[Tex/LaTex] Natural transformation arrow with TikZ

tikz-pgf

How do you draw natural transformation arrows as mentioned in this question but using TikZ rather than xy? More generally, is it possible to draw edges between edge labels, from an edge label to a node, and from a node to an edge label or is it necessary to explicitly construct a node in order to draw an edge? Does providing an edge label in the path environment implicitly construct a node that can be used as the source and target of another edge? I would like to do this using the matrix and path environments in TikZ if possible.

Example:

\documentclass{article} 
\usepackage{tikz}
\usetikzlibrary{matrix,arrows}

\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
    \matrix (m) [matrix of math nodes, row sep=3em,
    column sep=2.0em, text height=1.5ex, text depth=0.25ex]
    { \mathcal{D} & & \mathcal{J} \\
      \mathcal{C} & & \\ };
    \path[->,font=\scriptsize]
    (m-1-1) edge[loop left] node[auto] {$ F \circ G $} (m-1-1)
    (m-1-1.20) edge node[auto] {$ G $} (m-1-3.160)
    (m-1-3.200) edge node[auto] {$ F $} (m-1-1.340)
    (m-1-1) edge node[left] {$ H $} (m-2-1);    
\end{tikzpicture}

\end{document}

example diagram

How can I connect F◦G and H with a curved natural transformation arrow ⟹? Feel free to rearrange the diagram if you'd like.

Best Answer

As percusse mentions in his comment, the tikz-cd package offers you a convenient set of macros to draw commutative diagrams; here's a little example:

\documentclass{article} 
\usepackage{tikz-cd}

\begin{document}

\begin{tikzcd}[column sep=huge]
\textbf{CRing}
  \arrow[bend left=50]{r}[name=U,label=above:$\scriptstyle\mathrm{GL}_n$]{}
  \arrow[bend right=50]{r}[name=D,label=below:$\scriptstyle U$]{} &
\textbf{Grp}
  \arrow[shorten <=10pt,shorten >=10pt,Rightarrow,to path={(U) -- node[label=right:$\det$] {} (D)}]{}
\end{tikzcd}

\end{document}

enter image description here

Since originally the question asked for a TikZ solution using a matrix of nodes, here's a "pure" TikZ possible solution:

\documentclass{article} 
\usepackage{tikz}
\usetikzlibrary{matrix,arrows}

\begin{document}

\begin{tikzpicture}
\matrix[matrix of nodes,column sep=2cm] (cd)
{
   \textbf{CRing} & \textbf{Grp} \\
};
\draw[->] (cd-1-1) to[bend left=50] node[label=above:$\scriptstyle\mathrm{GL}_n$] (U) {} (cd-1-2);
\draw[->] (cd-1-1) to[bend right=50,name=D] node[label=below:$\scriptstyle U$] (V) {} (cd-1-2);
\draw[double,double equal sign distance,-implies,shorten >=10pt,shorten <=10pt] 
  (U) -- node[label=right:$\det$] {} (V);
\end{tikzpicture}

\end{document}

enter image description here

An answer to the edit to the original question, showing two possibilities (a curved double arrow, and a straight one):

\documentclass{article} 
\usepackage{tikz}
\usetikzlibrary{matrix,arrows}

\begin{document}

\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
    \matrix (m) [matrix of math nodes, row sep=3em,
    column sep=2.0em, text height=1.5ex, text depth=0.25ex]
    { \mathcal{D} & & \mathcal{J} \\
      \mathcal{C} & & \\ };
    \path[->,font=\scriptsize]
    (m-1-1) edge[loop left] node[auto] (fg) {$ F \circ G $} (m-1-1)
    (m-1-1.20) edge node[auto] {$ G $} (m-1-3.160)
    (m-1-3.200) edge node[auto] {$ F $} (m-1-1.340)
    (m-1-1) edge node[left] (h) {$ H $} (m-2-1);
    \draw[double,double equal sign distance,-implies] (fg.290) -- (h.150);    
\end{tikzpicture}

