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String Diagram Composition
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hooyuser committed Jan 12, 2024
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Expand Up @@ -340,6 +340,303 @@ \section{Category}
}

\section{String Diagram}
String diagrams are a convenient way to represent the composition of natural transformations. From top to bottom, a string diagram represents a series of vertical compositions of natural transformations.

Suppose $\varphi:F\Rightarrow G$ is a natural transformation between functors $F,G:\mathsf{C}\to \mathsf{D}$. The string diagram for $\varphi$ is just the Poincare dual of its 2-cell diagram.

\begin{center}
\begin{tikzpicture}[x=0.6cm,y=0.6cm, baseline=(current bounding box.center)]
% Define colors
\definecolor{leftcolor}{RGB}{255,255,204} % light yellow
\definecolor{rightcolor}{RGB}{204,204,255} % light purple

% Draw background
\fill[leftcolor] (0,0) rectangle (2,2);
\fill[rightcolor] (2,0) rectangle (4,2);

% Draw vertical line and point
\draw[line width=0.7pt] (2,0) -- (2,2);
\filldraw [black] (2,1) circle (2pt) node[left=1pt] {$\varphi$};

% Label axes
\draw (2,0) node[below] {$G$};
\draw (2,2) node[above] {$F$};
\draw (7,1) node {$=$};
\end{tikzpicture}
\hspace{1cm}
\begin{tikzcd}[ampersand replacement=\&]
\mathsf{C} \arrow[r, "F"{name=A, above}, bend left=40] \arrow[r, "G"'{name=B, below}, bend right=40] \&[+30pt] \mathsf{D}
\arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "\varphi"]
\end{tikzcd}
\end{center}

When we say two string diagrams are equal, we mean that the vertical compositions of natural transformations they represent are equal.

The point(0-cells), strings(1-cells), and 2-cells in a string diagram can be interpreted as follows
\begin{itemize}
\item 2-cells: categories. 2-cells with different colors represent different categories.
\item 1-cells: functors. Each 1-cell has exactly two adjacent 2-cells. The left and right adjacent 2-cells are the domain and codomain of the functor respectively. We can think each 1-cell has two end points connected to either a natural transformation or the point at infinity $\infty$. If an end point is connected to $\infty$, then we say that it is a \textbf{free end}. The end point on the top of a string and the end point on the bottom of a string are called the \textbf{top end} and \textbf{bottom end} respectively.
\item 0-cells: natural transformations. Each 0-cell has exactly two adjacent 1-cells. The top and bottom adjacent 1-cells are the domain and codomain of the natural transformation respectively.
\end{itemize}

\subsection{Basic Operations}
\subsubsection{Composition of Functors}
If two string $F:\mathsf{C}\to\mathsf{D}$ and $G:\mathsf{D}\to\mathsf{F}$ has the same top ends and bottom ends, then we can glue them together to form a new string $G\circ F$. Depending on whether the top ends and bottom ends are free or not, we can illustrate the following cases


\begin{center}
\begin{tikzpicture}[x=0.6cm,y=0.6cm]
% Define colors
\definecolor{leftcolor}{RGB}{255,255,204} % light yellow
\definecolor{midcolor}{RGB}{204,204,255} % light purple
\definecolor{rightcolor}{RGB}{204,255,204}
\begin{scope}
% Draw background
\fill[leftcolor] (0,0) rectangle (2,4);
\fill[midcolor] (2,0) rectangle (4,4);
\fill[rightcolor] (4,0) rectangle (6,4);

% Draw vertical line and point
\draw[line width=0.7pt] (2,0) -- (2,4);
\draw[line width=0.7pt] (4,0) -- (4,4);


% Label axes
\draw (2,4) node[above] {$F$};
\draw (4,4) node[above] {$G$};
\end{scope}

\draw (7,2) node {$=$};

\begin{scope}[shift={(8,0)}]
% Draw background
% Draw background
\fill[leftcolor] (0,0) rectangle (2,4);
\fill[rightcolor] (2,0) rectangle (4,4);

% Draw vertical line and point
\draw[line width=0.7pt] (2,0) -- (2,4);


