[Tex/LaTex] String diagrams in monoidal categories

diagramsstrings

I'm writing a paper which contains a chapter on a certain kind of monoidal categories (spherical, that is) and I would like to include the usual graphical calculus for morphisms. In particular, I'm looking for some way to draw diagrams like:

(this is taken from this paper which features many nice examples).
Now, I've been searching all day and just couldn't find a way to do it. The only thing I found was the braid package but I don't think it lets you do the caps and cups on the left and right side of the X arrow.
I'm starting to get really desperate, if anyone has an idea of how to draw them, I'd be glad about any suggestion!

Best Answer

Here's one option using TikZ:

The code:

\documentclass{article} 
\usepackage{tikz}
\usetikzlibrary{decorations.markings,arrows,arrows.meta}

\tikzset{
  myblock/.style={
    draw,text width=0.75cm,minimum height=2cm,align=center,
    append after command={node[fill,anchor=north west,minimum size=0.35cm] at (\tikzlastnode.north west) {}}
  },
  twoarrows/.style n args={4}{
    decoration={
      markings,
      mark=at position #1 with {\arrow{>}\node[above] {#3};}, 
      mark=at position #2 with {\arrow{>}\node[above] {#4};} 
    },
  postaction=decorate  
  },
  onearrow/.style 2 args={
    decoration={
      markings,
      mark=at position #1 with {\arrow{>}\node[above] {#2};}, 
    },
  postaction=decorate  
  },
}

\begin{document}

\begin{tikzpicture}
\node[myblock]
  (G) 
  {$g$};
\draw[rounded corners,twoarrows={0.07}{0.95}{$X$}{$X$}]
  ([shift={(0pt,10pt)}]G.south east) -|
  ([shift={(25pt,-20pt)}]G.south east) --
  ([shift={(-25pt,-20pt)}]G.south west) |-
  ([shift={(0pt,10pt)}]G.south west);
\draw[onearrow={0.5}{$A$}]
  ([shift={(-40pt,-15pt)}]G.north west) -- ([shift={(0pt,-15pt)}]G.north west);
\draw[onearrow={0.5}{$B$}]
  ([shift={(0pt,-15pt)}]G.north east) -- ([shift={(40pt,-15pt)}]G.north east);
\end{tikzpicture}

\end{document}

And another example, reproducing the table at the top of page 6 in the linked document:

\documentclass{article} 
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{decorations.markings,positioning}
\usepackage{array}

\DeclareMathOperator{\iden}{id}

\tikzset{
  myblock/.style={
    draw,text width=0.75cm,minimum height=2cm,align=center,
    append after command={node[fill,anchor=north west,minimum size=0.35cm] at (\tikzlastnode.north west) {}}
  },
  mysquare/.style={
    draw,minimum size=1cm,align=center,
    append after command={node[fill,anchor=north west,minimum size=0.35cm] at (\tikzlastnode.north west) {}}
  },
  twoarrows/.style n args={4}{
    decoration={
      markings,
      mark=at position #1 with {\arrow{>}\node[above] {#3};}, 
      mark=at position #2 with {\arrow{>}\node[above] {#4};} 
    },
  postaction=decorate  
  },
  onearrow/.style 2 args={
    decoration={
      markings,
      mark=at position #1 with {\arrow{>}\node[above] {#2};}, 
    },
  postaction=decorate  
  },
}

\begin{document}

\begin{table}

\renewcommand\arraystretch{3}
\begin{tabular}{l>{$}l<{$}c}
Object & A & 
  \begin{tikzpicture}[baseline]
  \draw[onearrow={0.5}{$A$}] (0,0) -- ++(40pt,0pt);
  \end{tikzpicture}
\\
Morphism & f\colon A\to B &
  \begin{tikzpicture}[baseline]
  \node[mysquare] (F) {$f$};
  \draw[onearrow={0.5}{$A$}]
    ([xshift=-40pt]F.west) -- (F.west);
  \draw[onearrow={0.5}{$A$}]
    (F.east) -- ([xshift=40pt]F.east);
  \end{tikzpicture}
\\
Identity &  \iden_{A}\colon A\to A &
  \begin{tikzpicture}[baseline]
  \draw[onearrow={0.5}{$A$}] (0,0) -- ++(40pt,0pt);
  \end{tikzpicture}
\\
Composition & t\circ s &
  \begin{tikzpicture}[baseline]
  \node[mysquare] (T) {$t$};
  \node[mysquare,right=40 pt of T] (S) {$s$};
  \draw[onearrow={0.5}{$A$}]
    ([xshift=-40pt]T.west) -- (T.west);
  \draw[onearrow={0.5}{$B$}]
    (T.east) -- (S.west);
  \draw[onearrow={0.5}{$C$}]
    (S.east) -- ([xshift=40pt]S.east);
  \end{tikzpicture}
\end{tabular}
\end{table}

