Drawing this string diagram using tikz

Question

I need to draw the following diagram

using the style of this second diagram

In other words, I need all boxes to be of the same shape (square or rectangular) and all wires to be as thin as those in the second diagram. Labels for the wires are not necessary. What would be the most efficient way to do this in tikz?

My code for the second picture is

\documentclass{book}
\usepackage{tikz}
\usetikzlibrary{decorations.markings,positioning}
\newcommand{\catname}[1]{\mathbf{#1}}
\begin{document}

\begin{tikzpicture}[arr/.style={
            postaction={decorate},          
            decoration={
            markings,
            mark=at position #1 with {\arrow{>}}}}]


\node at (0,0) [rectangle, draw, minimum size= 10 mm] (f) {\emph{John}};
\node at (0,2) [rectangle] (h) {};
\draw[arr=0.6] (0, 0.5) -- (0, 1.5) node[pos=0.3,right]{$\catname{n_m}$};
        \draw (0,1.5) to[out=90,in=90,looseness=2] (1,1.5);
        \draw[arr=0.5] (1,1.5) -- (1,0.5)node[pos=0.6,right]{$\catname{n^r}$};


\node at (1.5,0) [rectangle, draw, minimum size= 10 mm] (f) {\emph{likes}};
\node at (3,2) [rectangle] (h) {};
\draw [->] (1.6,0.5) -- (1.6,2) node[above, left] {$\catname{s_1}$};
\draw[arr=0.6] (2, 1.5) -- (2, 0.5) node[pos=0.6,right]{$\catname{n^l}$};
        \draw (2,1.5) to[out=90,in=90,looseness=2] (3,1.5);
        \draw[arr=0.5] (3,0.5) -- (3,1.5)node[pos=0.4,right]{$\catname{n^f}$};

\node at (3,0) [rectangle, draw, minimum size= 10 mm] (f) {\emph{Mary}};

\end{tikzpicture}
\end{document}

I would like to reproduce the first picture, for which I think I can use a similar technique to that of the second picture. However, what I find most difficult is to draw the the two arcs after the first triangle labelled with $\nu$, which I cannot make one on top of the other. Any hints?

Answer 1

Is there a particular reason why the wires connecting $\nu$ and $\Psi$ zig-zag so much? If you're considering string diagrams up to isotopy, you can considerably simplify the drawing by "straightening" all the wires in this way:

\documentclass[margin=4mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{decorations.markings,positioning}
\newcommand{\catname}[1]{\mathbf{#1}}
\begin{document}

\tikzset{
 arr/.style={ postaction={decorate}
            , decoration={ markings
                         , mark=at position #1 with {\arrow{>}}
                         }
            },
 arr/.default=0.5,
 rarr/.style 2 args={ rounded corners= #1
             , postaction={decorate}
             , decoration={ markings
                         , mark=at position #2 with {\arrow{>}}
                         }
            },
% the `rarr` style adds an argument for the roundedness of the corners
% `rarr` defaults at {4mm}{0.5}, and `arr` at 0.5:
 rarr/.default={4mm}{0.5},
 leaf/.style={ rectangle
             , draw
             %==
             , thick
             , fill=gray!10
             %== improves readability
             , minimum size= 10 mm
             },
}

\begin{tikzpicture}
\node[leaf] (nu) at (0,0) {$\nu$};
% ...is this a \nu or a `v`? Considering that the other is a `w`...
% anyway, first draw all the nodes:
\node[leaf] (not) at (4,0) {\emph{not}};
\node[leaf] (psi) at (6,0) {$\Psi$};
\node[leaf]   (w) at (8,0) {$w$};
% now the wires:
\draw[rarr] (psi.-50)
  %               ↑
  % note that the starting point of the wire is at -45 degress \pm an offset;
  % in this case, 5 degrees. Same for the left wire exiting from \Psi.
  |-      ++(1.5,-.5) 
  --        (w.south);
\draw[rarr] (psi.south) 
  |-         ++(-2,-.5) 
  --        (not.south);
\draw[rarr={2mm}{0.75}] (psi.-130) 
  % here's where the possibility to adjust `rarr` first parameter comes in handy.
  |-                  ++(-.5,-.25) 
  --                   ++(-.05, 3) 
  -|                        (2,-1) 
  --                        (0,-1) 
  --                    (nu.south);
\draw[rarr={1mm}{.75}] (not.north) 
  |-                      ++(-1,1) 
  -|                   ++(-.25,-3);
\end{tikzpicture}
\end{document}