\newcommand{\cat}[1]{\mathsf{#1}}

\newcommand{\Sets}[0]{{\mathsf{Set}}}
\newcommand{\Set}[0]{{\mathsf{Set}}}
\newcommand{\sets}[0]{{\mathsf{Set}}}
\newcommand{\set}{{\mathsf{Set}}}
\newcommand{\Poset}[0]{\mathsf{Poset}}
\newcommand{\GSets}[0]{{G\dash\mathsf{Set}}}
\newcommand{\Groups}[0]{{\mathsf{Group}}}
\newcommand{\Grp}[0]{{\mathsf{Grp}}}
\newcommand{\cC}[0]{{\mathsf{C}}}
\newcommand{\Ar}[0]{{\mathsf{Ar}}}
\newcommand{\Mot}[0]{{\mathsf{Mot}}}
\newcommand{\SW}[0]{{\mathsf{SW}}}
\newcommand{\cof}[0]{{\mathsf{cof}}}
\newcommand{\fib}[0]{{\mathsf{fib}}}

% Modifiers
\newcommand{\der}[0]{{\mathsf{d}}}
\newcommand{\dg}[0]{{\mathsf{dg}}}
\newcommand{\comm}[0]{{\mathsf{C}}}
\newcommand{\pre}[0]{{\mathsf{pre}}}
\newcommand{\fn}[0]{{\mathsf{fn}}}
\newcommand{\smooth}[0]{{\mathsf{sm}}}
\newcommand{\Aff}[0]{{\mathsf{Aff}}}
\newcommand{\Ab}[0]{{\mathsf{Ab}}}
\newcommand{\Add}[0]{{\mathsf{Add}}}
\newcommand{\Assoc}[0]{\mathsf{Assoc}}
\newcommand{\Ch}[0]{\mathsf{Ch}}
\newcommand{\Coh}[0]{{\mathsf{Coh}}}
\newcommand{\Comm}[0]{\mathsf{Comm}}
\newcommand{\Cor}[0]{\mathsf{Cor}}
\newcommand{\Corr}[0]{\mathsf{Cor}}
\newcommand{\Fin}[0]{{\mathsf{Fin}}}
\newcommand{\Free}[0]{\mathsf{Free}}
\newcommand{\Tors}[0]{\mathsf{Tors}}
\newcommand{\Perf}[0]{\mathsf{Perf}}
\newcommand{\Unital}[0]{\mathsf{Unital}}
\newcommand{\eff}[0]{\mathsf{eff}}
\newcommand{\Dc}[0]{\mathbf{D}}
\newcommand{\derivedcat}[1]{\Dc {#1}}
\newcommand{\Db}[0]{ \mathsf{D}^b }
\newcommand{\db}[0]{\Db}
\newcommand{\bderivedcat}[1]{\Db {#1}}
\newcommand{\Const}[0]{\mathsf{Const}}
\newcommand{\Cx}[0]{\mathsf{Ch}}
\newcommand{\Stable}[0]{\mathsf{Stab}}
\newcommand{\ChainCx}[1]{\mathsf{Ch}\qty{ #1 }}
\newcommand{\Vect}[0]{{ \mathsf{Vect}}}
\newcommand{\kvect}[0]{{ \mathsf{Vect}\slice{k}}}
\newcommand{\loc}[0]{{\mathsf{loc}}}
\newcommand{\locfree}[0]{{\mathsf{locfree}}}
\newcommand{\Bun}{{\mathsf{Bun}}}
\newcommand\Prinbun{{ \mathsf{PrinBun}}}
\newcommand{\bung}{{\mathsf{Bun}_G}}

% Rings
\newcommand{\Local}[0]{\mathsf{Local}}
\newcommand{\Fieldsover}[1]{{ \mathsf{Fields}_{#1}}}
\newcommand{\Field}[0]{\mathsf{Field}}
\newcommand{\Number}[0]{\mathsf{Number}}
\newcommand{\Numberfield}[0]{\Field\slice{\QQ}}
\newcommand{\NF}[0]{\Numberfield}
\newcommand{\Art}[0]{\mathsf{Art}}
\newcommand{\Global}[0]{\mathsf{Global}}
\newcommand{\Ring}[0]{\mathsf{Ring}}
\newcommand{\Mon}[0]{\mathsf{Mon}}
\newcommand{\CMon}[0]{\mathsf{CMon}}
\newcommand{\CRing}[0]{\mathsf{CRing}}
\newcommand{\DedekindDomain}[0]{\mathsf{DedekindDom}}
\newcommand{\IntDomain}[0]{\mathsf{IntDom}}
\newcommand{\Dom}[0]{\mathsf{Dom}}
\newcommand{\Domain}[0]{\mathsf{Domain}}
\newcommand{\DVR}[0]{\mathsf{DVR}}
\newcommand{\Dedekind}[0]{\mathsf{Dedekind}}

% Modules
\newcommand{\Quat}[0]{{\mathsf{Quat}}}
\newcommand{\torsors}[1]{{\mathsf{#1}\dash\mathsf{Torsors}}}
\newcommand{\torsorsright}[1]{\mathsf{Torsors}\dash\mathsf{#1}}
\newcommand{\torsorsleft}[1]{\mathsf{#1}\dash\mathsf{Torsors}}
\newcommand{\bimod}[2]{({#1}, {#2})\dash\mathsf{biMod}}
\newcommand{\bimods}[2]{({#1}, {#2})\dash\mathsf{biMod}}

\newcommand{\Mod}[0]{{\mathsf{Mod}}}
\newcommand{\modsleft}[1]{ {}_{#1}\Mod}
\newcommand{\modsright}[1]{ \Mod_{#1}}
\newcommand{\mods}[1]{ \modsleft{#1}}
\newcommand{\modr}[0]{ \modsright{R}}
\newcommand{\stmods}[1]{{\mathsf{#1}\dash\mathsf{stMod}}}
\newcommand{\grmods}[1]{{\mathsf{#1}\dash\mathsf{grMod}}}
\newcommand{\grMod}[0]{{\mathsf{grMod}}}
\newcommand{\comods}[1]{{\mathsf{#1}\dash\mathsf{coMod}}}

\newcommand{\gsetsleft}[1]{ {}_{#1}\Set}
\newcommand{\gsetsright}[1]{ \Set_{#1}}
\newcommand{\gset}[1]{\gsetsleft{#1}}
\newcommand{\gsets}[1]{\gset{#1}}

\newcommand{\zmod}[0]{ \modsleft{\ZZ}}
\newcommand{\qmod}[0]{ \modsleft{\QQ}}
\newcommand{\rmod}[0]{ \modsleft{R}}
\newcommand{\kmod}[0]{ \modsleft{k}}
\newcommand{\cmod}[0]{ \modsleft{ \CC }}
\newcommand{\fmod}[0]{ \modsleft{ \FF }}
\newcommand{\oxmods}[0]{ \mods{\OO_X}}
\newcommand{\amod}[0]{\mods{A}}
\newcommand{\gmod}[0]{\mods{G}}
\newcommand{\lmod}[0]{\mods{L}}
\newcommand{\Dmod}[0]{ \modsleft{\mathcal{D}} }
\newcommand{\dmod}[0]{ \Dmod }

\newcommand{\gr}[0]{{\mathsf{gr}\,}}
\newcommand{\mmod}[0]{{\dash\mathsf{Mod}}}
\newcommand{\Rep}[0]{{\mathsf{Rep}}}
\newcommand{\Irr}[0]{{\mathsf{Irr}}}
\newcommand{\Adm}[0]{{\mathsf{Adm}}}
\newcommand{\semisimp}[0]{{\mathsf{ss}}}
\newcommand{\Hodge}[0]{{\mathsf{Hodge}}}

% Vector Spaces and Bundles
\newcommand{\VectBundle}[0]{{ \Bun\qty{\GL_r}}}
\newcommand{\VectBundlerk}[1]{{ \Bun\qty{\GL_{#1}}}}
\newcommand{\VectSp}[0]{{ \VectSp }}
\newcommand{\VectBun}[0]{{ \VectBundle }}
\newcommand{\VectBunrk}[1]{{ \VectBundlerk{#1}}}
\newcommand{\Bung}[0]{{ \Bun\qty{G}}}

% Algebras
\newcommand{\Alg}[0]{ \mathsf{Alg}}
\newcommand{\algs}[1]{{ {}_{#1} \Alg }}
\newcommand{\Hopf}[0]{\mathsf{Hopf}}
\newcommand{\alg}[0]{\Alg}
\newcommand{\scalg}[0]{\mathsf{sCAlg}}
\newcommand{\cAlg}[0]{{\mathsf{cAlg}}}
\newcommand{\calg}[0]{\mathsf{CAlg}}
\newcommand{\liegmod}[0]{{\mathfrak{g}\dash\mathsf{Mod}}}
\newcommand{\liealg}[0]{{\mathsf{Lie}\dash\Alg}}
\newcommand{\Lie}[0]{\mathsf{Lie}}
\newcommand{\kalg}[0]{ {}_{k} \Alg }
\newcommand{\kAlg}[0]{\kalg}
\newcommand{\falg}[0]{ {}_{\FF} \Alg }
\newcommand{\ralg}[0]{ {}_{R} \Alg }
\newcommand{\rAlg}[0]{ \ralg }
\newcommand{\zalg}[0]{ {}_{\ZZ} \Alg }
\newcommand{\CCalg}[0]{ {}_{\\CC} \Alg }
\newcommand{\dga}[0]{{\mathsf{dg\Alg}}}
\newcommand{\cdga}[0]{{ \mathsf{c}\dga }}
\newcommand{\FMP}[0]{\mathsf{FMP}}
\newcommand{\dgla}[0]{\dg\Lie\Alg }
\newcommand{\DGLA}[0]{\dgla}
\newcommand{\Poly}[0]{{\mathsf{Poly}}}
\newcommand{\Hk}[0]{{\mathsf{Hk}}}
\newcommand{\Asm}[0]{{\mathsf{Asm}}}
\newcommand{\kSch}[0]{{\mathsf{Sch}_{/k}}}

\newcommand{\Grpd}[0]{{\mathsf{Grpd}}}
\newcommand{\inftyGrpd}[0]{{ \underset{\infty}{ \Grpd }}}
\newcommand{\Algebroid}[0]{{\mathsf{Algd}}}