\end{document}

The Code

\documentclass{article}
\usepackage{tikz}

\begin{document}

\tikzset{
    triple/.style args={[#1] in [#2] in [#3]}{#1,preaction={preaction={draw,#3},draw,#2}},
    McCurdy/.style={triple={[line width=0.5pt,black] in [line width=2mm,red!30] in [line width=2mm+1pt,black]}},
}

\newcommand{\trapezium}[1]% shift as x,y (lower left corner)
{ \draw[fill=white,shift={(#1)}](0,0) -- (0.3,0) -- (0.3,0.8) -- (0,0.95) -- cycle;
}        

\begin{tikzpicture}
  \draw[McCurdy,rounded corners=1mm] (0,0) coordinate (start1) -- (5,0) -- (5,1) -- (2,1) -- (2,2) -- (2.5,2) -- (2.7,2.2) coordinate (end1);
  \draw[McCurdy,rounded corners=1mm] (2.9,2.4) coordinate (start2) -- (3.5,3) -- (7,3) coordinate (end2);
  \node[left] at (start1) {x};
  \trapezium{4,0.8}
  \trapezium{6,2.8}
  \pgfmathsetmacro{\xydim}{sqrt(2)/2*(1mm+0.5pt)/28.4528}
  \draw (end1) ++ (-\xydim,\xydim) -- ++ (2*\xydim,-2*\xydim);
  \draw (start2) ++ (-\xydim,\xydim) -- ++ (2*\xydim,-2*\xydim);
  \draw[rounded corners=1mm] (2.8,2.3) -- (2,3.1) -- (0.8,3.1)
    (2.8,2.3) -- (3.6,1.5) -- (4,1.5);
\end{tikzpicture}
\end{document}

The Output

enter image description here

Edit 1: A concept for drawing "holes", but only up-down on parallel left to right lines. It draws over the existing lines:

enter image description here

The Code

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

\pgfmathsetmacro{\rlw}{0.2}
\pgfmathsetmacro{\blw}{0.02}

\pgfmathsetmacro{\hbw}{\blw/2}
\pgfmathsetmacro{\qbw}{\blw/4}
\pgfmathsetmacro{\hrw}{\rlw/2}

\tikzset{
    triple/.style args={[#1] in [#2] in [#3]}{#1,preaction={preaction={draw,#3},draw,#2}},
    McCurdy/.style={triple={[line width=\blw cm,black] in [line width=\rlw cm,red!30] in [line width=2*\blw cm+\rlw cm,black]}},
}

\newcommand{\trapezium}[1]% shift as x,y (lower left corner)
{ \draw[fill=white,shift={(#1)}](0,0) -- (0.3,0) -- (0.3,0.8) -- (0,0.95) -- cycle;
} 

