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separate appendices
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@hooyuser
hooyuser committed Jan 11, 2024
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122 changes: 2 additions & 120 deletions algebraic_construction.tex
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\input{preamble}

%\tikzexternalize % activate!

\usepackage[page,toc,titletoc,title]{appendix}
\usepackage{chemfig}
\usepackage{relsize}
\usepackage{multirow}
Expand Down Expand Up @@ -127,126 +127,8 @@ \chapter*{Notation Conventions}
\include{valuation_theory}
\include{number_theory}

\include{appendices}

\chapter*{Appendix}
\begin{table}[h]
\centering
\renewcommand{\arraystretch}{1.6}
\begin{tabular}{l|m{5.3cm}|m{5.3cm}}
\toprule
\textbf{Category} & \multicolumn{1}{c|}{\textbf{Initial Object}} & \multicolumn{1}{c}{\textbf{Terminal Object}} \\
\midrule
\midrule
$\mathsf{Set}$ & \multicolumn{1}{c|}{$\varnothing$} & \multicolumn{1}{c}{$\{*\}$} \\ \midrule
$\mathsf{Top}$ & \multicolumn{1}{c|}{$\varnothing$} & \multicolumn{1}{c}{$\{*\}$} \\ \midrule
$\mathsf{Grp}$ & \multicolumn{2}{c}{$\{0\}$} \\ \midrule
$R\text{-}\mathsf{Mod}$ & \multicolumn{2}{c}{$\{0\}$} \\ \midrule
$\mathsf{Ring}$ & \multicolumn{1}{c|}{$\mathbb{Z}$} & \multicolumn{1}{c}{$\{0\}$} \\ \midrule
$\mathsf{Sch}$ & \multicolumn{1}{c|}{$\spec\mathbb{Z}$} & \multicolumn{1}{c}{$\spec \{0\}=\varnothing$ } \\ \midrule
$\mathsf{Field}_p$ & \multicolumn{1}{c|}{\begin{minipage}{.3\linewidth}$$
\begin{array}{ll}
\mathbb{Q} & \text{if } p=0 \\
\mathbb{F}_p & \text{if } p>0
\end{array}
$$\end{minipage}} & \\ \bottomrule
\end{tabular}
\end{table}
\newpage

\begin{center}
\Large
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|P{4.4cm}|m{5.3cm}|m{5.3cm}|}
\hline
{\bf Diagram} &\multicolumn{1}{c|}{\bf Limit }& \multicolumn{1}{c|}{\bf Colimit } \\
\hline\hline
Empty Category $\varnothing$ & Terminal Object
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
\circ \arrow[r, dash pattern=on 4pt off 2pt, draw=arrowRed] & \varprojlim\\[-2em]
\end{tikzcd}
\end{center}
& Initial Object
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
\varinjlim\arrow[r, dash pattern=on 4pt off 2pt, draw=arrowRed]&\circ \\[-2em]
\end{tikzcd}
\end{center}
\\
\hline
Discrete Category $\bullet\;\bullet\;\bullet\;\bullet\;\cdots$& Product & Coproduct \\
\hline
$\bullet\qquad\bullet$ & Finite Product
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
& \circ \arrow[d, dash pattern=on 4pt off 2pt, draw=arrowRed] \arrow[ld, draw=cyan] \arrow[rd, draw=cyan] & \\
\bullet & \varprojlim \arrow[l, draw=arrowBlue] \arrow[r, draw=arrowBlue] & \bullet
\end{tikzcd}
\end{center} & Finite Coproduct
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
& \circ & \\
\bullet \arrow[r, draw=arrowBlue] \arrow[ru, draw=cyan] & \varinjlim \arrow[u, dash pattern=on 4pt off 2pt, draw=arrowRed] & \bullet\arrow[l, draw=arrowBlue]\arrow[lu, draw=cyan]
\end{tikzcd}
\end{center}\\
\hline
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
\bullet \arrow[r] &[-0.8em] \bullet &[-0.8em] \bullet \arrow[l]
\end{tikzcd}
& Pullback
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
\circ \arrow[rd, dash pattern=on 4pt off 2pt, draw=arrowRed] \arrow[rrd, draw=cyan, bend left] \arrow[rdd, draw=cyan, bend right] &[-2em] & [-0.4em] \\[-0.3cm]
& \varprojlim \arrow[d, draw=arrowBlue] \arrow[r, draw=arrowBlue] & \bullet \arrow[d] \\
& \bullet \arrow[r] & \bullet
\end{tikzcd}
\end{center}
& Pushout
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
\circ &[-2em] & [-0.4em] \\[-0.3cm]
& \varinjlim\arrow[lu, dash pattern=on 4pt off 2pt, draw=arrowRed] & \bullet \arrow[llu, draw=cyan, bend right] \arrow[l, draw=arrowBlue]\\
& \bullet \arrow[u, draw=arrowBlue] \arrow[luu, draw=cyan, bend left] & \bullet \arrow[l] \arrow[u]
\end{tikzcd}
\end{center} \\
\hline
$\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
\bullet \arrow[r, shift left] \arrow[r, shift right] & \bullet
\end{tikzcd}$ & Equalizer \begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
\circ \arrow[rd, draw=cyan] \arrow[d, dash pattern=on 4pt off 2pt, draw=arrowRed] & & \\
\varprojlim \arrow[r, draw=arrowBlue] & \bullet \arrow[r, shift left] \arrow[r, shift right] & \bullet
\end{tikzcd}
\end{center} & Coequalizer \begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
& & \circ \\
\bullet \arrow[r, shift left] \arrow[r, shift right] & \bullet \arrow[r, draw=arrowBlue] \arrow[ru, draw=cyan] & \varinjlim \arrow[u, dash pattern=on 4pt off 2pt, draw=arrowRed]
\end{tikzcd}
\end{center}\\
\hline
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt, shorten <=-3.5pt, shorten >=-3.5pt}]
\bullet &[-1.5em] \bullet \arrow[l] &[-1.5em] \bullet \arrow[l] &[-1.5em] \cdots \arrow[l]
\end{tikzcd}