% Schemes and Sheaves
\newcommand{\Loc}[0]{\mathsf{Loc}}
\newcommand{\Locsys}[0]{\mathsf{LocSys}}
\newcommand{\Ringedspace}[0]{\mathsf{RingSp}}
\newcommand{\RingedSpace}[0]{\mathsf{RingSp}}
\newcommand{\LRS}[0]{\Loc\RingedSpace}
\newcommand{\IndCoh}[0]{{\mathsf{IndCoh}}}
\newcommand{\dbcoh}[0]{\mathsf{D}^b\mathsf{Coh}}
\newcommand{\DbCoh}[0]{\dbcoh}
\newcommand{\DCoh}[0]{\mathsf{D}\mathsf{Coh}}
\newcommand{\dcoh}[0]{\DCoh}
\newcommand{\QCoh}[0]{{\mathsf{QCoh}}}
\newcommand{\qcoh}[0]{\QCoh}
\newcommand{\Ind}[0]{{\mathsf{Ind}}}
\newcommand{\ind}[0]{\Ind}
\newcommand{\Pro}[0]{\mathsf{pro}}
\newcommand{\pro}[0]{\Pro}
\newcommand{\Cov}[0]{{\mathsf{Cov}}}
\newcommand{\sch}[0]{{\mathsf{Sch}}}
\newcommand{\presh}[0]{ \underset{ \mathsf{pre}} {\mathsf{Sh}}}
\newcommand{\prest}[0]{ {\underset{ \mathsf{pre}} {\mathsf{St}} }}
\newcommand{\Descent}[0]{{\mathsf{Descent}}}
\newcommand{\Desc}[0]{{\mathsf{Desc}}}
\newcommand{\FFlat}[0]{{\mathsf{FFlat}}}
\newcommand{\Perv}[0]{\mathsf{Perv}}
\newcommand{\smsch}[0]{{ \smooth\Sch }}
\newcommand{\Sch}[0]{{\mathsf{Sch}}}
\newcommand{\Schf}[0]{{\mathsf{Schf}}}
\newcommand{\Sh}[0]{{\mathsf{Sh}}}
\newcommand{\St}[0]{{\mathsf{St}}}
\newcommand{\Stacks}[0]{{\mathsf{St}}}
\newcommand{\Var}[0]{{\mathsf{Var}}}
\newcommand{\Vark}[0]{{ \Var_{/k}}}
\newcommand{\kvar}[0]{{ \Var_{/k}}}
\newcommand{\Open}[0]{{\mathsf{Open}}}

% Homotopy
\newcommand{\CW}[0]{{\mathsf{CW}}}
\newcommand{\sset}[0]{{\mathsf{sSet}}}
\newcommand{\sSet}[0]{{\mathsf{sSet}}}
\newcommand{\ssets}[0]{\mathsf{sSet}}
\newcommand{\hoTop}[0]{{\mathsf{hoTop}}}
\newcommand{\hoType}[0]{{\mathsf{hoType}}}
\newcommand{\ho}[0]{{\mathsf{ho}}}
\newcommand{\SHC}[0]{{\mathsf{SHC}}}
\newcommand{\SH}[0]{{\mathsf{SH}}}
\newcommand{\Spaces}[0]{{\mathsf{Spaces}}}
\newcommand{\GSpaces}[1]{{G\dash\mathsf{Spaces}}}
\newcommand{\Spectra}[0]{{\mathsf{Sp}}}
\newcommand{\Sp}[0]{{\mathsf{Sp}}}
\newcommand{\Top}[0]{{\mathsf{Top}}}
\newcommand{\Bord}[0]{{\mathsf{Bord}}}
\newcommand{\TQFT}[0]{{\mathsf{TQFT}}}
\newcommand{\Kc}[0]{{\mathsf{K^c}}}
\newcommand{\triang}[0]{{\mathsf{triang}}}
\newcommand{\TTC}[0]{{\mathsf{TTC}}}
\newcommand{\dchrmod}{{\derivedcat{\Ch(\rmod)}}}

% Infty Cats
\newcommand{\Finset}[0]{{\mathsf{FinSet}}}
\newcommand{\Cat}[0]{\mathsf{Cat}}
\newcommand{\Fun}[0]{{\mathsf{Fun}}}
\newcommand{\Kan}[0]{{\mathsf{Kan}}}
\newcommand{\Monoid}[0]{\mathsf{Mon}}
\newcommand{\Arrow}[0]{\mathsf{Arrow}}
\newcommand{\quasiCat}[0]{{ \mathsf{quasiCat}} }
\newcommand{\inftycat}[0]{{ \underset{\infty}{ \Cat}  }}
\newcommand{\inftycatn}[1]{{ \underset{(\infty, {#1})}{ \Cat}  }}
\newcommand{\core}[0]{{ \mathsf{core}}}
\newcommand{\Indcat}[0]{ \mathsf{Ind}}

% New?
\newcommand{\Prism}[0]{\mathsf{Prism}}
\newcommand{\Solid}[0]{\mathsf{Solid}}
\newcommand{\WCart}[0]{\mathsf{WCart}}

% Motivic
\newcommand{\Torsor}[1]{{\mathsf{#1}\dash\mathsf{Torsor}}}
\newcommand{\Torsorleft}[1]{{\mathsf{#1}\dash\mathsf{Torsor}}}
\newcommand{\Torsorright}[1]{{\mathsf{Torsor}\dash\mathsf{#1}}}
\newcommand{\Quadform}[0]{{\mathsf{QuadForm}}}
\newcommand{\HI}[0]{{\mathsf{HI}}}
\newcommand{\DM}[0]{{\mathsf{DM}}}
\newcommand{\hoA}[0]{{\mathsf{ho}_*^{\scriptstyle \AA^1}}}
\newcommand\Tw[0]{\mathsf{Tw}}
\newcommand\SB[0]{\mathsf{SB}}
\newcommand\CSA[0]{\mathsf{CSA}}
\newcommand{\CSS}[0]{{ \mathsf{CSS}} }

% Unsorted
\newcommand{\FGL}[0]{\mathsf{FGL}}
\newcommand{\FI}[0]{{\mathsf{FI}}}
\newcommand{\CE}[0]{{\mathsf{CE}}}
\newcommand{\Fuk}[0]{{\mathsf{Fuk}}}
\newcommand{\Lag}[0]{{\mathsf{Lag}}}
\newcommand{\Mfd}[0]{{\mathsf{Mfd}}}
\newcommand{\Riem}[0]{\mathsf{Riem}}
\newcommand{\Wein}[0]{{\mathsf{Wein}}}
\newcommand{\gspaces}[1]{{#1}\dash{\mathsf{Spaces}}}
\newcommand{\deltaring}[0]{{\delta\dash\mathsf{Ring}}}
\newcommand{\terminal}[0]{{ 0_{\scriptscriptstyle \uparrow}}}
\newcommand{\initial}[0]{{ \mathscr \emptyset^{\scriptscriptstyle \downarrow}}}

% Universal guys

\newcommand{\coeq}[0]{\operatorname{coeq}}
\newcommand{\cocoeq}[0]{\operatorname{eq}}
\newcommand{\dgens}[1]{\gens{\gens{ #1 }}}
\newcommand{\ctz}[1]{\, {\converges{{#1} \to\infty}\longrightarrow 0} \, }
\newcommand{\conj}[1]{{\overline{{#1}}}}
\newcommand{\complex}[1]{{ {#1}_{\scriptscriptstyle \bullet}} }
\newcommand{\cocomplex}[1]{ { {#1}^{\scriptscriptstyle \bullet}} }
\newcommand{\bicomplex}[1]{{ {#1}_{\scriptscriptstyle \bullet, \bullet}} }
\newcommand{\cobicomplex}[1]{ { {#1}^{\scriptscriptstyle \bullet, \bullet}} }

\newcommand{\floor}[1]{{\left\lfloor #1 \right\rfloor}}
\newcommand{\ceiling}[1]{{\left\lceil #1 \right\rceil}}
\newcommand{\fourier}[1]{\widehat{#1}}
\newcommand{\embedsvia}[1]{\xhookrightarrow{#1}}
\newcommand{\openimmerse}[0]{\underset{\scriptscriptstyle O}{\hookrightarrow}}
\newcommand{\weakeq}[0]{\underset{\scriptscriptstyle W}{\rightarrow}}

\newcommand{\fromvia}[1]{\xleftarrow{#1}}
\newcommand{\generators}[1]{\left\langle{#1}\right\rangle}
\newcommand{\freeon}[1]{ \left[ {#1} \right]  }
\newcommand{\gens}[1]{\left\langle{#1}\right\rangle}
\newcommand{\globsec}[1]{{{\Gamma}\qty{#1} }}
\newcommand{\Globsec}[1]{{{\Gamma}\qty{#1} }}
\newcommand{\langL}[1]{ {}^{L}{#1} }
\newcommand{\modulo}[0]{{ \bigg/ }}

\newcommand{\equalsbecause}[1]{\overset{#1}{=}}
\newcommand{\congbecause}[1]{\overset{#1}{\cong}}
\newcommand{\congas}[1]{\underset{#1}{\cong}}
\newcommand{\isoas}[1]{\underset{#1}{\cong}}
\newcommand{\pwiso}[0]{\underset{\mathrm{pw}}{\cong}}
\newcommand{\addbase}[1]{{ {}_{\pt} }}
\newcommand{\ideal}[1]{\mathcal{#1}}
\newcommand{\adjoin}[1]{ { \left[ \scriptstyle {#1} \right] } }
\newcommand{\polynomialring}[1]{ { \left[ {#1} \right] } }
\newcommand{\htyclass}[1]{ { \left[ {#1} \right] } }
\newcommand{\qtext}[1]{{\quad \operatorname{#1} \quad}}
\newcommand{\abs}[1]{{\left\lvert {#1} \right\rvert}}
\newcommand{\stack}[1]{\mathclap{\substack{ #1 }}} 
\newcommand{\freezmod}[1]{\ZZ\left[ {#1} \right] } 

\newcommand{\functionfield}[1]{ { \left( {#1} \right) } }
\newcommand{\rff}[1]{ \functionfield{#1} }


\newcommand{\fps}[1]{{\llbracket #1 \rrbracket  }}
\newcommand{\formalseries}[1]{ \fps{#1} }
\newcommand{\formalpowerseries}[1]{ \fps{#1} }
\newcommand{\powerseries}[1]{ \fps{#1} }
\newcommand\fls[1]{(\hspace{-0.25em}( #1 )\hspace{-0.22em}) }

\newcommand\laurent[1]{\fls{#1}}
\newcommand\laurentseries[1]{\fls{#1}}

\newcommand\lshriek[0]{{}_{!}}
\newcommand\pushf[0]{{}^{*}}

\newcommand{\nilrad}[1]{{\sqrt{0_{#1}} }}
\newcommand{\jacobsonrad}[1]{{J ({#1}) }}
\newcommand{\localize}[1]{ \left[ { \scriptstyle { {#1}\inv}  } \right] }
\newcommand{\primelocalize}[1]{ \left[ { \scriptstyle { { ({#1}^c) }\inv}  } \right]}
\newcommand{\plocalize}[1]{\primelocalize{#1}}
\newcommand{\sheafify}[1]{ \left( #1 \right)^{\scriptscriptstyle \mathrm{sh}} }
\newcommand{\complete}[1]{{ {}_{ \hat{#1} } }}
\newcommand{\takecompletion}[1]{{ \overbrace{#1}^{\widehat{\hspace{4em}}}  }}
\newcommand{\pcomplete}[0]{{ {}^{ \wedge }_{p} }}
\newcommand{\procomplete}[0]{ {}^{ \wedge_{\scriptscriptstyle \pro }} } 
\newcommand{\kv}[0]{{ k_{\hat{v}} }}
\newcommand{\Lv}[0]{{ L_{\hat{v}} }}

