diff --git a/algebraic_construction.tex b/algebraic_construction.tex index f9e7671..f505d25 100644 --- a/algebraic_construction.tex +++ b/algebraic_construction.tex @@ -3,7 +3,7 @@ \input{preamble} %\tikzexternalize % activate! - +\usepackage[page,toc,titletoc,title]{appendix} \usepackage{chemfig} \usepackage{relsize} \usepackage{multirow} @@ -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} diff --git a/appendices.tex b/appendices.tex new file mode 100644 index 0000000..36e070d --- /dev/null +++ b/appendices.tex @@ -0,0 +1,122 @@ + +\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} \ No newline at end of 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