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# Commuting diagram examples

edited August 2018

These may not be pretty, and odd numbered diagrams require an $$id$$ morphism, but here are some examples:

A commuting "line" (aka a morphism)

X \overset{f}{\rightarrow}Y $$X \overset{f}{\rightarrow}Y$$ A commuting "triangle"

\begin{matrix} X & \overset{f}{\rightarrow } &Y \\ id_x \downarrow & & \downarrow h\\ X &\underset{g}{\rightarrow} &Z \end{matrix} $$\begin{matrix} X & \overset{f}{\rightarrow } &Y \\ id_x \downarrow & & \downarrow h\\ X &\underset{g}{\rightarrow} &Z \end{matrix}$$ A commuting square

\begin{matrix} X & \overset{f}{\rightarrow } &W \\ e \downarrow & & \downarrow h\\ Y &\underset{g}{\rightarrow} &Z \end{matrix} $$\begin{matrix} X & \overset{f}{\rightarrow } &W \\ e \downarrow & & \downarrow h\\ Y &\underset{g}{\rightarrow} &Z \end{matrix}$$ A commuting "pentagon"

\begin{matrix} X & \overset{f}{\rightarrow }& S & \overset{l}{\rightarrow } &W \\ id_x \downarrow & & & & \downarrow h\\ X &\underset{g}{\rightarrow}&T&\underset{k}{\rightarrow} &Z \end{matrix} $$\begin{matrix} X & \overset{f}{\rightarrow }& S & \overset{l}{\rightarrow } &W \\ id_x \downarrow & & & & \downarrow h\\ X &\underset{g}{\rightarrow}&T&\underset{k}{\rightarrow} &Z \end{matrix}$$ A commuting "hexagon"

\begin{matrix} X & \overset{f}{\rightarrow }& S & \overset{l}{\rightarrow } &W \\ e \downarrow & & & & \downarrow h\\ Y &\underset{g}{\rightarrow}&T&\underset{k}{\rightarrow} &Z \end{matrix} $$\begin{matrix} X & \overset{f}{\rightarrow }& S & \overset{l}{\rightarrow } &W \\ e \downarrow & & & & \downarrow h\\ Y &\underset{g}{\rightarrow}&T&\underset{k}{\rightarrow} &Z \end{matrix}$$ A commuting "$$(2n+1)-$$gon"

\begin{matrix} X & \overset{f}{\rightarrow }& S & \overset{l}{\rightarrow }&R&\overset{\cdots}{\rightarrow }&W \\ id_x \downarrow & & & & & & \downarrow h\\ X &\underset{g}{\rightarrow}&T&\underset{k}{\rightarrow}&Q&\underset{\cdots}{\rightarrow} &Z \end{matrix} $$\begin{matrix} X & \overset{f}{\rightarrow }& S & \overset{l}{\rightarrow }&R&\overset{\cdots}{\rightarrow }&W \\ id_x \downarrow & & & & & & \downarrow h\\ X &\underset{g}{\rightarrow}&T&\underset{k}{\rightarrow}&Q&\underset{\cdots}{\rightarrow} &Z \end{matrix}$$ A commuting "$$(2n)-$$gon"

\begin{matrix} X & \overset{f}{\rightarrow }& S & \overset{l}{\rightarrow }&R&\overset{\cdots}{\rightarrow }&W \\ e \downarrow & & & & & & \downarrow h\\ Y &\underset{g}{\rightarrow}&T&\underset{k}{\rightarrow}&Q&\underset{\cdots}{\rightarrow} &Z \end{matrix} $$\begin{matrix} X & \overset{f}{\rightarrow }& S & \overset{l}{\rightarrow }&R&\overset{\cdots}{\rightarrow }&W \\ e \downarrow & & & & & & \downarrow h\\ Y &\underset{g}{\rightarrow}&T&\underset{k}{\rightarrow}&Q&\underset{\cdots}{\rightarrow} &Z \end{matrix}$$

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1.

We could use diagonal arrows for the odd-numbered diagrams:

• \searrow produces an arrow pointing to south east $$\searrow$$
• \swarrow produces an arrow pointing to south west $$\swarrow$$
• \nearrow produces an arrow pointing to north east $$\nearrow$$
• \nwarrow produces an arrow pointing to north west $$\nwarrow$$

The commuting triangle example would look like this:

$$\begin{matrix} X & \xrightarrow{f} & Y \\ & \searrow & \downarrow \\ & & Z \end{matrix}$$ Unfortunately, I don't know of an easy and nice way of labelling the diagonal arrows; maybe we can draw some inspiration from this Maths Stackexchange thread.