\newcommand{\twistleft}[2]{{ {}^{#1} #2 }}
\newcommand{\twistright}[2]{{ #2 {}^{#1} }}
\newcommand{\liesover}[1]{{ {}_{/ {#1}} }}
\newcommand{\liesabove}[1]{{ {}_{/ {#1}} }}
\newcommand{\slice}[1]{_{/ {#1}}}
\newcommand{\coslice}[1]{_{{#1/}}}
\newcommand{\quotright}[2]{ {}^{#1}\mkern-2mu/\mkern-2mu_{#2} }
\newcommand{\quotleft}[2]{ {}_{#2}\mkern-.5mu\backslash\mkern-2mu^{#1} }
\newcommand{\invert}[1]{{ \left[ { \scriptstyle \frac{1}{#1} } \right] }}
\newcommand{\symb}[2]{{ \qty{ #1 \over #2 } }}
\newcommand{\squares}[1]{{ {#1}_{\scriptscriptstyle \square} }} 
\newcommand{\shift}[2]{{ \Sigma^{\scriptstyle[#2]} #1 }}

\newcommand\cartpower[1]{{ {}^{ \scriptscriptstyle\times^{#1} }  }}
\newcommand\disjointpower[1]{{ {}^{ \scriptscriptstyle\coprod^{#1} }  }}
\newcommand\sumpower[1]{{ {}^{ \scriptscriptstyle\oplus^{#1} }  }}
\newcommand\prodpower[1]{{ {}^{ \scriptscriptstyle\times^{#1} }  }}
\newcommand\tensorpower[2]{{ {}^{ \scriptstyle\otimes_{#1}^{#2} }  }}
\newcommand\tensorpowerk[1]{{ {}^{ \scriptscriptstyle\otimes_{k}^{#1} }  }}
\newcommand\derivedtensorpower[3]{{ {}^{ \scriptstyle {}_{#1} {\otimes_{#2}^{#3}} }  }}
\newcommand\smashpower[1]{{ {}^{ \scriptscriptstyle\smashprod^{#1} }  }}
\newcommand\wedgepower[1]{{ {}^{ \scriptscriptstyle\smashprod^{#1} }  }}
\newcommand\fiberpower[2]{{ {}^{ \scriptscriptstyle\fiberprod{#1}^{#2} }  }}
\newcommand\powers[1]{{ {}^{\cdot #1} }}
\newcommand\skel[1]{{ {}^{ (#1) } }}
\newcommand\transp[1]{{ \, {}^{t}{ \left( #1 \right) } }}

\newcommand\Restriction[2]{\mathrm{Res}^{#1}_{#2}}
\newcommand\coRestriction[2]{\mathrm{coRes}^{#1}_{#2}}
\newcommand\Induction[2]{\mathrm{Ind}^{#1}_{#2}}
\newcommand\coInduction[2]{\mathrm{coInd}^{#1}_{#2}}

\newcommand{\inner}[2]{{\left\langle {#1},~{#2} \right\rangle}}
\newcommand{\inp}[2]{{\left\langle {#1},~{#2} \right\rangle}}
\newcommand{\poisbrack}[2]{{\left\{ {#1},~{#2} \right\} }}
\newcommand\tl[2]{{ #1_1, \cdots, #1_{#2} }}
\newcommand\tlz[2]{{ #1_0, \cdots, #1_{#2} }}
\newcommand\tsl[3]{ \ts{ {#1}_{#2}, \cdots, {#1}_{#3} } }
\newcommand\tlset[2]{ \ts{ {#1}_{1}, \cdots, {#1}_{#2} } }

\newcommand{\cofinal}[0]{\mathsf{\emptyset} }
\newcommand{\final}[0]{\ts{\pt}}
\newcommand{\roof}[3]{ #1 {\, \scriptstyle {}^\swarrow\, } #2 {\, \scriptstyle {}^\searrow\,} #3 }
\newcommand{\catspan}[3]{\roof{#1}{#2}{#3}}

\newcommand{\norm}[1]{{\left\lVert {#1} \right\rVert}}
\newcommand{\normm}[1]{{\left\lVert \left\lVert {#1} \right\rVert\right\rVert}}
\newcommand{\open}[1]{\overset{\circ}{#1}}
\newcommand{\pnorm}[2]{{\left\lVert {#1} \right\rVert}_{#2}}

\newcommand{\fourcase}[4]{
\begin{cases}{#1} & {#2} \\ {#3} & {#4}\end{cases} 
}
\newcommand{\cvec}[2]{{ \begin{bmatrix} {#1} \\ {#2} \end{bmatrix} }}
\newcommand{\matt}[4]{{ \begin{bmatrix} {#1} & {#2}  \\ {#3} & {#4} \end{bmatrix} }}
\newcommand{\mattt}[9]{{
\begin{bmatrix}
  {#1} & {#2} & {#3}
\\
  {#4} & {#5} & {#6}
\\
  {#7} & {#8} & {#9}
\end{bmatrix}
}}

\newcommand\stacksymbol[3]{
  \mathrel{\stackunder[2pt]{\stackon[4pt]{$#3$}{$\scriptscriptstyle#1$}}{
  $\scriptscriptstyle#2$}}
}
\newcommand{\textoperatorname}[1]{
  \operatorname{\textnormal{#1}}
}

\newcommand{\bbcpt}[1]{\overline{ #1 }^{\bb}}
\newcommand{\ksbacpt}[1]{\overline{ #1 }^{\scriptsize \KSBA}}
\newcommand{\torcpt}[1]{\overline{ #1 }^{\scriptsize \mathrm{tor} }}
\newcommand{\torcptf}[1]{\overline{ #1 }^{\scriptsize \mathrm{tor}(F) }}
\newcommand{\rc}[1]{\overline{ #1 }^{\scriptsize \mathrm{rc}}}

\newcommand{\discgroup}[1]{A_{#1}}
\newcommand{\bracket}[1]{ \left\langle #1 \right\rangle }
\newcommand\tmf{ \mathrm{tmf} }
\newcommand\taf{ \mathrm{taf} }
\newcommand\TAF{ \mathrm{TAF} }
\newcommand\TMF{ \mathrm{TMF} }
\newcommand\String{ \mathrm{String} }

\newcommand{\BO}[0]{{\B \Orth}}
\newcommand{\EO}[0]{{\mathsf{E} \Orth}}
\newcommand{\BSO}[0]{{\B\SO}}
\newcommand{\ESO}[0]{{\mathsf{E}\SO}}
\newcommand{\BG}[0]{{\B G}}
\newcommand{\EG}[0]{{\mathsf{E} G}}
\newcommand{\BP}[0]{{\operatorname{BP}}}
\newcommand{\BU}[0]{\B{\operatorname{U}}}

\newcommand{\MO}[0]{{\operatorname{MO}}}
\newcommand{\MSO}[0]{{\operatorname{MSO}}}
\newcommand{\MSpin}[0]{{\operatorname{MSpin}}}
\newcommand{\MSp}[0]{{\operatorname{MSpin}}}
\newcommand{\MString}[0]{{\operatorname{MString}}}
\newcommand{\MStr}[0]{{\operatorname{MString}}}
\newcommand{\MU}[0]{{\operatorname{MU}}}


\newcommand{\KO}[0]{{\operatorname{KO}}}
\newcommand{\KU}[0]{{\operatorname{KU}}}

\newcommand{\smashprod}[0]{\wedge}

\newcommand{\ku}[0]{{\operatorname{ku}}}
\newcommand{\hofib}[0]{{\operatorname{hofib}}}
\newcommand{\cofib}[0]{{\operatorname{cofib}}}
\newcommand{\hocofib}[0]{{\operatorname{hocofib}}}


