Describe a braided monoidal category in terms of $\EE_n\dash$algebras. ✅ 2023-01-23 - [ ] An $\EE_2\dash$algebra object in $\Cat$, since $\EE_2(n) = \B B_n$ where $B_n$ is the pure braid group. - [x] What are the higher classifying spaces $\B^n A$ for $A\in \algs{\EE_n}$? ✅ 2023-01-23 - [ ] A $\cat{C}\dash$enriched $(\infty, n)\dash$category whose object is a point, a single $k\dash$morphism $\phi^k$ for each $1\leq k\leq n-1$, and $\B^nA(\phi^{n-1}, \phi^{n-1})\homotopic A$. - [x] Describe $\Omega_A$ algebraically. ✅ 2023-01-23 - [ ] $I/I^2$ where $I \da \ker(A\tensorpowerk 2\to A)$. - [ ] Corepresents derivations: $\Der_k(A, \wait) \cong \Hom_A(\Omega_A, \wait)$. - [x] Describe $\LL$ algebraically. ✅ 2023-01-23 - [ ] For $A$ nonsmooth, $\Der_k(A, \wait)$ is not right exact, and has Quillen right-derived functors corepresented by $\LL_A$. - [x] What is a presentable $\infty\dash$category? ✅ 2023-01-23 - [ ] Closed under small colimits and limits, and weakly generated by a small subcategory. - [ ] Given by localizations of $\infty\dash$categories of presheaves on a small $\infty\dash$category. - [x] What is a continuous functor? ✅ 2023-01-23 - [ ] Preserves all colimits. - [x] Give a precise definition of factorization homology. ✅ 2023-01-23 - [ ] ![](attachments/2023-01-23-factorization.png) - [x] What motivates construction of the Tate diagonal? ✅ 2023-01-23 - [ ] $\Spectra$ does not have a diagonal map, $X\to X\tensorpower{}{2}$ which factors through homotopy fixed points $(X\tensorpower{}{2})^{hC_2}\to X\tensorpower{}{2}$ (i.e. is symmetric), although this does exist for $\Sigma_+^\infty X$ for $X\in \Top$ induced by the diagonal of spaces. - [x] Define the Tate diagonal. ✅ 2023-01-23 - [ ] The natural transformation $$ \Delta_p: \mathrm{id}_{\mathrm{Sp}} \rightarrow T_p: X \rightarrow(X \otimes \ldots \otimes X)^{t C_p}$$of endofunctors of $\mathrm{Sp}$ which corresponds to the map $$\mathbb{S} \rightarrow T_p(\mathbb{S})=\mathbb{S}^{t C_p}$$ which is the composition $\mathbb{S} \rightarrow \mathbb{S}^{h C_p} \rightarrow \mathbb{S}^{t C_p}$. - [x] What is the right analog of $\ZZpadic$ in $\Spectra$? ✅ 2023-01-23 - [ ] $\SS^{tC_p}$, since $H\ZZ^{tC_p}$ is not connective. - [x] Motivate the passage from schemes to derived stacks. ✅ 2023-01-23 - [ ] $\Sch\to\St$: closure under taking quotients and other colimits - [ ] $\St\to \mathsf{dSt}$: closure under taking fiber products and other limits - [ ] $\Sch\to \mathsf{dSt}:$ closure under "imposing equations". - [ ] Think of $X\in \dSt$ as a functor $F_X: \mathsf{dCRing}\to \Top$. - [ ] Think of source as connective cdgas, more generally connective $\EE_\infty\dash$ring spectra. - [x] Give a notion of "smallness" for $F\in \QCoh(X)$. ✅ 2023-01-23 - [ ] Perfect, dualizable, or compact objects. - [x] What is the derived version of the center of an algebra $Z(A)$? ✅ 2023-01-23 - [ ] $\HoH(A)$, computes$\RHom_{\bimods{A}{A}}(A, A)$ (the derived $(A,A)\dash$bimodule endomorphism algebra). - [x] What is the free loop space of a derived stack? ✅ 2023-01-23 - [ ] $\mcl X \da\Map(X, S^1) = X\fiberprod{X^e} X$ where $X^e \da X\fiberprod{k}X$. - [x] What is $\mcl X$ when $X\in \Sch\slice k$ in characteristic zero? ✅ 2023-01-23 - [ ] $\mcl X = \T_X[-1] = \spec \sym \Omega_X[1]$. - [x] What is $\mcl \BG$? ✅ 2023-01-23 - [ ] The adjoint quotient $G/G$. - [x] What is the stacky version of a $G\dash$local system on $X$? ✅ 2023-01-23 - [ ] $\Loc_G(X) \cong \Map(X, \BG)$. - [x] What is a stable $\infty\dash$category? ✅ 2023-01-23 - [ ] Zero object - [ ] Closed under finite limits and colimits - [ ] Pushouts equal pullbacks - [x] How are triangulated categories related to infinity categories? ✅ 2023-01-23 - [ ] $\ho\cat{C}$ has a canonical triangulated structure for $\cat{C}$ a stable infty cat. - [x] What does it mean for a functor $F:\cat{C}\times \cat{D}\to \cat{E}$ to be bilinear? ✅ 2023-01-23 - [ ] Factors through $\cat{C}\tensor \cat{D}$, commutes with colimits in both variables separately. - [x] What is the sheaf condition for a derived stack? ✅ 2023-01-23 - [ ] $X\in \Fun(\kalg, \Top)$ where if $A\to B$ is an etale cover, $X(A) \iso \lim X(\tilde B)$ is an equivalence where $\tilde B\covers B$ is the cosimplicial bar resolution induced by $\wait\tensor_A B$. - [x] What is a compact object? ✅ 2023-01-23 - [ ] An object $M$ of a stable $\infty$-category $\mathcal{C}$ is said to be compact if $\operatorname{Hom}_{\mathcal{C}}(M,-)$ commutes with all coproducts (equivalently, with all colimits). - [ ] $M$ is compact iff any map $M\to A$ where $A$ is a small coproduct factors through a finite coproduct, when $\cat{C}$ is stable presentable. - [x] What is a dualizable object? ✅ 2023-01-23 - [ ] (2) An object $M$ of a stable symmetric monoidal $\infty$-category $\mathcal{C}$ is said to be (strongly) dualizable if there is an object $M^{\vee}$ and unit and trace maps $$1 \stackrel{u}{\longrightarrow} M \otimes M^{\vee} \stackrel{\tau}{\longrightarrow} 1$$such that the composite map $$M \stackrel{u \otimes \mathrm{id}}{\longrightarrow} M \otimes M^{\vee} \otimes M \stackrel{\mathrm{id} \otimes \tau}{\longrightarrow} M$$is the identity. - [x] What is the usefulness of dualizable objects? ✅ 2023-01-23 - [ ] ![](attachments/2023-01-23-canonical.png) - [ ] What is a topological gauge theory? - [ ] What is descent? - [ ] What is the height $n$ Lubin-Tate spectrum? - [ ] What is $\BP$? - [ ] What is $\KU$? - [ ] What is $\K(n)$? - [ ] What is a conservative functor? - [ ] What is a compactly generated category? - [ ] What is the Drinfeld center of a monoidal category?