HTML math symbols

See http://www.dionysia.org/html/entities/symbols.html.

HTML test:
← → ↓ ↑ ↔
⇐ ⇒ ⇓ ⇑ ⇔

← → ↓ ↑ ↔ ⇐ ⇒ ⇓ ⇑ ⇔

×

Scaling factors for printing:
TAC: 112%
JPAA: 128%
CTGDC: 124%

$P(E) = {n \choose k} p^k (1-p)^{ n-k}$ $$\int_{-\infty}^{\infty} e^{-x^2}dx.$$

$$\begin{CD} A @>a>> B\\ @VVbV @VVcV\\ C @>d>> D \end{CD}$$

Chrome bug : extra unrequested border in table

The comparison map into the fibre product, i.e., pullback
$\homst \setX \Set \setY \longrightarrow \source(\symsX \source\backslash \homst \setX \Set \setY\source) \times_{(\symsX \source\backslash \homst \setX \Set \setY \target/ \symtY)} \target(\homst \setX \Set \setY \target/ \symtY\target)$
when $\source{(\setX=[4]=\{0,1,2,3\})}$ and $\target{(\setY=\{\elta,\eltb\})}$

$\SetXY \rlap{\target{{} \xrightarrow[\mkern28em]{\displaystyle t} {}}}$							$\homst \setX \Set \setY \target/ \symtY$
	$\elta^4$	$\eltb^4$					$0123\mid{}$		$4+0$
$\smash{\llap{s}\Bigg\downarrow}$			$\elta^3\eltb$	$\eltb^3\elta$			$012\mid3$		$3+1$
			$\elta^2\eltb\elta$	$\eltb^2\elta\eltb$			$013\mid2$
			$\elta\eltb\elta^2$	$\eltb\elta\eltb^2$			$023\mid1$
			$\eltb\elta^3$	$\elta\eltb^3$			$123\mid0$
					$\elta^2\eltb^2,\eltb^2\elta^2$		$01\mid23$		$2+2$
					$\elta\eltb\elta\eltb,\eltb\elta\eltb\elta$		$02\mid13$
					$\elta\eltb\eltb\elta,\eltb\elta\elta\eltb$		$03\mid12$
$\begin{array}{} \symsX \source\backslash \homst \setX \Set \setY\\ \displaystyle \bigg(\mkern-.35em {\target{{\setY=\{\elta,\eltb\}}} \choose \source{|\setX|=4}} \mkern-.35em\bigg)\\ \end{array}$	$\begin{array}{}4+0\\ \elta^4\\\end{array}$	$\begin{array}{}0+4\\ \eltb^4\\\end{array}$	$\begin{array}{}3+1\\ \elta^3\eltb\\\end{array}$	$\begin{array}{}1+3\\ \elta\eltb^3\\\end{array}$	$\begin{array}{}2+2\\ \elta^2\eltb^2\\\end{array}$
	$4+0$		$3+1$		$2+2$				$\symsX \source\backslash \homst \setX \Set \setY \target/ \symtY$