\newcommand{\Loop}[0]{{\Omega}}
\newcommand{\Loops}[0]{\Loop}
\newcommand{\Suspend}[0]{{\Sigma}}
\DeclareMathOperator{\Suspendpinf}{{\Sigma_+^\infty}}
\DeclareMathOperator{\Loopinf}{{\Omega^\infty}}
\newcommand{\Loopsinf}[0]{\Loopinf}
% Names and Symbols
\newcommand*\dif{\mathop{}\!\operatorname{d}}
\newcommand*{\horzbar}{\rule[.5ex]{2.5ex}{0.5pt}}
\newcommand*{\vertbar}{\rule[-1ex]{0.5pt}{2.5ex}}
\newcommand\Fix{ \mathrm{Fix}}
\newcommand\CS{ \mathrm{CS}}
\newcommand\FP{ \mathrm{FP}}
\newcommand\places[1]{ \mathrm{Pl}\qty{#1}}
\newcommand\Ell{ \mathrm{Ell}}
\newcommand\homog{ { \mathrm{homog} }}
\newcommand\Kahler[0]{\operatorname{Kähler}}
\newcommand\aug{\fboxsep=-\fboxrule\!\!\!\fbox{\strut}\!\!\!}
\newcommand\compact[0]{\operatorname{cpt}}
\newcommand\hyp[0]{{\operatorname{hyp}}}
\newcommand\jan{\operatorname{Jan}}
\newcommand\curl{\operatorname{curl}}
\newcommand\EZ{\operatorname{EZ}}
\newcommand\depth{\operatorname{depth}}
\newcommand\Sim{\operatorname{Sim}}
\newcommand\kbar{{ \overline{k} }}
\newcommand\ksep{{ k\sep }}
\newcommand\mypound{\scalebox{0.8}{\raisebox{0.4ex}{\#}}}
\newcommand\rref{\operatorname{RREF}}
\newcommand\RREF{\operatorname{RREF}}
\newcommand{\Tatesymbol}{\operatorname{TateSymb}}
\newcommand{\Vor}{\operatorname{Vor}}
\newcommand{\Star}{\operatorname{Star}}
\newcommand\tilt[0]{ {}^{ \flat }}
\newcommand\vecc[2]{\textcolor{#1}{\textbf{#2}}}
\newcommand{\Af}[0]{{\mathbf{A}}}
\newcommand{\Ag}[0]{{\mathcal{A}_g}}
\newcommand{\Mg}[0]{{\mathcal{M}_g}}
\newcommand{\mgbar}[0]{\bar{\Mg}}
\newcommand{\Mgbar}[0]{\bar{\Mg}}
\newcommand{\agbar}[0]{\bar{\Ag}}
\newcommand{\Ahat}[0]{\hat{ \operatorname{A}}_g }
\newcommand{\Ann}[0]{\operatorname{Ann}}
\newcommand{\sinc}[0]{\operatorname{sinc}}
\newcommand{\Banach}[0]{\mathcal{B}}
\newcommand{\Arg}[0]{\operatorname{Arg}}
\newcommand{\BB}[0]{{\mathbb{B}}}
\newcommand{\Betti}[0]{{\operatorname{Betti}}}
\newcommand{\CC}[0]{{\mathbf{C}}}
\newcommand{\CCstar}[0]{{\CC\units }}
\newcommand{\cstar}[0]{{\CC\units }}
\newcommand{\CF}[0]{\operatorname{CF}}
\newcommand{\CH}[0]{{\operatorname{CH}}}
\newcommand{\Chow}[0]{{\operatorname{Ch}}}
\newcommand{\CP}[0]{{\mathbf{CP}}}
\newcommand{\CY}{{ \text{CY} }}
\newcommand{\Cl}[0]{ \operatorname{Cl}}
\newcommand{\Crit}[0]{\operatorname{Crit}}
\newcommand{\DD}[0]{\mathbb{D}}
\newcommand{\UU}[0]{\mathbb{U}}
\newcommand{\DSt}[0]{{ \operatorname{DSt}}}
\newcommand{\Def}{\operatorname{Def}}
\newcommand{\MT}{\operatorname{MT}}
\newcommand{\Diffeo}[0]{{\operatorname{Diffeo}}}
\newcommand{\Diff}[0]{\operatorname{Diff}}
\newcommand{\Disjoint}[0]{\amalg}
\newcommand{\resprod}[0]{\prod^{\res}}
\newcommand{\restensor}[0]{\bigotimes^{\res}}
\newcommand{\Disk}[0]{{\operatorname{Disk}}}
\newcommand{\Dist}[0]{\operatorname{Dist}}
\newcommand{\EE}[0]{{\mathbb{E}}}
\newcommand{\EKL}[0]{{\mathrm{EKL}}}
\newcommand{\vir}[0]{{\mathrm{vir}}}
\newcommand{\Nil}[0]{{\mathrm{Nil}}}
\newcommand{\QH}[0]{{\mathrm{QH}}}
\newcommand{\AMGM}[0]{{\mathrm{AMGM}}}
\newcommand{\Hasse}[0]{{\mathrm{Hasse}}}
\newcommand{\resultant}[0]{{\mathrm{res}}}
\newcommand{\tame}[0]{{\mathrm{tame}}}
\newcommand{\primetop}[0]{{\scriptscriptstyle \mathrm{prime-to-}p}}
\newcommand{\VHS}[0]{{\mathrm{VHS} }}
\newcommand{\ZVHS}[0]{{ \ZZ\mathrm{VHS} }}
\newcommand{\CR}[0]{{\mathrm{CR}}}
\newcommand{\unram}[0]{{\scriptscriptstyle\mathrm{un}}}
\newcommand{\Emb}[0]{{\operatorname{Emb}}}
\newcommand{\minor}[0]{{\operatorname{minor}}}
\newcommand{\fet}{\text{fét}}
\newcommand{\et}{\text{ét}}
\newcommand{\Et}{\text{Ét}}
\newcommand{\trace}{\operatorname{tr}}
\newcommand{\Trace}{\operatorname{Trace}}
\newcommand{\Kl}{\operatorname{Kl}}
\newcommand{\Rel}{\operatorname{Rel}}
\newcommand{\Norm}{\operatorname{Nm}}
\newcommand{\Extpower}[0]{\bigwedge\nolimits}
\newcommand{\Extalgebra}[0]{\cocomplex{\bigwedge}}
\newcommand{\Extalg}[0]{\Extalgebra}
\newcommand{\Extcomplex}[0]{\cocomplex{ \Extalgebra}}
\newcommand{\Extprod}[0]{\bigwedge\nolimits}
\newcommand{\Ext}{ \operatorname{Ext}}
\newcommand{\FFbar}[0]{{\bar\FF}}
\newcommand{\FFpn}[0]{{\FF_{p^n}}}
\newcommand{\FFp}[0]{{\FF_p}}
\newcommand{\FF}[0]{{ \mathbf{F} }}
\newcommand{\FS}{{ \text{FS} }}
\newcommand{\Fil}[0]{\operatorname{Fil}}
\newcommand{\Flat}[0]{{\operatorname{Flat}}}
\newcommand{\Fpbar}[0]{\bar{\FF_p}}
\newcommand{\Fpn}[0]{{\FF_{p^n} }}
\newcommand{\Fppf}[0]{\mathrm{\operatorname{Fppf}}}
\newcommand{\Fp}[0]{{\FF_p}}
\newcommand{\Frac}[0]{\operatorname{Frac}}
\newcommand{\GF}[0]{{\mathbf{GF}}}
\newcommand{\GG}[0]{{\mathbf{G}}}
\newcommand{\GL}[0]{\operatorname{GL}}
\newcommand{\GW}[0]{{\operatorname{GW}}}
\newcommand{\Gal}[0]{{ \mathsf{Gal}}}
\newcommand{\bigo}[0]{{ \mathsf{O}}}
\newcommand{\Gl}[0]{\operatorname{GL}}
\newcommand{\Gr}[0]{{\operatorname{Gr}}}
\newcommand{\GGr}[0]{ {\mathbb{Gr}}}
\newcommand{\HC}[0]{{\operatorname{HC}}}
\newcommand{\HFK}[0]{\operatorname{HFK}}
\newcommand{\HF}[0]{\operatorname{HF}}
\newcommand{\HHom}{\mathscr{H}\kern-2pt\operatorname{om}}
\newcommand{\HH}[0]{{\mathbb{H}}}
\newcommand{\HP}[0]{{\operatorname{HP}}}
\newcommand{\HT}[0]{{\operatorname{HT}}}
\newcommand{\HZ}[0]{{H\ZZ}}
\newcommand{\Hilb}[0]{\operatorname{Hilb}}
\newcommand{\Homeo}[0]{{\operatorname{Homeo}}}
\newcommand{\Honda}[0]{\mathrm{\operatorname{Honda}}}
\newcommand{\Hsh}{{ \mathcal{H} }}
\newcommand{\Id}[0]{\operatorname{Id}}
\newcommand{\Intersect}[0]{\bigcap}
\newcommand{\JCF}[0]{\operatorname{JCF}}
\newcommand{\RCF}[0]{\operatorname{RCF}}
\newcommand{\Jac}[0]{\operatorname{Jac}}
\newcommand{\II}[0]{{\mathbb{I}}}
\newcommand{\KK}[0]{{\mathbb{K}}}
\newcommand{\KH}[0]{ \K^{\scriptscriptstyle \mathrm{H}}}
\newcommand{\KMW}[0]{ \K^{\scriptscriptstyle \mathrm{MW}}}
\newcommand{\KMimp}[0]{ \hat{\K}^{\scriptscriptstyle \mathrm{M}}}
\newcommand{\KM}[0]{ \K^{\scriptstyle\mathrm{M}}}
\newcommand{\Kah}[0]{{ \operatorname{Kähler} }}
\newcommand{\LC}[0]{{\mathrm{LC}}}
\newcommand{\LL}[0]{{\mathbb{L}}}
\newcommand{\Log}[0]{\operatorname{Log}}
\newcommand{\MC}[0]{\operatorname{MC}}
\newcommand{\tw}[0]{\mathrm{tw}}
\newcommand{\MCG}[0]{{\operatorname{MCG}}}
\newcommand{\MM}[0]{{\mathcal{M}}}
\newcommand{\mbar}[0]{\bar{\mathcal{M}}}
\newcommand{\MW}[0]{\operatorname{MW}}
\newcommand{\Mat}[0]{\operatorname{Mat}}
\newcommand{\NN}[0]{{\mathbb{N}}}
\newcommand{\NS}[0]{{\operatorname{NS}}}
\newcommand{\NE}[0]{{\operatorname{NE}}}
\newcommand{\Amp}[0]{{\operatorname{Amp}}}
\newcommand{\OO}[0]{{\mathcal{O}}}
\newcommand{\OP}[0]{{\mathbb{OP}}}
\newcommand{\OX}[0]{{\mathcal{O}_X}}
\newcommand{\Obs}{\operatorname{Obs}}
\newcommand{\obs}{\operatorname{obs}}
\newcommand{\Ob}[0]{{\operatorname{Ob}}}
\newcommand{\Op}[0]{{\operatorname{Op}}}
\newcommand{\Orb}[0]{{\mathrm{Orb}}}
\newcommand{\Conj}[0]{{\mathrm{Conj}}}
\newcommand{\Orth}[0]{{\operatorname{O}}}
\newcommand{\PD}[0]{\mathrm{PD}}
\newcommand{\GU}[0]{\operatorname{GU}}
\newcommand{\PP}[0]{{\mathbf{P}}}
\newcommand{\WP}[0]{{\mathbf{WP}}}
\newcommand{\bb}[0]{\operatorname{BB}}
\newcommand{\En}[0]{\operatorname{En}}
\newcommand{\he}[0]{\operatorname{h.e.}}
\newcommand{\PGL}[0]{\operatorname{PGL}}
\newcommand{\PSL}[0]{{\operatorname{PSL}}}
\newcommand{\PSU}[0]{{\operatorname{PSU}}}
\newcommand{\Pic}[0]{\operatorname{Pic}}
\newcommand{\Alb}[0]{\operatorname{Alb}}
\newcommand{\Num}[0]{\operatorname{Num}}
\newcommand{\Pin}[0]{{\operatorname{Pin}}}
\newcommand{\Places}[0]{{\operatorname{Places}}}
\newcommand{\Presh}[0]{\presh}
\newcommand{\QQpadic}[0]{{ \QQ_p }}
\newcommand{\ZZelladic}[0]{{ \ZZ_\ell }}
\newcommand{\QQ}[0]{{\mathbf{Q}}}
\newcommand{\QQbar}[0]{{ \bar{ \mathbf{Q} } }}
\newcommand{\Quot}[0]{\operatorname{Quot}}
\newcommand{\RP}[0]{{\mathbf{RP}}}
\newcommand{\RR}[0]{{\mathbf{R}}}
\newcommand{\VV}[0]{{\mathbf{V}}}
\newcommand{\Rat}[0]{\operatorname{Rat}}
\newcommand{\Reg}[0]{\operatorname{Reg}}
\newcommand{\Ric}[0]{\operatorname{Ric}}
\newcommand{\SF}[0]{\operatorname{SF}}
\newcommand{\SL}[0]{{\operatorname{SL}}}
\newcommand{\SNF}[0]{\mathrm{SNF}}
\newcommand{\slc}[0]{\mathrm{slc}}
\newcommand{\nef}[0]{\mathrm{nef}}
\newcommand{\SO}[0]{{\operatorname{SO}}}
\newcommand{\Symp}[0]{{\operatorname{Sp}}}
\newcommand{\SU}[0]{{\operatorname{SU}}}
\newcommand{\F}[0]{{\operatorname{F}}}
\newcommand{\Sgn}[0]{{ \Sigma_{g, n} }}
\newcommand{\Sm}[0]{{\operatorname{Sm}}}
\newcommand{\SpSp}[0]{{\mathbb{S}}}
\newcommand{\Spec}[0]{\operatorname{Spec}}
\newcommand{\relspec}[0]{\ul{ \operatorname{Spec}} }
\newcommand{\Spf}[0]{\operatorname{Spf}}
\newcommand{\Spc}[0]{\operatorname{Spc}}
\newcommand{\spc}[0]{\operatorname{Spc}}
\newcommand{\Spinc}[0]{\mathrm{Spin}^{{ \scriptscriptstyle \mathbf C} }}
\newcommand{\Spin}[0]{{\operatorname{Spin}}}
\newcommand{\Mp}[0]{{\operatorname{Mp}}}
\newcommand{\Herm}[0]{{\mathsf{Herm}}}
\newcommand{\coinv}[0]{{\operatorname{coinv}}}
\newcommand{\Sq}[0]{\operatorname{Sq}}
\newcommand{\Stab}[0]{{\operatorname{Stab}}}
\newcommand{\stable}[0]{\mathrm{st}}
\newcommand{\lkt}[0]{\Lambda_{\mathrm{K3}}}
\newcommand{\unstable}[0]{\mathrm{unst}}
\newcommand{\Sum}[0]{ \sum }
\newcommand{\Syl}[0]{{\operatorname{Syl}}}
\newcommand{\Sym}[0]{\operatorname{Sym}}
\newcommand{\Tor}[0]{\operatorname{Tor}}
\newcommand{\Homcx}[0]{\operatorname{Hom}^{\bullet}}
\newcommand{\Tr}[0]{\operatorname{Tr}}
\newcommand{\Ug}[0]{{\mathcal{U}(\mathfrak{g}) }}
\newcommand{\Uh}[0]{{\mathcal{U}(\mathfrak{h}) }}
\newcommand{\Union}[0]{\bigcup}
\newcommand{\U}[0]{{\operatorname{U}}}
\newcommand{\Wedge}[0]{\bigwedge}
\newcommand{\Wittvectors}[0]{{\mathbb{W}}}
\newcommand{\ZHB}[0]{\operatorname{ZHB}}
\newcommand{\ZHS}[0]{\ZZ\operatorname{HS}}
\newcommand{\QHB}[0]{\operatorname{QHB}}
\newcommand{\PHS}[0]{\operatorname{PHS}}
\newcommand{\QHS}[0]{\mathbf{Q}\kern-0.5pt\operatorname{HS}}
\newcommand{\ZZG}[0]{{\ZZ G}}
\newcommand{\ZZH}[0]{{\ZZ H}}
\newcommand{\ZZlocal}[1]{{ \ZZ_{\hat{#1}} }}
\newcommand{\plocal}[0]{{ \scriptsize {}_{ \localize{p}  } }}
\newcommand{\ZZpadic}[0]{{ \ZZ_{\hat p} }}
\newcommand{\ZZladic}[0]{{ \ZZ_{\hat \ell} }}
\newcommand{\ZZpcomplete}[0]{{ \ZZpadic }}
\newcommand{\ZZplocal}[0]{{ L_p \ZZ }}
\newcommand{\ZZprof}[0]{{ \hat{\ZZ} }}
\newcommand{\QQladic}[0]{{ \QQ_\ell }}
\newcommand{\CCpadic}[0]{{ \CC_p }}
\newcommand{\ZZ}[0]{{\mathbf{Z}}}
\newcommand{\ZZbar}[0]{{ \bar{ \ZZ } }}
\newcommand{\ZZhat}[0]{{ \widehat{ \ZZ } }}
\newcommand{\Zar}[0]{{\mathrm{Zar}}}
\newcommand{\ZpZ}[0]{\ZZ/p}
\newcommand{\abuts}[0]{\Rightarrow}
\newcommand{\ab}[0]{{\operatorname{ab}}}
\newcommand{\Rees}[0]{{\operatorname{Rees}}}
\newcommand{\actsonl}[0]{\curvearrowleft}
\newcommand{\actson}[0]{\curvearrowright}
\newcommand{\adjoint}[0]{\dagger}
\newcommand{\adj}[0]{\operatorname{adj}}
\newcommand{\ad}[0]{ { \operatorname{ad}}}
\newcommand{\Ad}[0]{{ \operatorname{Ad} }}
\newcommand{\afp}[0]{A_{/\FF_p}}
\newcommand{\annd}[0]{{\operatorname{ and }}}
\newcommand{\ann}[0]{\operatorname{Ann}}
\newcommand{\csch}[0]{\operatorname{csch}}
\newcommand{\arccot}[0]{\operatorname{arccot}}
\newcommand{\arccsc}[0]{\operatorname{arccsc}}
\newcommand{\arcsec}[0]{\operatorname{arcsec}}
\newcommand{\bP}[0]{\operatorname{bP}}
\newcommand{\barz}{\bar{z}}
\newcommand{\bbm}[0]{{\mathbb{M}}}
\newcommand{\bd}[0]{{\del}}
\newcommand{\bigast}[0]{{\mathop{\text{\Large $\ast$}}}}
\newcommand{\bmgn}[0]{{ \bar{\mathcal{M}}_{g, n} }}
\newcommand{\bundle}[1]{\mathcal{#1}}
\newcommand{\by}[0]{\times}
\newcommand{\candim}[0]{\operatorname{candim}}
\newcommand{\chp}[0]{\operatorname{ch. p}}
\newcommand{\ch}[0]{\operatorname{ch}}
\newcommand{\AJ}[0]{\operatorname{AJ}}
\newcommand{\chr}[0]{\operatorname{ch}}
\newcommand{\characteristic}[0]{\operatorname{ch}}
\newcommand{\character}[0]{\operatorname{ch}}
\newcommand{\cl}[0]{{ \operatorname{cl}}}
\newcommand{\codim}[0]{\operatorname{codim}}
\newcommand{\cohdim}[0]{\operatorname{cohdim}}
\newcommand{\coim}[0]{\operatorname{coim}}
\newcommand{\coker}[0]{\operatorname{coker}}
\newcommand{\cok}[0]{\operatorname{coker}}
\newcommand{\cone}[0]{\operatorname{cone}}
\newcommand{\conjugate}[1]{{\overline{{#1}}}}
\newcommand{\connectsum}[0]{\mathop{ \Large\mypound }}
\newcommand{\const}[0]{{\operatorname{const.}}}
\newcommand{\converges}[1]{\overset{#1}}
\newcommand{\convergesto}[1]{\overset{#1}\too}
\newcommand{\convolve}[0]{\ast}
\newcommand{\correspond}[1]{\theset{\substack{#1}}}
\newcommand{\covers}[0]{\rightrightarrows}
\newcommand{\cocovers}[0]{\leftleftarrows}
\newcommand{\coveredby}[0]{\leftleftarrows}
\newcommand{\projresolve}[0]{\rightrightarrows}
\newcommand{\injresolve}[0]{\leftleftarrows}
\newcommand{\covol}[0]{\operatorname{covol}}
\newcommand{\cpt}[0]{{ \operatorname{compact} }}
\newcommand{\crit}[0]{\operatorname{crit}}
\newcommand{\cross}[0]{\times}
\newcommand{\dR}[0]{\mathrm{dR}}
\newcommand{\dV}{\,dV}
\newcommand{\dash}[0]{{\hbox{-}}}
\newcommand{\da}[0]{\coloneqq}
\newcommand{\ddd}[2]{{\frac{d #1}{d #2}\,}}
\newcommand{\ddim}[0]{\operatorname{ddim}}
\newcommand{\area}[0]{\operatorname{area}}
\newcommand{\ddt}{\tfrac{\dif}{\dif t}}
\newcommand{\ddx}{\tfrac{\dif}{\dif x}}
\newcommand{\dd}[2]{{\frac{\partial #1}{\partial #2}\,}}
\newcommand{\definedas}[0]{\coloneqq}
\newcommand{\del}[0]{{\partial}}
\newcommand{\diagonal}[1]{\Delta}
\newcommand{\Diagonal}[1]{\Delta}
\newcommand{\diag}[0]{\operatorname{diag}}
\newcommand{\diam}[0]{{\operatorname{diam}}}
\newcommand{\diff}[0]{\operatorname{Diff}}
\newcommand{\discriminant}[0]{{\Delta}}
\newcommand{\disc}[0]{{\operatorname{disc}}}
\newcommand{\disjoint}[0]{{\textstyle\coprod}}
\newcommand{\dist}[0]{\operatorname{dist}}
\newcommand{\dlog}[0]{\operatorname{dLog}}
\newcommand{\logd}[0]{{ \del^{\scriptsize \log} }}
\newcommand{\dom}[0]{\operatorname{dom}}
\newcommand{\BM}[0]{{\operatorname{BM}}}
\newcommand{\dual}[0]{ {}^{ \vee }}
\newcommand{\shriek}[0]{ { ! }}
\newcommand{\dmu}{\,d\mu}
\newcommand{\dr}{\,dr}
\newcommand{\dtau}{\,d\tau}
\newcommand{\ds}{\,ds}
\newcommand{\dtheta}{\,d\theta}
\newcommand{\dt}{\,dt}
\newcommand{\du}{\,du}
\newcommand{\dw}{\,dw}
\newcommand{\dxi}{\,d\xi}
\newcommand{\dx}{\,dx}
\newcommand{\dm}{\,dm}
\newcommand{\dA}{\,dA}
\newcommand{\dy}{\,dy}
\newcommand{\dn}{\,dn}
\newcommand{\dalpha}{\,d\alpha}
\newcommand{\dzbar}{\,d\bar{z}}
\newcommand{\dzeta}{\,d\zeta}
\newcommand{\dz}{\,dz}
\newcommand{\drho}{\,d\rho}
\newcommand{\dphi}{\,d\phi}
\newcommand{\embeds}[0]{\hookrightarrow}
\newcommand{\eo}[0]{{\operatorname{eo}}}
\newcommand{\eps}[0]{{\varepsilon}}
\newcommand{\essdim}[0]{\operatorname{essdim}}
\newcommand{\eul}[0]{{\operatorname{eul}}}
\newcommand{\evalfrom}[0]{\Big|}
\newcommand{\ext}{\operatorname{Ext}}
\newcommand{\ff}[0]{\operatorname{ff}}
\newcommand{\fppf}[0]{{\operatorname{fppf}}}
\newcommand{\fpqc}[0]{{\operatorname{fpqc}}}
\newcommand{\fp}[0]{{ \operatorname{fp} }}
\newcommand{\fqr}[0]{{\FF_{q^r}}}
\newcommand{\fq}[0]{{\FF_{q}}}
\newcommand{\fqbar}[0]{\bar{\FF_{q}}}
\newcommand{\freeprod}[0]{\ast}
\newcommand{\from}[0]{\leftarrow}
\newcommand{\gal}[0]{{ \operatorname{Gal}}}
\newcommand{\gl}[0]{{\mathfrak{gl}}}
\newcommand{\gp}[0]{ {\operatorname{gp} }}
\newcommand{\grad}[0]{\operatorname{grad}}
\newcommand{\graded}[0]{\operatorname{gr}}
\newcommand{\grdim}[0]{{\operatorname{gr\,dim}}}
\newcommand{\height}[0]{\operatorname{ht}}
\newcommand{\homotopic}[0]{\simeq}
\newcommand{\id}[0]{\operatorname{id}}
\newcommand{\im}[0]{\operatorname{im}}
\newcommand{\decreasesto}[0]{\searrow}
\newcommand{\increasesto}[0]{\nearrow}
\newcommand{\injectivelim}[0]{\varinjlim}
\newcommand{\injects}[0]{\hookrightarrow}
\newcommand{\interior}[0]{^\circ}
\newcommand{\intersect}[0]{\cap}
\newcommand{\Todd}[0]{\operatorname{Td}}
\newcommand{\into}[0]{\to}
\newcommand{\inverselim}[0]{\varprojlim}
\newcommand{\inv}[0]{^{-1}}
\newcommand{\ip}[2]{{\left\langle {#1},~{#2} \right\rangle}}
\newcommand{\kG}[0]{{kG}}
\newcommand{\kfq}[0]{K_{/\FF_q}}
\newcommand{\kk}[0]{{\mathbf{k}}}
\newcommand{\ko}[0]{{\operatorname{ko}}}
\newcommand{\krulldim}[0]{\operatorname{krulldim}}
\newcommand{\ks}[0]{\operatorname{ks}}
\newcommand{\kxn}[0]{k[x_1, \cdots, x_{n}]}
\newcommand{\kxnz}[0]{k[x_0, \cdots, x_{n}]}
\newcommand{\kx}[1]{k[x_1, \cdots, x_{#1}]}
\newcommand{\lci}[0]{\mathrm{lci}}
\newcommand{\SC}[0]{\mathrm{SC}}
\newcommand{\tb}[0]{\mathrm{tb}}
\newcommand{\lcm}[0]{\operatorname{lcm}}
% Lie theory
\newcommand{\liealgk}[0]{{ \liealg_{/k} }}
\newcommand{\liea}[0]{{\mathfrak{a}}}
\newcommand{\lieb}[0]{{\mathfrak{b}}}
\newcommand{\mfe}[0]{{\mathfrak{e}}}
\newcommand{\lied}[0]{{\mathfrak{d}}}
\newcommand{\lief}[0]{{\mathfrak{f}}}
\newcommand{\liegl}[0]{{\mathfrak{gl}}}
\newcommand{\liep}[0]{{\mathfrak{p}}}
\newcommand{\liesu}[0]{{\mathfrak{su}}}
\newcommand{\lieg}[0]{{\mathfrak{g}}}
\newcommand{\lieh}[0]{{\mathfrak{h}}}
\newcommand{\liel}[0]{{\mathfrak{l}}}
\newcommand{\lien}[0]{{\mathfrak{n}}}
\newcommand{\lieo}[0]{{\mathfrak{o}}}
\newcommand{\lieq}[0]{{\mathfrak{q}}}
\newcommand{\lier}[0]{{\mathfrak{r}}}
\newcommand{\lies}[0]{{\mathfrak{s}}}
\newcommand{\liesl}[0]{{\mathfrak{sl}}}
\newcommand{\lieso}[0]{{\mathfrak{so}}}
\newcommand{\liesp}[0]{{\mathfrak{sp}}}
\newcommand{\liet}[0]{{\mathfrak{t}}}
\newcommand{\lieu}[0]{{\mathfrak{u}}}
\newcommand{\lieW}[0]{{\mathfrak{W}}}
\newcommand{\liey}[0]{{\mathfrak{y}}}
\newcommand{\mfX}[0]{{\mathfrak{X}}}
\newcommand{\mfS}[0]{{\mathfrak{S}}}
\newcommand{\gitquot}{{ \mathbin{/\mkern-6mu/}}}
\newcommand{\modmod}{\gitquot}
\newcommand{\htyquot}{\gitquot}
\newcommand{\lk}[0]{\operatorname{lk}}
\newcommand{\rot}[0]{\operatorname{rot}}
\newcommand{\mTHH}[0]{{\operatorname{THH}}}
\newcommand{\mHH}[0]{{\operatorname{HH}}}
\newcommand{\TCH}[0]{{\operatorname{TCH}}}
\newcommand{\TC}[0]{{\operatorname{TC}}}
\newcommand{\THC}[0]{{\operatorname{THC}}}
\newcommand{\THoH}[0]{{\operatorname{THH}}}
\newcommand{\HoH}[0]{{\operatorname{HH}}}
\newcommand{\TP}[0]{{\operatorname{TP}}}
\newcommand{\TT}[0]{{\mathbb{T}}}
\newcommand{\mapscorrespond}[2]{\mathrel{\operatorname*{\rightleftharpoons}_{#2}^{#1}}}
\newcommand{\mapstofrom}[0]{\rightleftharpoons}
\newcommand{\ofrom}[0]{\overset{\circ}{\leftarrow}}
\newcommand{\oto}[0]{\overset{\circ}{\rightarrow}}
\newcommand{\wdot}[0]{{ \mathsf{w}_{\cdot} }}
\newcommand{\sdot}[0]{{ \mathsf{S}_{\cdot} }}
\newcommand{\mapstovia}[1]{\xmapsto{#1}}
\newcommand{\mapsvia}[1]{\xrightarrow{#1}}
\newcommand{\tovia}[1]{\xrightarrow{#1}}
\newcommand{\torational}[0]{\dashrightarrow}
\newcommand{\rational}[0]{\torational}
\newcommand{\birational}[0]{\overset{\sim}{\torational}}
\newcommand{\birationaliso}[0]{\overset{\sim}{\torational}}
\newcommand{\sbirational}[0]{\overset{\sim_{ \stab} }{\torational}}
\newcommand{\isomorphic}{{ \, \mapsvia{\sim}\, }}
\newcommand{\iso}{ \isomorphic  }
\newcommand{\qiso}{ \homotopic }
\newcommand{\isovia}[1]{\underset{#1}{\iso}}
\newcommand{\isoin}[1]{\overset{#1}{\iso}}
\newcommand{\injectsvia}[1]{\xhookrightarrow{#1}}
\newcommand{\injectsfrom}[0]{\hookleftarrow}
\newcommand{\embedsdense}[0]{{ \underset{ {\scriptscriptstyle\mathrm{dense}} }{\hookrightarrow} }}
\newcommand{\injectsfromvia}[1]{\xhookleftarrow{#1}}
\newcommand{\maps}[0]{\operatorname{Maps}}
\newcommand{\mat}[0]{\operatorname{Mat}}
\newcommand{\maxspec}[0]{{\operatorname{maxSpec}}}
\newcommand{\mcTop}[0]{\mathcal{T}\mathsf{op}}
\newcommand{\mca}[0]{{\mathcal{A}}}
\newcommand{\mcb}[0]{{\mathcal{B}}}
\newcommand{\mcc}[0]{{\mathcal{C}}}
\newcommand{\mcd}[0]{{\mathcal{D}}}
\newcommand{\mce}[0]{{\mathcal{E}}}
\newcommand{\mcf}[0]{{\mathcal{F}}}
\newcommand{\mcg}[0]{{\mathcal{G}}}
\newcommand{\mch}[0]{{\mathcal{H}}}
\newcommand{\mci}[0]{{\mathcal{I}}}
\newcommand{\mcj}[0]{{\mathcal{J}}}
\newcommand{\mck}[0]{{\mathcal{K}}}
\newcommand{\mcl}[0]{{\mathcal{L}}}
\newcommand{\mcm}[0]{{\mathcal{M}}}
\newcommand{\mcn}[0]{{\mathcal{N}}}
\newcommand{\mco}[0]{{\mathcal{O}}}
\newcommand{\mcp}[0]{{\mathcal{P}}}
\newcommand{\mcr}[0]{{\mathcal{R}}}
\newcommand{\mcs}[0]{{\mathcal{S}}}
\newcommand{\mct}[0]{{\mathcal{T}}}
\newcommand{\mcu}[0]{{\mathcal{U}}}
\newcommand{\mcv}[0]{{\mathcal{V}}}
\newcommand{\mcw}[0]{{\mathcal{W}}}
\newcommand{\mcx}[0]{{\mathcal{X}}}
\newcommand{\mcX}[0]{{\mathcal{X}}}
\newcommand{\mcy}[0]{{\mathcal{Y}}}
\newcommand{\mcz}[0]{{\mathcal{Z}}}
\newcommand{\mfa}[0]{{\mathfrak{a}}}
\newcommand{\mfb}[0]{{\mathfrak{b}}}
\newcommand{\mfc}[0]{{\mathfrak{c}}}
\newcommand{\mff}[0]{{\mathfrak{f}}}
\newcommand{\mfi}[0]{{\mathfrak{I}}}
\newcommand{\mfh}[0]{{\mathfrak{h}}}
\newcommand{\mfm}[0]{{\mathfrak{m}}}
\newcommand{\mfn}[0]{{\mathfrak{n}}}
\newcommand{\mfp}[0]{{\mathfrak{p}}}
\newcommand{\mfo}[0]{{\mathfrak{o}}}
\newcommand{\mfq}[0]{{\mathfrak{q}}}
\newcommand{\mfr}[0]{{\mathfrak{r}}}
\newcommand{\mfs}[0]{{\mathfrak{s}}}
\newcommand{\mfy}[0]{{\mathfrak{Y}}}
\newcommand{\mfx}[0]{{\mathfrak{X}}}
\newcommand{\mg}[0]{{ \mathcal{M}_{g} }}
\newcommand{\mgn}[0]{{ \mathcal{M}_{g, n} }}
\newcommand{\Mgn}[0]{{ \mathcal{M}_{g, n} }}
\newcommand{\Mell}[0]{{ \mathcal{M}_{\mathrm{ell}} }}
\newcommand{\cusp}[0]{{ \mathrm{cusp} }}
\newcommand{\Noeth}[0]{{ \mathrm{Noeth} }}
\newcommand{\minpoly}[0]{{\operatorname{minpoly}}}
\newcommand{\mltext}[1]{\left\{\begin{array}{c}#1\end{array}\right\}}
\newcommand{\mm}[0]{{\mathfrak{m}}}
\newcommand{\mot}[0]{{ \mathrm{mot}}}
\newcommand{\cell}[0]{{ \mathrm{cell}}}
\newcommand{\mspec}[0]{\operatorname{mSpec}}
\newcommand{\mproj}[0]{\operatorname{mProj}}
\newcommand{\ms}[0]{\xrightarrow{\sim}}
\newcommand{\multinomial}[1]{\left(\!\!{#1}\!\!\right)}
\newcommand{\mult}[0]{{\operatorname{mult}}}
\newcommand{\mveq}[0]{{\mapsvia{\sim}}} 
\newcommand{\mviso}[0]{{\mapsvia{\sim}}} 
\newcommand{\nd}[0]{\operatorname{nd}}
\newcommand{\nil}[0]{{\operatorname{nil}}}
\newcommand{\nonzero}[0]{^{\bullet}}
\newcommand{\normalneq}{\mathrel{\reflectbox{$\trianglerightneq$}}}
\newcommand{\normal}[0]{{~\trianglelefteq~}}
\newcommand{\notimplies}[0]{\centernot\implies}
\newcommand{\onto}[0]{\twoheadhthtarrow}