\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
    \matrix (m) [matrix of math nodes, row sep=3em,
    column sep=2.0em, text height=1.5ex, text depth=0.25ex]
    { \mathcal{D} & & \mathcal{J} \\
      \mathcal{C} & & \\ };
    \path[->,font=\scriptsize]
    (m-1-1) edge[loop left] node[auto] (fg) {$ F \circ G $} (m-1-1)
    (m-1-1.20) edge node[auto] {$ G $} (m-1-3.160)
    (m-1-3.200) edge node[auto] {$ F $} (m-1-1.340)
    (m-1-1) edge node[left] (h) {$ H $} (m-2-1);
    \draw[double,double equal sign distance,-implies] (fg.290) to[out=-90,in=180] (h.180);    
\end{tikzpicture}

\end{document}

enter image description here

And here's the corresponding code using tikz-cd:

\documentclass{article} 
\usepackage{tikz-cd}

\begin{document}

\begin{tikzcd}[column sep=huge,row sep=huge]
\mathcal{D}
  \arrow[loop left]{}[name=fg]{F \circ G}
  \rar[start anchor=30, end anchor=151]{G}
  \arrow{d}[name=h,swap]{H} & 
\mathcal{J}\lar[start anchor=196, end anchor=-14]{F} \\
\mathcal{C}
\arrow[shorten >=4pt,Rightarrow,to path={(fg.290) -- (h.175)}]{}
\end{tikzcd}

\begin{tikzcd}[column sep=huge,row sep=huge]
\mathcal{D}
  \arrow[loop left]{}[name=fg]{F \circ G}
  \rar[start anchor=30, end anchor=151]{G}
  \arrow{d}[swap,name=h]{H} & 
\mathcal{J}\lar[start anchor=196, end anchor=-14]{F} \\
\mathcal{C}
\arrow[shorten >=3pt,Rightarrow,to path={(fg.290) to[out=-90,in=180] (h)}]{}
\end{tikzcd}

\end{document}

enter image description here

To add a label to the double arrow (as requested in a comment), you can use an additional node; here's an example using both approaches (the first one using tikz-cd and the second one using "pure" TikZ):

\documentclass{article} 
\usepackage{tikz-cd}
\usepackage{tikz}
\usetikzlibrary{matrix,arrows}

\begin{document}

\begin{tikzcd}[column sep=huge,row sep=huge]
\mathcal{D}
  \arrow[loop left]{}[name=fg]{F \circ G}
  \rar[start anchor=30, end anchor=151]{G}
  \arrow{d}[swap,name=h]{H} & 
\mathcal{J}\lar[start anchor=196, end anchor=-14]{F} \\
\mathcal{C}
\arrow[shorten >=1pt,Rightarrow,to path={(fg.290) to[out=-90,in=180] node[xshift=-3.5mm] {$\tau$} (h)}]{}
\end{tikzcd}

\begin{tikzpicture}[description/.style={fill=white,inner sep=2pt}]
    \matrix (m) [matrix of math nodes, row sep=3em,
    column sep=2.0em, text height=1.5ex, text depth=0.25ex]
    { \mathcal{D} & & \mathcal{J} \\
      \mathcal{C} & & \\ };
    \path[->,font=\scriptsize]
    (m-1-1) edge[loop left] node[auto] (fg) {$ F \circ G $} (m-1-1)
    (m-1-1.20) edge node[auto] {$ G $} (m-1-3.160)
    (m-1-3.200) edge node[auto] {$ F $} (m-1-1.340)
    (m-1-1) edge node[left] (h) {$ H $} (m-2-1);
    \draw[double,double equal sign distance,-implies] (fg.290) to[out=-90,in=180] node[xshift=-3.5mm] {$\tau$} (h.180);    
\end{tikzpicture}

\end{document}

enter image description here

Related Solutions

[Tex/LaTex] Odd behaviour with arrows and multipart rectangles

Well, this one was a little strange but I think I know what is happening. Let me take a brief detour: Consider the following construction

\draw (0,0) -- (1,1) node (a) {A};.