Test 3

The five-quotient diagram and the $3\times3$ lemma

$$\begin{array}{} \boxed{\begin{array}{} \text{2-cells} \\ \ssigma,\functionf,\ttau \\ \symsX \mathrel{\source\times} \SetXY \mathrel{\target\times} \symtY \\ \end{array}} & \mathop\rightrightarrows\limits^{\symsX \times \text{proj}}_{\symsX \target{\times \text{comp}}} & \boxed{\begin{array}{} \ssigma,\functionf \\ \ssigma,\functionf\ttau \\ \symsX \mathrel{\source\times} \SetXY \\ \end{array}} & {}\rlap{\mkern-7em\target{\xrightarrow[\mkern25em]{\displaystyle \symsX \mathrel{\source\times} \functiont}}} &&& \boxed{\begin{array}{} \ssigma, \target[\functionf\symtY\target] \\ \symsX \mathrel{\source\times} \target( \SetXY\target/\symtY \target) \\ \end{array}} \\ \smash{\lower8ex{\llap{\scriptstyle \text{proj}\times\symtY} \downdownarrows \rlap{\scriptstyle \source{\text{comp}}\times\symtY}}} && \llap{\scriptstyle \text{proj}} \downdownarrows \rlap{\scriptstyle \source{\text{comp}}} && && \smash{\lower8ex{\llap{\scriptstyle \text{proj}} \downdownarrows \rlap{\scriptstyle \source{\text{action}}}}} \\ \boxed{\begin{array}{} \functionf,\ttau\mkern.5em;\mkern.5em\ssigma\functionf,\ttau \\ \SetXY \mathrel{\target\times} \symtY \\ \end{array}} & \mathop\rightrightarrows\limits^{\text{proj}}_{\target{\text{comp}}} & \boxed{\begin{array}{} \scriptstyle \text{proj} \\ \scriptstyle (\symsX \times \text{proj})\text{proj} &&&& \scriptstyle (\symsX \times \target{\text{comp}})\text{proj} \\ \scriptstyle (\text{proj} \times \symtY)\text{proj} &&&& \scriptstyle (\text{proj} \times \symtY)\target{\text{comp}} \\ \functionf & {}\rlap{\mkern-2em \target{\xrightarrow[\mkern17em]{\displaystyle \ttau}}} &&& \functionf\ttau \\ &&&& \llap\ssigma \downarrow \\ \smash{\source{\llap\ssigma \Bigg\downarrow}} && \SetXY && {\displaystyle \ssigma\target(\functionf\ttau\target) \atop \source(\symsX \source\times \target{\text{comp}}\source)\source{\text{comp}}} \\ &&&& \Vert \\ \ssigma\functionf & {}\rlap{\mkern-2em \target{\xrightarrow[\displaystyle \ttau]{\mkern4.5em}}} & \source(\ssigma\functionf\source)\ttau & {}\rlap{\smash{\mkern-2em\xlongequal{\mkern5.5em}}} & \ssigma\functionf\ttau \\ \scriptstyle (\symsX \times \text{proj})\source{\text{comp}} && \scriptstyle (\source{\text{comp}} \times \symtY)\target{\text{comp}} && \scriptstyle \text{comp} \\ \scriptstyle (\source{\text{comp}} \times \symtY)\text{proj} \\ \end{array}} & {}\rlap{\target{\xrightarrow[\mkern15em]{\displaystyle \functiont}}} &&& \boxed{\begin{array}{} \target[\functionf\symtY\target], \ssigma\target[\functionf\symtY\target] = \target[\ssigma\functionf\symtY\target] \\\SetXY\target/\symtY \\ \end{array}} \\ &&&&&& \raise11ex{\smash{\Bigg\downarrow}} \\ \smash{\raise10ex{\smash{ \source{\llap\functions\Bigg\downarrow} \rlap{\mathrel{\target\times} \symtY} }}} && \smash{\raise6ex{\smash{ \source{\llap\functions\Bigg\downarrow} }}} && \searrow \rlap\functionr && \boxed{\begin{array}{} \symsX\target[\functionf\symtY\target] \\ \symsX\source\backslash \target( \SetXY \target{{/}} \symtY \target) \\ \end{array}} \\ &&&&&& \wr\Vert \\ \boxed{\begin{array}{} \source[\symsX\functionf\source], \ttau \\ \source( \symsX\source\backslash\SetXY \source) \mathrel{\target\times} \symtY \\ \end{array}} & \mathop\rightrightarrows\limits^{\text{proj}}_{\target{\text{action}}} & \boxed{\begin{array}{}\source[\symsX\functionf\source] \\ \source[\symsX\functionf\source]\ttau = \source[\symsX\functionf\ttau\source] \\ \symsX\source\backslash\SetXY \\ \end{array}} & {}\rlap{\mkern-7em\xrightarrow[\mkern9em]{}} & \boxed{\begin{array}{} \source[\symsX\functionf\source]\symtY \\ \source( \symsX\source\backslash\SetXY \source) \target{{/}} \symtY \\ \end{array}} & \cong & \boxed{\begin{array}{} \source[\symsX\functionf\symtY\source] \\ \symsX \source\backslash \SetXY \target{{/}} \symtY \\ \end{array}} \\ \end{array}$$