Comment Source:We could use diagonal arrows for the odd-numbered diagrams: - \searrow produces an arrow pointing to south east \$$\searrow\$$ - \swarrow produces an arrow pointing to south west \$$\swarrow\$$ - \nearrow produces an arrow pointing to north east \$$\nearrow\$$ - \nwarrow produces an arrow pointing to north west \$$\nwarrow\$$ The commuting triangle example would look like this: $\begin{matrix} X & \xrightarrow{f} & Y \\\\ & \searrow & \downarrow \\\\ & & Z \end{matrix}$ Unfortunately, I don't know of an easy and nice way of labelling the diagonal arrows; maybe we can draw some inspiration from [this Maths Stackexchange thread](https://math.meta.stackexchange.com/questions/2324/how-to-draw-a-commutative-diagram).
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2.
edited May 2018

Here's a somewhat more complicated way, but the results look pretty good.

You will need to install:

First, there's a website for making commutative diagrams with TikZ: https://tikzcd.yichuanshen.de/

After you are done making your diagram, click the link icon, which will copy the $$\LaTeX$$ for the diagram to the clipboard.

Then, insert the code that was copied into this template:

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{cd}
\begin{document}

% Paste your copied diagram here!

\end{document}


Save the file to commutative_diagram.tex. Then type at the command line:

# pdflatex commutative_diagram.tex


Next convert the resulting pdf into an svg:

# pdf2svg commutative_diagram.pdf commutative_diagram.svg


The result has some obnoxious white blocks, so we'll want to filter those out. Type:

# sed -i.bak -e s/fill:rgb$$100\%,100\%,100\%$$\;fill-opacity:1/fill-opacity:0/ \
commutative_diagram.svg


Finally, the result is a little small. To resize the svg, type:

rsvg-convert -a -w 500 -f svg commutative_diagram.svg -o enlarged_diagram.svg


To make a smaller diagram use a number less than 500.

Finally, you can share SVG image files with svgur.com. Here is the result of this little tutorial:

Comment Source:Here's a somewhat more complicated way, but the results look pretty good. You will need to install: - [pdflatex](https://www.tug.org/texlive/) - [pdf2svg](https://github.com/dawbarton/pdf2svg) - [rsvg-convert](https://github.com/GNOME/librsvg) - [sed](https://www.gnu.org/software/sed/manual/sed.html) First, there's a website for making commutative diagrams with TikZ: https://tikzcd.yichuanshen.de/ ![Screen shot](https://i.imgur.com/OQCyb1D.png) After you are done making your diagram, click the link icon, which will copy the \$$\LaTeX\$$ for the diagram to the clipboard. Then, insert the code that was copied into this template: <pre> \documentclass{standalone} \usepackage{tikz} \usetikzlibrary{cd} \begin{document} % Paste your copied diagram here! \end{document} </pre> Save the file to commutative_diagram.tex. Then type at the command line: <pre> # pdflatex commutative_diagram.tex </pre> Next convert the resulting pdf into an svg: <pre> # pdf2svg commutative_diagram.pdf commutative_diagram.svg </pre> The result has some obnoxious white blocks, so we'll want to filter those out. Type: <pre> # sed -i.bak -e s/fill:rgb$$100\%,100\%,100\%$$\;fill-opacity:1/fill-opacity:0/ \ commutative_diagram.svg </pre> Finally, the result is a little small. To resize the svg, type: <pre> rsvg-convert -a -w 500 -f svg commutative_diagram.svg -o enlarged_diagram.svg </pre> To make a smaller diagram use a number less than 500. Finally, you can share SVG image files with [svgur.com](http://svgur.com/). Here is the result of this little tutorial: ![Product Diagram](https://svgshare.com/i/6qX.svg)
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3.


Examples:

Morphism:

\begin{CD} A @>f>> B \end{CD}\begin{CD} A @>f>> B \end{CD} "Triangle":

\begin{CD} X @>g>> Y \\\ @| {}@VfVV \\\ X @>{f\circ g}>> Z \end{CD}\begin{CD} X @>g>> Y \\ @| {}@VfVV \\ X @>{f\circ g}>> Z \end{CD} Square:

\begin{CD} X @>g>> W \\\@VhVV {}@VVfV \\\ Y @>>k> Z \end{CD}\begin{CD} X @>g>> W \\@VhVV {}@VVfV \\ Y @>>k> Z \end{CD} Stretching:

\begin{CD} X @>g>> W \\\@VhVV {}@VVfV \\\ Y @>\text{This is a long label.}>> Z \end{CD}\begin{CD} X @>g>> W \\@VhVV {}@VVfV \\ Y @>\text{This is a long label.}>> Z \end{CD} $$\endgroup$$