\newcommand{\op}[0]{^{\operatorname{op}}}
\newcommand{\ord}[0]{{\operatorname{Ord}}}
\newcommand{\order}[0]{{\operatorname{Ord}}}
\newcommand{\oriented}[0]{{ \operatorname{oriented} }}
\newcommand{\orr}[0]{{\operatorname{ or }}}
\newcommand{\padic}[0]{p\dash\text{adic}}
\newcommand{\pic}[0]{{\operatorname{Pic}}}
\newcommand{\poly}[0]{\mathrm{poly}}
\newcommand{\prim}[0]{{\operatorname{prim}}}
\newcommand{\projectivelim}[0]{\varprojlim}
\newcommand{\pr}[0]{{\operatorname{pr}}}
\newcommand{\pt}[0]{{\operatorname{pt}}}
\newcommand{\qc}[0]{{\operatorname{qc}}}
\newcommand{\qst}[0]{{\quad \operatorname{such that} \quad}}
\newcommand{\rank}[0]{\operatorname{rank}}
\newcommand{\realpart}[1]{{\mathcal{Re}({#1})}}
\newcommand{\red}[0]{{ \text{red} }}
\newcommand{\reg}[0]{\mathrm{reg}}
\newcommand{\reldim}[0]{\operatorname{reldim}}
\newcommand{\hilbdim}[0]{\operatorname{hilbdim}}
\newcommand{\gendim}[0]{\operatorname{gendim}}
\newcommand{\restrictionof}[2]{ {\left.{{#1}} \right|_{{#2}} }}
\newcommand{\rk}[0]{{\operatorname{rank}}}
\newcommand{\rotate}[2]{{\style{display: inline-block; transform: rotate(#1deg)}{#2}}}
\newcommand{\ro}[2]{{ \left.{{#1}} \right|_{{#2}} }}
\newcommand{\selfmap}[0]{{\circlearrowleft}}
\newcommand{\semidirect}[0]{\rtimes}
\newcommand{\wreath}[0]{\wr}
\newcommand{\size}[0]{{\sharp}}
\newcommand{\sep}[0]{{ {}^{ \operatorname{sep} } }}
\newcommand{\sgn}[0]{\operatorname{sgn}}
\newcommand{\shom}{ {\mathcal{H}}\kern-0.5pt{\operatorname{om}}}
\newcommand{\signature}[0]{\operatorname{sig}}
\newcommand{\sign}[0]{\operatorname{sign}}
\newcommand{\Sing}[0]{{\operatorname{Sing}}}
\newcommand{\sing}[0]{{\mathrm{sing}}}
\newcommand{\Pet}[0]{{\mathrm{Pet}}}
\newcommand{\CM}[0]{{\mathrm{CM}}}
\newcommand{\slope}[0]{{\mathrm{slope}}}
\newcommand{\smpt}[1]{\setminus\theset{#1}}
\newcommand{\smts}[1]{\setminus\theset{ #1 }}
\newcommand{\smz}[0]{\setminus\theset{0}}
\newcommand{\sm}[0]{\setminus}
\newcommand{\spec}[0]{\operatorname{Spec}}
\newcommand{\stab}[0]{{\operatorname{Stab}}}
\newcommand{\stirlingfirst}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}}
\newcommand{\stirling}[2]{\genfrac\{\}{0pt}{}{#1}{#2}}
\newcommand{\surjectsvia}[2][]{ \xrightarrow[#1]{#2} { \mathrel{\mkern-16mu}\rightarrow }\, }
\newcommand{\surjects}[0]{\twoheadrightarrow}
\newcommand{\syl}[0]{{\operatorname{Syl}}}
\newcommand{\sym}[0]{\operatorname{Sym}^*}
\newcommand{\Cox}[0]{\operatorname{Cox}}
\newcommand{\Eff}[0]{\operatorname{Eff}}
\newcommand{\Symalg}[0]{\sym}
\newcommand{\symalg}[0]{\sym}
\newcommand{\td}[0]{\mathrm{td}}
\newcommand{\Li}[0]{\mathrm{Li}}
\newcommand{\VMHS}[0]{\mathrm{VMHS}}
\newcommand{\MHS}[0]{\mathrm{MHS}}
\newcommand{\Tensor}[0]{\bigotimes}
\newcommand{\tensor}[0]{\otimes}
\newcommand{\boxtensor}[0]{\boxtimes}
\newcommand{\Ltensor}[0]{ \overset{\mathbb{L}}{ \otimes}}
\newcommand{\dtensor}[0]{ \overset{\mathbb{L}}{ \otimes}}
\newcommand{\tgn}[0]{{ \mathcal{T}_{g, n} }}
\newcommand{\theset}[1]{\left\{{#1}\right\}}
\newcommand{\thetaref}[0]{{ \theta_{\mathrm{Ref} } }}
\newcommand{\thevector}[1]{{\left[ {#1} \right]}}
\newcommand{\thinrank}[0]{T_n\dash\operatorname{rank}}
\newcommand{\too}[1]{{\xrightarrow{#1}}}
\newcommand{\tors}[0]{{\operatorname{tors}}}
\newcommand{\tor}[0]{\operatorname{Tor}}
\newcommand{\transverse}[0]{\pitchfork}
\newcommand{\trdeg}[0]{{\mathrm{trdeg}}}
\newcommand{\trianglerightneq}{\mathrel{\ooalign{\raisebox{-0.5ex}{\reflectbox{\rotatebox{90}{$\nshortmid$}}}\cr$\triangleright$\cr}\mkern-3mu}}
\newcommand{\tr}[0]{{\mathrm{tr}}}
\newcommand{\tspt}[0]{{\{\operatorname{pt}\}}}
\newcommand{\ts}[1]{\left\{{#1}\right\}}
\newcommand{\tv}[1]{{\left[ {#1} \right]}}
\newcommand{\txand}[0]{{\operatorname{ and }}}
\newcommand{\txor}[0]{{\operatorname{ or }}}
\newcommand{\type}[0]{{\operatorname{type}}}
\newcommand{\uniformlyconverges}[0]{\rightrightarrows}
\newcommand{\union}[0]{\cup}
\newcommand{\unital}[0]{{\operatorname{unital}}}
\newcommand{\units}[0]{^{\times}}
\newcommand{\up}[0]{\uparrow}
\newcommand{\vhat}[1]{\widehat{ \vector{#1} }}
\newcommand{\vol}[0]{\operatorname{vol}}
\newcommand{\wait}[0]{{-}}
\newcommand{\wt}[0]{{\operatorname{wt}}}
\newcommand{\zar}[0]{{\mathrm{zar}}}
\newcommand{\zbar}{\bar{z}}
\newcommand{\zlnz}[0]{\ZZ/\ell^n\ZZ}
\newcommand{\zlz}[0]{\ZZ/\ell\ZZ}
\newcommand{\znz}[0]{\ZZ/n\ZZ}
\newcommand{\zpz}[0]{\ZZ/p\ZZ}
\newcommand{\divergence}[0]{{ \nabla\cdot }}
\newcommand{\normalizer}[0]{{ N }}
\newcommand{\fracId}[0]{{ \ddot{\Id} }}
\newcommand{\Ord}[0]{{ \mathrm{Ord} }}
\newcommand{\join}[0]{{ \ast }}