\def\connector(#1,#2,#3)% midposition one, midposition two, radius
{   \path (#1);
    \pgfgetlastxy{\xtl}{\ytl}
    \path (#2);
    \pgfgetlastxy{\xbr}{\ybr}
    \pgfmathsetmacro{\xmin}{min(\xtl,\xbr)/28.453}
    \pgfmathsetmacro{\xmax}{max(\xtl,\xbr)/28.453}
    \pgfmathsetmacro{\ymin}{min(\ytl,\ybr)/28.453}
    \pgfmathsetmacro{\ymax}{max(\ytl,\ybr)/28.453}
    \fill[red!30] ($(\xmin,\ymax)+(-#3,-\hrw)$)
        arc (90:0:#3) -- 
        ($(\xmin,\ymin)+(0,#3+\hrw)$)
        arc (360:270:#3) --
        ($(\xmax,\ymin)+(#3,\hrw)$)
        arc (270:180:#3) --
        ($(\xmax,\ymax)+(0,-#3-\hrw)$)
        arc (180:90:#3) --
        cycle ;
    \draw[line width=\blw cm] ($(\xmin,\ymax)+(-#3,-\hrw)+(0,-\hbw)$)
        arc (90:0:#3) -- 
        ($(\xmin,\ymin)+(0,#3+\hrw+\hbw)$)
        arc (360:270:#3);
    \draw[line width=\blw cm] ($(\xmax,\ymax)+(#3,-\hrw)+(0,-\hbw)$)
        arc (90:180:#3) -- 
        ($(\xmax,\ymin)+(0,#3+\hrw+\hbw)$)
        arc (180:270:#3);   
}       

\begin{tikzpicture}
  \draw[McCurdy,rounded corners=1mm] (0,0) -- (1,0) coordinate (a) -- (3,0) coordinate (b) -- (4,0);
  \draw[McCurdy,rounded corners=1mm] (0,-1) -- (1.2,-1) coordinate (c) -- (2.7,-1) coordinate (d) -- (4,-1);
  \draw[McCurdy,rounded corners=1mm] (0,-2) -- (1.6,-2) coordinate (e) -- (2.0,-2) coordinate (f) -- (4,-2);
    \connector(a,c,0.15)
  \connector(b,d,0.2)
  \connector(c,e,0.05)
  \connector(d,f,0.1)
 \end{tikzpicture} 

\end{document}

The Output

enter image description here

[Tex/LaTex] Typesetting inline Penrose/Cvitanovic tensor diagrams (birdtracks)

I figured out enough about tikz to produce the following:

tensors written in birdtracks notation with tikz

My code is below. Comments would be appreciated, since I don't really know what I'm doing with tikz. Is there a good way to avoid hardcoding the distances between nodes and the size of the circle?

\documentclass{article}
\RequirePackage{tikz}
\begin{document}

\pgfdeclarelayer{nodelayer}
\pgfdeclarelayer{edgelayer}
\pgfsetlayers{edgelayer,nodelayer,main}
\tikzstyle{plain}=[rectangle,fill=none,draw=black,scale=1.0,inner sep=1.8pt] % circle also works
\tikzstyle{arrow}=[draw=black,arrows=-latex]

The notation
\begin{tikzpicture}[baseline=(n0.base),thick]
        \begin{pgfonlayer}{nodelayer}
                \node [style=plain] (n0) at (0, 0) {$u$};
                \node [style=plain] (n1) at +(2.3em, 0) {$v$};
        \end{pgfonlayer}
        \begin{pgfonlayer}{edgelayer}
                \draw [style=arrow] (n0) to (n1);
        \end{pgfonlayer}
\end{tikzpicture}
means the inner product $u_a v^a$.

The following displayed picture represents the tensor expression $u^a_{abc}\nabla_d v^{bc}$:

\begin{tikzpicture}[thick]
        \begin{pgfonlayer}{nodelayer}
                \node [style=plain] (n0) at (0, 0) {$u$};
                \node [style=plain] (n1) at (3.1em, 0) {$v$};
                \node [style=circle,draw=black,minimum size=2.1\baselineskip] (c1) at (n1) {};
        \end{pgfonlayer}
        \begin{pgfonlayer}{edgelayer}
                \draw [style=arrow,out=10,in=170] (n0) to (n1);
                \draw [style=arrow,out=-10,in=190] (n0) to (n1);
                \draw [style=arrow,out=30,in=150, distance=1.8\baselineskip, loop] (n0) to ();
                \draw [style=arrow] (c1.east) to +(1.4em,0);
        \end{pgfonlayer}
\end{tikzpicture}

\end{document}

Best Answer

Related Solutions

[Tex/LaTex] String diagrams for monoidal functors a la McCurdy

The Code

The Output

The Code

The Output

[Tex/LaTex] Typesetting inline Penrose/Cvitanovic tensor diagrams (birdtracks)

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