% Label axes
\draw (2,4) node[above] {$G\circ F$};
\end{scope}



\begin{scope}[shift={(16,0)}]
% Draw background

\fill[leftcolor] (-1,0) rectangle (2,4);
\fill[rightcolor] (2,0) rectangle (5,4);
\draw[fill=midcolor, rounded corners=0.6cm, line width=0.5pt] (1,1) rectangle (3, 3);

% Draw vertical line and point
\draw[line width=0.7pt] (2,0) -- (2,1);
\draw[line width=0.7pt] (2,3) -- (2,4);
\filldraw [black] (2,1) circle (2pt);
\filldraw [black] (2,3) circle (2pt);

% Label axes
\draw (1,2) node[left] {$F$};
\draw (3,2) node[right] {$G$};
\end{scope}
\draw (22,2) node {$=$};
\begin{scope}[shift={(23,0)}]
% Draw background
% Draw background
\fill[leftcolor] (0,0) rectangle (2,4);
\fill[rightcolor] (2,0) rectangle (4,4);

% Draw vertical line and point
\draw[line width=0.7pt] (2,0) -- (2,4);
\filldraw [black] (2,1) circle (2pt);
\filldraw [black] (2,3) circle (2pt);

% Label axes
\draw (2,2) node[left] {$G\circ F$};
\end{scope}
\end{tikzpicture}
\end{center}
\begin{center}
\begin{tikzpicture}[x=0.6cm,y=0.6cm]
\definecolor{leftcolor}{RGB}{255,255,204} % light yellow
\definecolor{midcolor}{RGB}{204,204,255} % light purple
\definecolor{rightcolor}{RGB}{204,255,204}

\begin{scope}[shift={(0,0)}]
\begin{scope}
\clip (0,0) rectangle (6,4);
% Draw background
\fill[leftcolor] (0,0) rectangle (3,4);
%\fill[midcolor] (2,0) rectangle (4,4);
\fill[rightcolor] (3,0) rectangle (6,4);
\filldraw[fill=blue!20] (2,5)-- (2,4) .. controls (2,2) and (2.5, 1) .. (3,1).. controls (3.5,1) and (4, 2) .. (4,4)--(4,5)--cycle;
\end{scope}

% Draw vertical line and point
\draw[line width=0.7pt] (3,0) -- (3,1);
\filldraw [black] (3,1) circle (2pt);


% Label axes
\draw (2,4) node[above] {$F$};
\draw (4,4) node[above] {$G$};
\end{scope}

\draw (7,2) node {$=$};

\begin{scope}[shift={(8,0)}]
% Draw background
% Draw background
\fill[leftcolor] (0,0) rectangle (2,4);
\fill[rightcolor] (2,0) rectangle (4,4);

% Draw vertical line and point
\draw[line width=0.7pt] (2,0) -- (2,4);
\filldraw [black] (2,1) circle (2pt);

% Label axes
\draw (2,4) node[above] {$G\circ F$};
\end{scope}

\begin{scope}[shift={(15,0)}]
\begin{scope}
\clip (0,0) rectangle (6,4);
% Draw background
\fill[leftcolor] (0,0) rectangle (3,4);
\fill[rightcolor] (3,0) rectangle (6,4);
\filldraw[fill=blue!20] (2,-1)-- (2,0) .. controls (2,2) and (2.5, 3) .. (3,3).. controls (3.5,3) and (4, 2) .. (4,0)--(4,-1)--cycle;
\end{scope}

% Draw vertical line and point
\draw[line width=0.7pt] (3,4) -- (3,3);
\filldraw [black] (3,3) circle (2pt);


% Label axes
\draw (2,0) node[below] {$F$};
\draw (4,0) node[below] {$G$};
\end{scope}

\draw (22,2) node {$=$};

\begin{scope}[shift={(23,0)}]
% Draw background
% Draw background
\fill[leftcolor] (0,0) rectangle (2,4);
\fill[rightcolor] (2,0) rectangle (4,4);

% Draw vertical line and point
\draw[line width=0.7pt] (2,0) -- (2,4);
\filldraw [black] (2,3) circle (2pt);

% Label axes
\draw (2,0) node[below] {$G\circ F$};
\end{scope}
\end{tikzpicture}
\end{center}