\renewcommand{\AA}[0]{{\mathbf{A}}}
\renewcommand{\SS}[0]{{\mathbf{S}}}
\renewcommand{\ss}[0]{{\mathrm{ss}}}
\renewcommand{\bar}[1]{\overline{#1}}
\renewcommand{\det}{\operatorname{det}}
\renewcommand{\div}[0]{\operatorname{div}}
\newcommand{\IAS}[0]{\mathrm{IAS}}
\newcommand{\KSBA}[0]{\mathrm{KSBA}}
\newcommand{\isotrop}[0]{\mathrm{isotrop}}
\newcommand{\Div}[0]{\operatorname{Div}}
\newcommand{\CDiv}[0]{\operatorname{CDiv}}
\newcommand{\WDiv}[0]{\operatorname{WDiv}}
\newcommand{\Bs}[0]{\operatorname{Bs}}
\newcommand{\Nef}[0]{\operatorname{Nef}}
\renewcommand{\hat}[1]{\widehat{#1}}
\renewcommand{\labelitemiii}{$\diamondsuit$}
\renewcommand{\labelitemiv}{$\diamondsuit$}
\renewcommand{\mid}[0]{\mathrel{\Big|}}
\renewcommand{\mod}{\operatorname{mod}}
\renewcommand{\qed}[0]{\hfill\blacksquare}
\renewcommand{\too}[0]{\longrightarrow}
\renewcommand{\vector}[1]{\mathbf{#1}}
\newcommand{\sheafhom}[0]{\mathop{\mathcal{H}\! \mathit{om}}}