& Inverse Limit
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
& [-1.5em] &[-1em] \circ \arrow[rdd, draw=cyan] \arrow[ldd, draw=cyan] \arrow[d, dash pattern=on 4pt off 2pt, draw=arrowRed] & [-1em] \\
& & \varprojlim \arrow[ld, draw=arrowBlue, shorten <=-4pt] \arrow[rd, draw=arrowBlue, shorten <=-3pt] & \\[-0.3cm]
\bullet & \bullet \arrow[l, shorten <=-3pt, shorten >=-3pt] & & \cdots \arrow[ll]
\end{tikzcd}
\end{center}
& Direct Limit
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
& [-1.5em] &[-1em] \circ & [-1em] \\
& & \varinjlim \arrow[u, dash pattern=on 4pt off 2pt, draw=arrowRed] & \\[-0.3cm]
\bullet \arrow[r, shorten <=-3pt, shorten >=-3pt]& \bullet \arrow[ru, draw=arrowBlue, shorten >=-4.7pt] \arrow[rr] \arrow[ruu, draw=cyan] & & \cdots \arrow[lu, draw=arrowBlue, shorten >=-2.5pt] \arrow[luu, draw=cyan]
\end{tikzcd}
\end{center}\\
\hline
\end{tabular}
\end{center}
% \begin{Theorem}{Title}{label}
% This is the statement of the theorem.
% \end{Theorem}
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122 changes: 122 additions & 0 deletions appendices.tex
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\begin{appendices}
\chapter{Category Theory Facts}
\begin{table}[h]
\centering
\renewcommand{\arraystretch}{1.6}
\begin{tabular}{l|m{5.3cm}|m{5.3cm}}
\toprule
\textbf{Category} & \multicolumn{1}{c|}{\textbf{Initial Object}} & \multicolumn{1}{c}{\textbf{Terminal Object}} \\
\midrule
\midrule
$\mathsf{Set}$ & \multicolumn{1}{c|}{$\varnothing$} & \multicolumn{1}{c}{$\{*\}$} \\ \midrule
$\mathsf{Top}$ & \multicolumn{1}{c|}{$\varnothing$} & \multicolumn{1}{c}{$\{*\}$} \\ \midrule
$\mathsf{Grp}$ & \multicolumn{2}{c}{$\{0\}$} \\ \midrule
$R\text{-}\mathsf{Mod}$ & \multicolumn{2}{c}{$\{0\}$} \\ \midrule
$\mathsf{Ring}$ & \multicolumn{1}{c|}{$\mathbb{Z}$} & \multicolumn{1}{c}{$\{0\}$} \\ \midrule
$\mathsf{Sch}$ & \multicolumn{1}{c|}{$\spec\mathbb{Z}$} & \multicolumn{1}{c}{$\spec \{0\}=\varnothing$ } \\ \midrule
$\mathsf{Field}_p$ & \multicolumn{1}{c|}{\begin{minipage}{.3\linewidth}$$
\begin{array}{ll}
\mathbb{Q} & \text{if } p=0 \\
\mathbb{F}_p & \text{if } p>0
\end{array}
$$\end{minipage}} & \\ \bottomrule
\end{tabular}
\end{table}
\newpage