What we expect from this piece of code is to put a node after the main path is created. Notice that the node has no idea of the nature of the path. Even if we use [pos=0.xx] it just looks for the last available path so there is no organic connection between the node placement and the path creation.

It turns out that edge is a to operation added in a similar manner without any relation whatsoever to the main path constructed before that. Another example (zoomed in)

\begin{tikzpicture}
\path[->,
     draw,
     line width=1mm % To make the arrowhead bigger
     ] (0,0);
\end{tikzpicture}

enter image description here

So, an arrowhead with a path of zero length. Same happens with the edge if we dissect one of your paths

\path [line] (leftrow1.two north) % This is the main path as the example above 
     edge[out=90, in=90] node {}(tripletoprow); % This is added afterwards without the 
                                                % line option in place creating the
                                                % illusion that the path is having 
                                                % a disconnected arrowhead

so shorten makes things even worse because it's shortening a zero length path taking the arrowhead even further. Once we get the problem right, then, it's easy to fix the problem via shifting the line option to the edge;

\documentclass[preview,tikz,border=3mm]{standalone}
\usetikzlibrary{shapes, arrows}
\tikzset{line/.style={draw, latex'-},
     seq/.style={rectangle split, rectangle split horizontal, rectangle split parts=#1, draw}
}
\begin{document}
\begin{tikzpicture}
\node [seq=3] (leftrow1) at (0cm, 4cm){};
\node [seq=3] (tripletoprow)  at (4cm, 4cm){};
\path (leftrow1.two north) edge[out=90, in=90,line] (tripletoprow);
\end{tikzpicture}
\end{document}

enter image description here

Also see the manual for the \tikztonodes operation to avoid the extra node{} before the target point.

[Tex/LaTex] How to define the default vertical distance between nodes

I don't really get the question so I hope this is what you wanted. If you include a full document (such that we copy paste and see the problem on our systems) things are much more easier.

Here, you can change the default setting within a scope but your block style had a node distance which was resetting every time it is issued. I've made it 2mm such that we can see the difference easier.

\documentclass[tikz]{standalone}
\usetikzlibrary{arrows,shapes.geometric,positioning}

\begin{document}
\begin{tikzpicture}[decision/.style={diamond, draw, text width=4.5em, text badly centered, node distance=3.5cm, inner sep=0pt},
                    block/.style   ={rectangle, draw, text width=6em, text centered, rounded corners, minimum height=4em, minimum height=2em},
                    cloud/.style   ={draw, ellipse, minimum height=2em},
                    line/.style    ={draw,-latex'},
                    node distance = 1cm, 
                    auto]
    \node [block] (1st) {1st};
    \node [block, right= of 1st]  (2nd1) {2nd1};
\begin{scope}[node distance=2mm and 10mm]%Here we change it for everything inside this scope
    \node [block, above= of 2nd1] (2nd2) {2nd2};
    \node [block, below= of 2nd1] (2nd3) {2nd3};
    \node [block, right= of 2nd1] (3rd1) {3rd1};
    \node [block, above= of 3rd1] (3rd2) {3rd2};
    \node [block, above= of 3rd2] (3rd3) {3rd3};
\end{scope}
    \node [block, below= of 3rd1] (3rd4) {3rd4};
    \node [block, below= of 3rd4] (3rd5) {3rd5};

    \path [line] (1st) -- (2nd1);
    \path [line] (2nd1) -- (2nd2);
    \path [line] (2nd1) -- (2nd3);
    \path [line] (2nd2) -- (3rd3);
    \path [line] (2nd1) -- (3rd1);
    \path [line] (1st) -- (2nd1);
\end{tikzpicture}
\end{document}

enter image description here

Best Answer

Related Solutions

[Tex/LaTex] Odd behaviour with arrows and multipart rectangles

[Tex/LaTex] How to define the default vertical distance between nodes

Related Question