% Declare math operators
\DeclareMathOperator*{\spanof}{span}
\DeclareMathOperator*{\Mor}{Mor}
\DeclareMathOperator*{\Sec}{Sec}
\DeclareMathOperator*{\supp}{supp}
\DeclareMathOperator*{\eq}{=}
\DeclareMathOperator*{\hocolim}{hocolim}
\DeclareMathOperator{\Index}{Index}
\DeclareMathOperator{\aut}{Aut}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Inn}{Inn}
\DeclareMathOperator{\Out}{Out}
\DeclareMathOperator{\BiHol}{BiHol}
\DeclareMathOperator{\Curv}{Curv}
\DeclareMathOperator{\Deck}{Deck}
\DeclareMathOperator{\Der}{Der}
\DeclareMathOperator{\Exists}{\exists}
\DeclareMathOperator{\Forall}{\forall}
\DeclareMathOperator{\Forget}{Forget}
\DeclareMathOperator{\Frame}{Frame}
\DeclareMathOperator{\Fr}{Fr}
\DeclareMathOperator{\Griff}{Griff}
\DeclareMathOperator{\Hol}{Hol}
\DeclareMathOperator{\Ld}{{\mathbb{L} }}
\DeclareMathOperator*{\Map}{Maps}
\DeclareMathOperator*{\Maps}{Maps}
\DeclareMathOperator{\OFrame}{OFrame}
\DeclareMathOperator{\Prin}{Prin}
\DeclareMathOperator{\Cart}{Cart}
\DeclareMathOperator{\Proj}{Proj}
\DeclareMathOperator{\RHom}{\mathbb{R}Hom}
\DeclareMathOperator{\Rad}{Rad}
\DeclareMathOperator{\Rd}{{\mathbb{R} }}
\DeclareMathOperator{\Symb}{Symb}
\DeclareMathOperator{\Taut}{Taut}
\DeclareMathOperator{\Th}{Th}
\DeclareMathOperator{\triv}{triv}
\DeclareMathOperator{\Triv}{Triv}
\DeclareMathOperator{\UFrame}{UFrame}
\DeclareMathOperator{\Isom}{Isom}
\DeclareMathOperator{\codom}{codom}
\DeclareMathOperator{\coh}{coh}
\DeclareMathOperator{\coinfl}{coinf}
\DeclareMathOperator{\colspace}{colspace}
\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\cores}{coRes}
\DeclareMathOperator{\hd}{Head}
\DeclareMathOperator{\hilb}{Hilb}
\DeclareMathOperator{\infl}{inf}
\DeclareMathOperator{\len}{len}
\DeclareMathOperator{\length}{length}
\DeclareMathOperator{\nullity}{nullspace}
\DeclareMathOperator{\nullspace}{nullspace}
\DeclareMathOperator{\per}{per}
\DeclareMathOperator{\prin}{prin}
\DeclareMathOperator{\projection}{Proj}
\DeclareMathOperator{\proj}{proj}
\DeclareMathOperator{\range}{range}
\DeclareMathOperator{\rowspace}{rowspace}
\DeclareMathOperator{\soc}{Soc}
\DeclareMathOperator{\submfds}{SubMfds}

\newcommand{\st}[0]{{~\mathrel{\Big\vert}~}}
\newcommand{\suchthat}[0]{\st}
\newcommand{\Suchthat}[0]{\middle\vert}

\newcommand{\delbar}[0]{{ \bar{\del}}}

\newcommand{\containedin}[0]{\subseteq}
\newcommand{\contains}[0]{\supseteq}
\newcommand{\containing}[0]{\supseteq}
\newcommand{\iscontainedin}[0]{\supseteq}

\newcommand{\rad}[1]{\sqrt{#1}}
\newcommand{\thecat}[1]{\mathbf{#1}}
\newcommand{\sheaf}[1]{\operatorname{\mathcal{#1}}}
\newcommand{\IC}[0]{\sheaf{IC}}
\newcommand{\rightderive}[0]{{\mathbf{R}}}
\newcommand{\leftderive}[0]{{\mathbf{L}}}
\newcommand{\rderive}[0]{\rightderive}
\newcommand{\lderive}[0]{\leftderive}

% More involved commands
\newcommand\rrarrows{\rightrightarrows}
\newcommand\rrrarrows{
    \mathrel{\substack{\textstyle\rightarrow\\[-0.6ex]
        \textstyle\rightarrow \\[-0.6ex]
        \textstyle\rightarrow}}
}