\begin{center}
\Large
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|P{4.4cm}|m{5.3cm}|m{5.3cm}|}
\hline
{\bf Diagram} &\multicolumn{1}{c|}{\bf Limit }& \multicolumn{1}{c|}{\bf Colimit } \\
\hline\hline
Empty Category $\varnothing$ & Terminal Object
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
\circ \arrow[r, dash pattern=on 4pt off 2pt, draw=arrowRed] & \varprojlim\\[-2em]
\end{tikzcd}
\end{center}
& Initial Object
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
\varinjlim\arrow[r, dash pattern=on 4pt off 2pt, draw=arrowRed]&\circ \\[-2em]
\end{tikzcd}
\end{center}
\\
\hline
Discrete Category $\bullet\;\bullet\;\bullet\;\bullet\;\cdots$& Product & Coproduct \\
\hline
$\bullet\qquad\bullet$ & Finite Product
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
& \circ \arrow[d, dash pattern=on 4pt off 2pt, draw=arrowRed] \arrow[ld, draw=cyan] \arrow[rd, draw=cyan] & \\
\bullet & \varprojlim \arrow[l, draw=arrowBlue] \arrow[r, draw=arrowBlue] & \bullet
\end{tikzcd}
\end{center} & Finite Coproduct
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
& \circ & \\
\bullet \arrow[r, draw=arrowBlue] \arrow[ru, draw=cyan] & \varinjlim \arrow[u, dash pattern=on 4pt off 2pt, draw=arrowRed] & \bullet\arrow[l, draw=arrowBlue]\arrow[lu, draw=cyan]
\end{tikzcd}
\end{center}\\
\hline
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
\bullet \arrow[r] &[-0.8em] \bullet &[-0.8em] \bullet \arrow[l]
\end{tikzcd}
& Pullback
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
\circ \arrow[rd, dash pattern=on 4pt off 2pt, draw=arrowRed] \arrow[rrd, draw=cyan, bend left] \arrow[rdd, draw=cyan, bend right] &[-2em] & [-0.4em] \\[-0.3cm]
& \varprojlim \arrow[d, draw=arrowBlue] \arrow[r, draw=arrowBlue] & \bullet \arrow[d] \\
& \bullet \arrow[r] & \bullet
\end{tikzcd}
\end{center}
& Pushout
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
\circ &[-2em] & [-0.4em] \\[-0.3cm]
& \varinjlim\arrow[lu, dash pattern=on 4pt off 2pt, draw=arrowRed] & \bullet \arrow[llu, draw=cyan, bend right] \arrow[l, draw=arrowBlue]\\
& \bullet \arrow[u, draw=arrowBlue] \arrow[luu, draw=cyan, bend left] & \bullet \arrow[l] \arrow[u]
\end{tikzcd}
\end{center} \\
\hline
$\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
\bullet \arrow[r, shift left] \arrow[r, shift right] & \bullet
\end{tikzcd}$ & Equalizer \begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
\circ \arrow[rd, draw=cyan] \arrow[d, dash pattern=on 4pt off 2pt, draw=arrowRed] & & \\
\varprojlim \arrow[r, draw=arrowBlue] & \bullet \arrow[r, shift left] \arrow[r, shift right] & \bullet
\end{tikzcd}
\end{center} & Coequalizer \begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
& & \circ \\
\bullet \arrow[r, shift left] \arrow[r, shift right] & \bullet \arrow[r, draw=arrowBlue] \arrow[ru, draw=cyan] & \varinjlim \arrow[u, dash pattern=on 4pt off 2pt, draw=arrowRed]
\end{tikzcd}
\end{center}\\
\hline
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt, shorten <=-3.5pt, shorten >=-3.5pt}]
\bullet &[-1.5em] \bullet \arrow[l] &[-1.5em] \bullet \arrow[l] &[-1.5em] \cdots \arrow[l]
\end{tikzcd}

& Inverse Limit
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
& [-1.5em] &[-1em] \circ \arrow[rdd, draw=cyan] \arrow[ldd, draw=cyan] \arrow[d, dash pattern=on 4pt off 2pt, draw=arrowRed] & [-1em] \\
& & \varprojlim \arrow[ld, draw=arrowBlue, shorten <=-4pt] \arrow[rd, draw=arrowBlue, shorten <=-3pt] & \\[-0.3cm]
\bullet & \bullet \arrow[l, shorten <=-3pt, shorten >=-3pt] & & \cdots \arrow[ll]
\end{tikzcd}
\end{center}
& Direct Limit
\begin{center}
\begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}]
& [-1.5em] &[-1em] \circ & [-1em] \\
& & \varinjlim \arrow[u, dash pattern=on 4pt off 2pt, draw=arrowRed] & \\[-0.3cm]
\bullet \arrow[r, shorten <=-3pt, shorten >=-3pt]& \bullet \arrow[ru, draw=arrowBlue, shorten >=-4.7pt] \arrow[rr] \arrow[ruu, draw=cyan] & & \cdots \arrow[lu, draw=arrowBlue, shorten >=-2.5pt] \arrow[luu, draw=cyan]
\end{tikzcd}
\end{center}\\
\hline
\end{tabular}
\end{center}
\end{appendices}

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