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hooyuser committed Jan 12, 2024
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Expand Up @@ -204,7 +204,7 @@ \section{Category}
\mathsf{C}\&[-25pt]\&[+10pt]\&[-30pt][\mathsf{J},\mathsf{C}]\&[-30pt]\&[-30pt] \&\&\&\\ [-15pt]
x \arrow[dd, "f"{name=L, left}]
\&[-25pt] \& [+10pt]
\& [-30pt]\Delta x\arrow[dd, Rightarrow, "f_*"{name=R}] \& \hspace{30pt}\& x\arrow[r, "\mathrm{id}_x"]\arrow[dd, "f"']\&x\arrow[dd, "f"]\\ [-10pt]
\& [-30pt]\Delta x\arrow[dd, Rightarrow, "f_\bullet"{name=R}] \& \hspace{30pt}\& x\arrow[r, "\mathrm{id}_x"]\arrow[dd, "f"']\&x\arrow[dd, "f"]\\ [-10pt]
\& \phantom{.}\arrow[r, "\Delta", squigarrow]\&\phantom{.} \& \\[-10pt]
y \& \& \& \Delta y\&\& y\arrow[r, "\mathrm{id}_y"']\&y
\end{tikzcd}
Expand Down Expand Up @@ -339,7 +339,128 @@ \section{Category}
\]
}

\section{String Diagram}
\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}
\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[anchor=east] {$f$};

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

Suppose $\varphi:F\Rightarrow G$ is a natural transformation between functors $F,G:\mathsf{C}\to \mathsf{D}$. Then the functoriality of $\varphi$ can be described by the following string diagram
\[
\begin{tikzpicture}[x=1cm,y=1cm, 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 (1,3);
\fill[green!20] (1,0) rectangle (3,3);
\begin{scope}
\clip (1,0) rectangle (3, 3);
\draw[fill=blue!20, rounded corners=1cm, line width=0.7pt] (0,2) rectangle (2, 5);
\draw[fill=blue!20, rounded corners=1cm, line width=0.7pt] (0,2) rectangle (2, -1);
%\fill[color=green!20] (2,0) rectangle (3,3);
%\filldraw[fill=green!20, rounded corners=1cm] (2, 4) -- (2, 1) -- (1, 1) -- (2, 1) -- (2, -1) -- (4, -1) -- (4, 4) -- cycle;
\end{scope}
% Draw vertical line and point
\draw[line width=0.7pt] (1,0) -- (1,3);
\filldraw [black] (1,1) circle (2pt) node[left=2pt] {$f$};
\filldraw [black] (1,2) circle (2pt) node[left=2pt] {$\varphi_X$};
% Label axes
\draw (1,0) node[below] {$Y$};
\draw (1,3) node[above] {$X$};
\draw (2,0) node[below] {$G$};
\draw (2,3) node[above] {$F$};
\end{tikzpicture}
\hspace{5pt}=\hspace{5pt}
\begin{tikzpicture}[x=1cm,y=1cm, 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 (1,3);
\fill[blue!20] (1,0) rectangle (2,3);
\fill[green!20] (2,0) rectangle (3,3);
% Draw vertical line and point
\draw[line width=0.7pt] (1,0) -- (1,3);
\draw[line width=0.7pt] (2,0) -- (2,3);
\filldraw [black] (1,1.5) circle (2pt) node[left=2pt] {$f$};
\filldraw [black] (2,1.5) circle (2pt) node[left=2pt] {$\varphi$};
% Label axes
\draw (1,0) node[below] {$Y$};
\draw (1,3) node[above] {$X$};
\draw (2,0) node[below] {$G$};
\draw (2,3) node[above] {$F$};
\end{tikzpicture}
\hspace{5pt}=\hspace{5pt}
\begin{tikzpicture}[x=1cm,y=1cm, 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 (1,3);
\fill[green!20] (1,0) rectangle (3,3);
\begin{scope}
\clip (1,0) rectangle (3, 3);
\draw[fill=blue!20, rounded corners=1cm, line width=0.7pt] (0,1) rectangle (2, 4);
\draw[fill=blue!20, rounded corners=1cm, line width=0.7pt] (0,1) rectangle (2, -2);
%\fill[color=green!20] (2,0) rectangle (3,3);
%\filldraw[fill=green!20, rounded corners=1cm] (2, 4) -- (2, 1) -- (1, 1) -- (2, 1) -- (2, -1) -- (4, -1) -- (4, 4) -- cycle;
\end{scope}
% Draw vertical line and point
\draw[line width=0.7pt] (1,0) -- (1,3);
\filldraw [black] (1,1) circle (2pt) node[left=2pt] {$\varphi_Y$};
\filldraw [black] (1,2) circle (2pt) node[left=2pt] {$f$};
% Label axes
\draw (1,0) node[below] {$Y$};
\draw (1,3) node[above] {$X$};
\draw (2,0) node[below] {$G$};
\draw (2,3) node[above] {$F$};
\end{tikzpicture}
\]
Note that appending the string diagram \begin{tikzpicture}[x=0.3cm,y=0.3cm, 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[anchor=east] {$f$};