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$\begin{array}{} \text{$2$-cells} \\ \ssigma,\functionf,\ttau \\ \symsX \mathrel{\source\times} \SetXY \mathrel{\target\times} \symtY \\ \end{array}$	$\mkern2em\mathop\rightrightarrows\limits^{\symsX \times \text{proj}}_{\symsX \target{\times \text{comp}}}$	$\begin{array}{} \source{\text{v-$1$-cells}} \\ \ssigma,\functionf \\ \ssigma,\functionf\ttau \\ \symsX \mathrel{\source\times} \SetXY \\ \end{array}$	$\mkern-6em\target{\xrightarrow[\mkern20em]{\displaystyle \symsX \mathrel{\source\times} \functiont}}$			$\begin{array}{} \ssigma, \target[\functionf\symtY\target] \\ \symsX \mathrel{\source\times} \target( \SetXY\target/\symtY \target) \\ \end{array}$
$\smash{\lower8ex{\llap{\scriptstyle \text{proj}\times\symtY} \downdownarrows \rlap{\scriptstyle \source{\text{comp}}\times\symtY}}}$		$\llap{\scriptstyle \text{proj}} \downdownarrows \rlap{\scriptstyle \source{\text{comp}}}$				$\smash{\lower8ex{\llap{\scriptstyle \text{proj}} \downdownarrows \rlap{\scriptstyle \source{\text{action}}}}}$
$\begin{array}{} \target{\text{h-$1$-cells}} \\ \functionf,\ttau\mkern.5em;\mkern.5em\ssigma\functionf,\ttau \\ \SetXY \mathrel{\target\times} \symtY \\ \end{array}$	$\mkern-2em\mathop\rightrightarrows\limits^{\text{proj}}_{\target{\text{comp}}}$	$\begin{array}{} && \text{$0$-cells} \\ \scriptstyle \text{proj} \\ \scriptstyle (\symsX \times \text{proj})\text{proj} &&&& \scriptstyle (\symsX \times \target{\text{comp}})\text{proj} \\ \scriptstyle (\text{proj} \times \symtY)\text{proj} &&&& \scriptstyle (\text{proj} \times \symtY)\target{\text{comp}} \\ \functionf & {}\rlap{\mkern-2em \target{\xrightarrow[\mkern17em]{\displaystyle \ttau}}} &&& \functionf\ttau \\ &&&& \llap\ssigma \downarrow \\ \smash{\source{\llap\ssigma \Bigg\downarrow}} && \SetXY && {\displaystyle \ssigma\target(\functionf\ttau\target) \atop \source(\symsX \source\times \target{\text{comp}}\source)\source{\text{comp}}} \\ &&&& \Vert \\ \ssigma\functionf & {}\rlap{\mkern-2em \target{\xrightarrow[\displaystyle \ttau]{\mkern4.5em}}} & \source(\ssigma\functionf\source)\ttau & {}\rlap{\smash{\mkern-2em\xlongequal{\mkern5.5em}}} & \ssigma\functionf\ttau \\ \scriptstyle (\symsX \times \text{proj})\source{\text{comp}} && \scriptstyle (\source{\text{comp}} \times \symtY)\target{\text{comp}} && \scriptstyle \text{comp} \\ \scriptstyle (\source{\text{comp}} \times \symtY)\text{proj} \\ \end{array}$	$\mkern.5em\target{\xrightarrow[\mkern11em]{\displaystyle \functiont}}$			$\begin{array}{} \target[\functionf\symtY\target], \ssigma\target[\functionf\symtY\target] = \target[\ssigma\functionf\symtY\target] \\\SetXY\target/\symtY \\ \end{array}$
						$\smash{\raise8ex{\Bigg\downarrow}}$
$\smash{\raise6ex{\source{\llap\functions\Bigg\downarrow} \rlap{\mathrel{\target\times} \symtY}}}$		$\source{\llap\functions\Bigg\downarrow}$			$\searrow \rlap\functionr$	$\begin{array}{} \symsX\target[\functionf\symtY\target] \\ \symsX\source\backslash \target( \SetXY \target{{/}} \symtY \target) \\ \end{array}$
						$\wr\Vert$
$\begin{array}{} \source[\symsX\functionf\source], \ttau \\ \source( \symsX\source\backslash\SetXY \source) \mathrel{\target\times} \symtY \\ \end{array}$	$\mkern2em\mathop\rightrightarrows\limits^{\text{proj}}_{\target{\text{action}}}$	$\begin{array}{}\source[\symsX\functionf\source] \\ \source[\symsX\functionf\source]\ttau = \source[\symsX\functionf\ttau\source] \\ \symsX\source\backslash\SetXY \\ \end{array}$	$\mkern-6.5em\xrightarrow[\mkern7em]{}$	$\begin{array}{} \source[\symsX\functionf\source]\symtY \\ \source( \symsX\source\backslash\SetXY \source) \target{{/}} \symtY \\ \end{array}$	$\cong$	$\begin{array}{} \source[\symsX\functionf\symtY\source] \\ \symsX \source\backslash \SetXY \target{{/}} \symtY \\ \end{array}$