\subsubsection{Vertical Composition}
The vertical composition of natural transformations $\varphi:F\Rightarrow G$ and $\psi:G\Rightarrow H$ collapses two adjacent points in a string into one single point and ``eats" the intermediate segment, which is illustrated as follows

\begin{center}
\begin{tikzpicture}[x=0.6cm,y=0.6cm, baseline=(current bounding box.center)]
% Define colors
\definecolor{leftcolor}{RGB}{255,255,204} % light yellow
\definecolor{rightcolor}{RGB}{204,204,255} % light purple
\begin{scope}
% Draw background
\fill[leftcolor] (0,-2) rectangle (2,2);
\fill[rightcolor] (2,-2) rectangle (4,2);


% Draw vertical line and point
\draw[line width=0.7pt] (2,-2) -- (2,2);
\filldraw [black] (2,1) circle (2pt) node[left=1pt] {$\varphi$};
\filldraw [black] (2,-1) circle (2pt) node[left=1pt] {$\psi$};

% Label axes
\draw (2,-2) node[below] {$H$};
\draw (2,0) node[left] {$G$};
\draw (2,2) node[above] {$F$};
\end{scope}

\draw (6,0) node {$=$};

\begin{scope}[shift={(8,0)}]
% Draw background
\fill[leftcolor] (0,-2) rectangle (2,2);
\fill[rightcolor] (2,-2) rectangle (4,2);


% Draw vertical line and point
\draw[line width=0.7pt] (2,-2) -- (2,2);
\filldraw [black] (2,0) circle (2pt) node[left=1pt] {$\psi\circ \varphi$};

% Label axes
\draw (2,-2) node[below] {$H$};
\draw (2,2) node[above] {$F$};
\end{scope}
\end{tikzpicture}
\end{center}
\subsubsection{Horizontal Composition}
The horizontal composition of natural transformations $\varphi:F\Rightarrow G$ and $\theta:H\Rightarrow I$ stick two natural transformations in a horizontal line together, which is illustrated as follows


\begin{center}
\begin{tikzpicture}[x=0.6cm,y=0.6cm, baseline=(current bounding box.center)]
% Define colors
\definecolor{leftcolor}{RGB}{255,255,204} % light yellow
\definecolor{midcolor}{RGB}{204,204,255} % light purple
\definecolor{rightcolor}{RGB}{204,255,204}
\begin{scope}
% Draw background
\fill[leftcolor] (0,0) rectangle (2,4);
\fill[midcolor] (2,0) rectangle (4,4);
\fill[rightcolor] (4,0) rectangle (6,4);

% Draw vertical line and point
\draw[line width=0.7pt] (2,0) -- (2,4);
\filldraw [black] (2,2) circle (2pt) node[anchor=east] {$\varphi$};
\draw[line width=0.7pt] (4,0) -- (4,4);
\filldraw [black] (4,2) circle (2pt) node[anchor=east] {$\theta$};

% Label axes
\draw (2,0) node[below] {$G$};
\draw (2,4) node[above] {$F$};
\draw (4,0) node[below] {$I$};
\draw (4,4) node[above] {$H$};
\end{scope}

\draw (7.5,2) node {$=$};

\begin{scope}[shift={(9,0)}]
% Draw background
% Draw background

\begin{scope}
\clip (0,0) rectangle (6,4);
\fill[leftcolor] (0,0) rectangle (3,4);
\fill[rightcolor] (3,0) rectangle (6,4);
\filldraw[fill=blue!20, line width=0.7pt] (2,5)-- (2,4) .. controls (2, 2.5) and (2.5, 2) .. (3,2).. controls (3.5,2) and (4, 2.5) .. (4,4)--(4,5)--cycle;
\filldraw[fill=blue!20, line width=0.7pt] (2,-1)-- (2,0) .. controls (2,1.5) and (2.5, 2) .. (3,2).. controls (3.5,2) and (4, 1.5) .. (4,0)--(4,-1)--cycle;
\end{scope}

% Draw vertical line and point
\filldraw [black] (3,2) circle (2pt) node[left=9pt] {$\theta\circ\varphi$};

% Label axes
\draw (2,0) node[below] {$G$};
\draw (2,4) node[above] {$F$};
\draw (4,0) node[below] {$I$};
\draw (4,4) node[above] {$H$};
\end{scope}
\end{tikzpicture}
\end{center}



\subsection{Morphism as Natural Transformation}
If $f:X\to Y$ is a morphism in $\mathsf{C}$, then the following string diagram should be understood as follows
\begin{center}
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