% Label axes
\draw (2,0) node[below] {$Y$};
\draw (2,2) node[above] {$X$};
\end{tikzpicture} to the left of the string diagram of $\varphi$ is equivalent to evaluating $\varphi$ at $f$.

\section{Limit and Colimit}
\dfn{Cone}{
Expand Down Expand Up @@ -368,7 +489,7 @@ \section{Limit and Colimit}
\item Morphisms: The morphisms in $\mathsf{Cone}(\textsf{C},F)$ are commutative triangles shown as follows
\[
\begin{tikzcd}[ampersand replacement=\&, row sep=9pt]
\Delta C \arrow[rd, Rightarrow, "h"] \arrow[dd, Rightarrow, "f_*"'] \& \\
\Delta C \arrow[rd, Rightarrow, "h"] \arrow[dd, Rightarrow, "f_\bullet"'] \& \\
\& F\\
\Delta C' \arrow[ru, Rightarrow, "h^{\prime}"'] \&
\end{tikzcd}
Expand Down Expand Up @@ -419,7 +540,7 @@ \section{Limit and Colimit}
\item Morphisms: The morphisms in $\mathsf{Cocone}(F,\textsf{C})$ are commutative triangles shown as follows
\[
\begin{tikzcd}[ampersand replacement=\&, row sep=9pt]
\&\Delta C \arrow[dd, Rightarrow, "f_*"] \\
\&\Delta C \arrow[dd, Rightarrow, "f_\bullet"] \\
F \arrow[ru, Rightarrow, "h"]\arrow[rd, Rightarrow, "h^{\prime}"']\& \\
\&\Delta C'
\end{tikzcd}
Expand Down Expand Up @@ -987,9 +1108,190 @@ \section{Adjoint Functor}
\end{tikzcd}
\]
}
\prop{Snake Equations}{
\prop{Adjunction Isomorphism Determined by Unit or Counit}{
Let $\left(L,R,\varphi\right)$ be an adjoint pair of functors and $\eta$ and $\varepsilon$ be the adjunction unit and counit respectively. Then
\begin{align*}
\varphi_{X,Y}(f) & =G f \circ \eta_X: X \rightarrow G Y, \quad \forall f: F X \rightarrow Y , \\
\varphi_{X,Y}^{-1}(g) & =\varepsilon_Y \circ F g: F X \rightarrow Y, \quad \forall g: X \rightarrow G Y .
\end{align*}
}
\pf{
\[
\begin{tikzcd}[ampersand replacement=\&, row sep = 3.5em]
\mathrm{id}_A\arrow[d, mapsto]\&[-25pt]\in\&[-25pt]\mathrm{Hom}_{\mathsf{C}}(A,A) \arrow[d, "{\phi_A}"'] \arrow[rr, "{f_*}"] \&[-20pt] \& {\mathrm{Hom}_{\mathsf{C}}(A,X)} \arrow[d, "\phi_X"]\&[-25pt]\ni\&[-25pt]f \arrow[d, mapsto]\\
u=\phi_A(\mathrm{id}_A)\&\in\&{F(A)}\arrow[rr, "{F(f)}"'] \& \& {F(X)} \&\ni\& \phi_X(f)
\end{tikzcd}
\]
}
\lemm{Snake Equations}{
Let $\left(L,R,\varphi\right)$ be an adjoint pair of functors and $\eta$ and $\varepsilon$ be the adjunction unit and counit respectively. Then we have the following equalities of natural transformations
\begin{center}
\begin{tikzpicture}[x=0.8cm,y=0.8cm]
\node at (-7.4,1.5) {${\left[L \xRightarrow{L \eta}LR L\xRightarrow{\varepsilon L} L\right]=\mathrm{id}_L}$};
\begin{scope}
\clip (-2.4,0) rectangle (2.4,3);
\fill[fill=green!20] (-2.5,0) rectangle (2.5,3);
\filldraw[fill=blue!20, rounded corners=0.48cm, line width=0.5pt] (-1.2,-1) -- (-1.2, 2.1)-- (0,2.1)-- (0,0.9)-- (1.2,0.9)
-- (1.2,4)-- (4,4)-- (4,-1)--cycle;
\end{scope}