Test2

The math I want for mathjax to display: $$\begin{array}{} & \rightadj{\xrightarrow{\textstyle \rightcat f \rightadj K \leftadj Q \rightadj K}} \atop \rightadj{\xrightarrow[\phantom{\textstyle \rightcat f \rightadj K \leftadj Q \rightadj K}]{}} & X & \leftadj{\xtwoheadrightarrow{\textstyle \rightcat f \rightadj K \leftadj Q}}\\ \llap{\rightcat f \rightadj K \leftadj\eta} \leftadj{\Bigg\uparrow} \lower1.7ex\hbox{$\rightadj\parallel$} \rightadj{\Bigg\downarrow} \rlap{\rightcat f \rightadj\epsilon \rightadj K} && \Bigg\Vert && \rightadj{\Bigg\downarrow} \rlap{\rightcat f \rightadj\epsilon}\\ & \lower4pt\hbox{$\rightadj{\begin{smallmatrix} \xrightarrow{\Space{6em}{0ex}{0ex}} \\ \xrightarrow[\textstyle \rightcat f\rightadj K]{} \end{smallmatrix}}$} & X & \rightcat{\xrightarrow[\textstyle f]{}} & \\ \end{array}$$

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A simplified fragment of that math, not using the mathjax array construction: $$\xrightarrow{\textstyle f K Q K} \atop \xrightarrow{\textstyle f K Q K}$$

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The same, but wrapped in the array construction: $$\begin{array}{} \xrightarrow{\textstyle f K Q K} \atop \xrightarrow{\textstyle f K Q K} \end{array}$$

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Further experiment: $${a \atop b} \begin{array}{} {a \atop b} \end{array}$$

Test

Misc. macros: $\Newextarrow{\xLongrightarrow}{3,2}{0x21D2} \Newextarrow{\xrightrightarrows}{5,5}{0x21C9}$

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123, 1\mathop2 3, 1\mathop{2}3, 1{\mathop2}3, 1{\mathop{2}}3 $$123, 1\mathop2 3, 1\mathop{2}3, 1{\mathop2}3, 1{\mathop{2}}3$$

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${\bf Span}({\bf Graph})$ $S^{{\mathcal{W}}_\Lambda}\otimes T$
$$\begin{equation}\begin{CD} S^{{\mathcal{W}}_\Lambda}\otimes T @>j>> T\\ @VVV @VV{{\rm End}\, P}V\\ (S\otimes T)/I @= (Z\otimes T)/J \end{CD}\end{equation}$$ $$\begin{equation}\begin{CD} A @>a>> B\\ @VbVV @VVcV\\ C @>d>> D \end{CD}\end{equation}$$
A significant principal $S_n$-bundle ($n \in \bf\text{N}$, $X \in \bf\text{Set}$): $$\begin{equation}\begin{CD} S_{[n]} @>>> \homst {[n]} \Inj X\\ \text{ } @VV\text{image}V\\ \text{} @. \mathcal{P}^{[n]} (X)\\ \end{CD}\end{equation}$$ An important adjunction (in combinatorics and elsewhere; $X \in \bf\text{Set}$): $$\begin{equation}\text{\substack:}\qquad \mathrel{\mathcal{SPart}(X) \substack{\xrightarrow{X/-} \\\xleftarrow[\kernelpart]{}} X\downarrow\Surj} \end{equation}$$

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$$\binom {n} {k} \left\{ \begin{aligned} n\\ k \end{aligned} \right\}$$ \buildrel: $\buildrel \text{action associativity} \over =$ and \stackrel: $\stackrel {\text{action associativity}} =$ $$\text{\Newextarrow:}\qquad \Newextarrow{\xequals}{5,5}{0x003D} \xequals{\text{long message}}$$ $$\text{\Newextarrow:}\qquad \Newextarrow{\xequals}{5,5}{0x21C9} \xequals{\text{long message}}$$ $$\text{\xlongequal:}\qquad \xlongequal{\text{long message}}$$ $\Aut X \quad \homst X\Set Y _{subscript1}text_{subscript2}$ $$\text{\atop:}\qquad X {\longrightarrow \atop \longrightarrow} Y$$ $$\hbox{text 1} \qquad \raise10pt{\hbox{text 2}}$$ $\def\graphsoverx{\leftcat{\Set\mathord\downdownarrows}X} \def\setsunderx{X\rightcat{\mathord\downarrow\Set}}$ $\graphsoverx \quad \setsunderx$

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$$\begin{array}{} \Sub (\N) & \xrightarrow{\textstyle \text{least}} & \bftwo^\op \\ & \llap{{?}\notin{?}'} \searrow & \Bigg\Updownarrow \rlap{\mkern-57mu \text{least}{+}{+}PCON} & \searrow \rlap\neg \\ && [\N,\bftwo] & \xrightarrow[\textstyle PCON]{} & \bftwo \\ \end{array}$$