\newcommand\ul[1]{\underline{#1}}
\newcommand\constantsheaf[1]{\underline{#1}}
\newcommand\holomorphic[0]{\text{holo}}
\newcommand\hol[0]{\text{hol}}
\newcommand\cts[0]{\text{cts}}
\newcommand\std[0]{\text{std}}
\newcommand\Fl[0]{\operatorname{Fl}}
\newcommand\Mero[0]{\operatorname{Mero}}
\newcommand\Bl[0]{\operatorname{Bl}}
\newcommand\res[0]{\operatorname{res}}
\newcommand\coind[0]{\operatorname{coInd}}
%\newcommand\ind[0]{\operatorname{Ind}}
\newcommand\Sel[0]{\operatorname{Sel}}
\newcommand\Frob[0]{\operatorname{Frob}}
\newcommand\transfer[0]{\operatorname{transfer}}
\newcommand\modiso[0]{{_{\scriptstyle / \sim} }}
\newcommand\even[0]{\text{even}}
\newcommand\odd[0]{\text{odd}}
\newcommand\hodgestar[0]{\star}
\newcommand\dirac[0]{\mkern-3mu \not{ \partial}}
\newcommand\laplacian[0]{\Delta}
\newcommand\Laplacian[0]{\Delta}
\newcommand\stardstar[0]{\hodgestar \mathrm{d} \mkern-1mu \hodgestar}
\newcommand\covariant[0]{\nabla}
\newcommand\ol[1]{\overline{#1}}
\newcommand\univcover[1]{\overline{#1}}
\newcommand\closure[1]{\overline{#1}}
\newcommand\capprod{\frown}
\newcommand\cupprod{\smile}
\newcommand\Path{\mathcal{P}}
\newcommand\gradient{\nabla}

\newcommand\cechH[0]{{\check{H}}}
\newcommand\Hc[0]{{\check{H}}}
\newcommand\Cc[0]{{\check{C}}}
\newcommand\cupp[0]{\smile}
\newcommand\capp[0]{\frown}
\newcommand\sig[0]{\operatorname{sig}}
\newcommand\ev[0]{\operatorname{ev}}
\newcommand\coev[0]{\operatorname{coev}}
\newcommand\period[0]{\operatorname{period}}

\newcommand{\fractional}[1]{\theset{#1}}
\newcommand{\fractionalpart}[1]{\theset{#1}}
\newcommand{\integerpart}[1]{\left[ {#1}\right] }
\newcommand{\zadjoin}[1]{\ZZ\left[ {#1} \right]}
\newcommand{\wedgeprod}[0]{\vee}
\newcommand{\Prod}[0]{\displaystyle\prod}
\newcommand{\Wedgepower}[0]{\bigwedge\nolimits}
\def\multichoose#1#2{{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{#1}{#2}\right)\kern-.3em\right)}}
\def\rising#1#2{ \qty{#1}^{ (#2) }}
\def\falling#1#2{ \qty{#1}_{ (#2) }}
\newcommand\elts[2]{{ {#1}_1, {#1}_2, \cdots, {#1}_{#2}}}
\newcommand\tselts[2]{{ \theset{ {#1}_1, {#1}_2, \cdots, {#1}_{#2} } }}
\newcommand\mix[1]{\overset{\scriptscriptstyle {#1} }{\times}}
\newcommand\fiberproduct[1]{\underset{\scriptscriptstyle {#1} }{\times}}
\newcommand\fiberprod[1]{ \fiberproduct{#1}}
\newcommand\basechange[1]{{ \fiberproduct{k} {#1} }}
\newcommand\fprod[1]{{ \fiberproduct{#1} }}
\newcommand\smallprod[0]{{ \scriptscriptstyle\prod }}
\newcommand\eqLH[0]{{ \equalsbecause{\scriptscriptstyle\text{LH}} }}
\newcommand\drcomplex[0]{{\cocomplex{\Omega}}}
\newcommand{\cxH}[0]{{\complex{H}}}
\newcommand{\ccxH}[0]{{\cocomplex{H}}}
\newcommand{\chern}[0]{{\mathrm{ch}}}
\newcommand{\qsymb}[2]{{ \left( {#1} \over {#2} \right)  }}
\newcommand{\spinornorm}[0]{{\mathrm{spinornorm}}}
\newcommand{\RT}[0]{{\mathrm{RT}}}
\newcommand{\amp}[0]{{\mathrm{amp}}}
\newcommand{\vamp}[0]{{\mathrm{v.amp}}}

\newcommand{\Tot}[0]{{ \operatorname{Tot} }}
\newcommand{\Ram}[0]{ \operatorname{Ram}}
\newcommand{\weight}[0]{{ \operatorname{weight} }}
\newcommand{\Verts}[0]{{ \operatorname{Verts} }}
\newcommand{\Endo}[0]{{ \operatorname{End} }}
\newcommand{\Hom}[0]{{ \operatorname{Hom} }}
%\DeclareMathOperator*{\Hom}{Hom}
\newcommand{\PV}[0]{{ \operatorname{PV} }}
\newcommand{\Totsum}[0]{\Tot^{\oplus}}
\newcommand{\Totprod}[0]{\Tot^{\Pi}}
\newcommand{\xpn}[0]{ { x^{p^n} }}
\newcommand{\Qbar}[0]{{ \bar{ \mathbf{Q} } }}
% mathrms
\newcommand{\ptd}{{\scriptstyle { \ast } }}
\newcommand{\charpoly}[0]{{\mathrm{charpoly}}}
\newcommand{\Sw}[0]{{\mathrm{Sw}}}
\newcommand{\Sieg}[0]{{\mathrm{Sieg}}}
\newcommand{\Inertia}[0]{{\mathrm{In}}}
\newcommand{\generic}[0]{{\mathrm{gen}}}
\newcommand{\Conf}[0]{{\mathrm{Conf}}}
\newcommand{\Sub}[0]{{\mathrm{Sub}}}
\newcommand{\fin}[0]{{\mathrm{fin}}}
\newcommand{\can}[0]{{\mathrm{can}}}
\newcommand{\ess}[0]{{\mathrm{ess}}}
\newcommand{\fd}[0]{{\mathrm{fd}}}
\newcommand{\fg}[0]{{\mathrm{fg}}}
\newcommand{\free}[0]{{\mathrm{free}}}
\newcommand{\lf}[0]{{\mathrm{lf}}}
\newcommand{\holonomy}[0]{{\mathrm{holon}}}
\newcommand{\qproj}[0]{{\mathrm{qproj}}}
\newcommand{\irr}[0]{{\mathrm{irr}}}
\newcommand{\lft}[0]{\mathrm{lft}}
\newcommand{\ft}[0]{\mathrm{ft}}
\newcommand{\Brauer}[0]{\mathrm{Br}}
\newcommand{\smol}[0]{{\mathrm{small}}}
\newcommand{\alev}[0]{{\,\mathrm{a.e.}}}
\newcommand{\semisimple}[0]{{\mathrm{ss}}}
\newcommand{\semisimplification}[0]{{\mathrm{ss}}}
\newcommand{\val}[0]{{\operatorname{val}}}
\newcommand{\gon}[0]{{\dash\mathrm{gon}}}
\newcommand{\semi}[0]{{\mathrm{semi}}}
\newcommand{\inc}[0]{{\mathrm{inc}}}
\newcommand{\Ball}[0]{{B}}
\newcommand{\hq}[0]{{/}}
\newcommand{\unioninfty}[0]{{\union\ts{\infty}}}
\newcommand{\dualnumbers}[0]{{ [\eps] / \eps^2 }}
\newcommand{\crys}[0]{{\mathrm{crys}}}
\newcommand{\Xff}[0]{{X_\mathrm{FF}}}
\newcommand{\an}[0]{{\mathrm{an}}}
\newcommand{\prop}[0]{{\mathrm{prop}}}
\newcommand{\ram}[0]{{\mathrm{ram}}}
\newcommand{\Nis}[0]{{\mathrm{Nis}}}
\newcommand{\perf}[0]{{\mathrm{perf}}}
\newcommand{\exist}[0]{{\exists}}

\newcommand{\quillenplus}[0]{{ {}^{+} }}
\newcommand{\glue}[1]{{ \Disjoint_{#1} }}
\newcommand{\normcomplex}[1]{{\norm{\complex{#1}}}}
\newcommand{\nerve}[1]{{ \mathcal{N}({#1}) }}
\newcommand{\realize}[1]{{ \abs{#1} }}
\newcommand{\opcat}[1]{{ {#1}\op }}
\newcommand{\intcl}[0] { \operatorname{cl}^{\mathrm{int}}}
\newcommand{\algcl}[0] { \operatorname{cl}^{\mathrm{alg}}}
\newcommand{\sepcl}[0] { \operatorname{cl}^{\mathrm{sep}}}

\newcommand{\B}[0]{{\mathbf{B}}}
\newcommand{\E}[0]{{\mathbf{E}}}
\newcommand{\T}[0]{{\mathbf{T}}}
\newcommand{\TX}[0]{{\T X}}
\newcommand{\TM}[0]{{\T M}}
\newcommand{\K}[0]{{\mathsf{K}}}
\newcommand{\G}[0]{{\mathsf{G}}}
%\newcommand{\H}[0]{{\mathsf{H}}}
\newcommand{\D}{{ \mathsf{D} }}
\newcommand{\mH}{{ \mathsf{H} }}
\newcommand{\BGL}[0]{ \mathbf{B}\mkern-3mu \operatorname{GL}}
\newcommand{\proportional}{ \propto }
\newcommand{\asymptotic}{ \ll }
% More complicated math commands
\newcommand{\RM}[1]{%
  \textup{\uppercase\expandafter{\romannumeral#1}}%
}
\DeclareMathOperator{\righttriplearrows} {{\; \tikz{ \foreach \y in {0, 0.1, 0.2} { \draw [-stealth] (0, \y) -- +(0.5, 0);}} \; }}
\DeclareMathOperator*{\mapbackforth}{\rightleftharpoons}
% idk man
\newcommand\caniso[0]{{ \underset{\can}{\iso} }}
\renewcommand{\ae}[0]{{ \text{a.e.} }}
\newcommand\eqae[0]{\underset{\ae}{=}}

\newcommand{\sech}[0]{{ \mathrm{sech} }}
\newcommand{\Cone}[0]{{ \mathrm{Cone} }}
\newcommand{\Cyl}[0]{{ \mathrm{Cyl} }}

%\newcommand{\strike}[1]{{\enclose{\horizontalstrike}{#1}}}
\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil}
\newcommand{\qty}[1]{ \left( #1 \right) }
\newcommand{\one}[0]{\mathbb{1}} 
\newcommand{\dcoset}[3]{{  _{ { #1} \diagdown} {\scriptstyle #2} {}_{\diagup {#3} }  }} 
\newcommand{\dcosetl}[2]{{  _{ { #1} \diagdown} {\scriptstyle #2}  }} 
\newcommand{\dcosetr}[2]{{   {\scriptstyle #1} {}_{\diagup {#2} }  }} 
\newcommand{\colim}[0]{\mathrm{colim}} 
\newcommand{\cocolim}[0]{\mathrm{lim}} 
\newcommand{\prism}[0]{{\Delta_{\mathrm{prism}}} }
\newcommand{\llbracket}[0]{[[}
\newcommand{\rrbracket}[0]{]]}