\node[above] at (1.2,3) {$L$};
\node[below] at (-1.2,0) {$L$};

\draw[fill=black] (0.6, 0.9) circle (0.07);
\node[above=2pt] at (0.6, 0.9) {$\varepsilon$};

\draw[fill=black] (-0.6, 2.1) circle (0.07);
\node[above=2pt] at (-0.6, 2.1) {$\eta$};

\node at (3.2, 1.5) {$=$};

\begin{scope}
\clip (4,0) rectangle (6.4,3);
\fill[fill=green!20] (4,0) rectangle (6.4,3);
\filldraw[fill=blue!20, line width=0.5pt] (5.2,-1) rectangle (7,4);
\end{scope}

\node[below] at (5.2,0) {$L$};
\node[above] at (5.2,3) {$L$};
\end{tikzpicture}
\end{center}
\begin{center}
\begin{tikzpicture}[x=0.8cm,y=0.8cm]
\node at (-7.3,1.5) {${\left[R \xRightarrow{\eta R}RLR \xRightarrow{R \varepsilon}R\right]=\mathrm{id}_R}$};
\begin{scope}
\clip (-2.4, 0) rectangle (2.4, 3);
\fill[fill=blue!20] (-2.5, 0) rectangle (2.5,3);
\filldraw[fill=green!20, rounded corners=0.48cm] (-1.2, 4) -- (-1.2, 0.9)-- (0, 0.9)-- (0, 2.1)-- (1.2, 2.1)
-- (1.2, -1)-- (4, -1)-- (4, 4)--cycle;
\end{scope}

\node[below] at (1.2,0) {$R$};
\node[above] at (-1.2,3) {$R$};

\draw[fill=black] (0.6, 2.1) circle (0.07);
\node[above=2pt] at (0.6, 2.1) {$\eta$};

\draw[fill=black] (-0.6, 0.9) circle (0.07);
\node[above=2pt] at (-0.6, 0.9) {$\varepsilon$};

\node at (3.2, 1.5) {$=$};

\begin{scope}
\clip (4,0) rectangle (6.4,3);
\fill[fill=blue!20] (4,0) rectangle (6.4,3);
\filldraw[fill=green!20] (5.2,-1) rectangle (7,4);
\end{scope}

\node[below] at (5.2,0) {$R$};
\node[above] at (5.2,3) {$R$};
\end{tikzpicture}
\end{center}
}
\dfn{Wire Bending}{
Let $\mathsf{C}$, $\mathsf{D}$, $\mathsf{C'}$, $\mathsf{D'}$ be categories and $L,R,L',R',H,K$ be functors shown in the following diagram
\[
\begin{tikzcd}[ampersand replacement=\&]
\mathsf{C} \arrow[dd, "L"', bend right=38] \arrow[r, "H"] \&[+36pt] \mathsf{C'} \arrow[dd, "L'"', bend right=38] \\[-5pt]
\dashv \& \dashv \\[-5pt]
\mathsf{D} \arrow[uu, "R"', bend right=38] \arrow[r, "K"'] \& \mathsf{D'} \arrow[uu, "R'"', bend right=38]
\end{tikzcd}
\]
There are bijections between the following sets of natural transformations
\[
\begin{tikzcd}[ampersand replacement=\&]
{\mathrm{Hom}_{[\mathsf{C}, \mathsf{D'}]}\left(L'H, KL\right) } \arrow[r, shift left=1pt, rightharpoonup, "\triangleright"]\arrow[r, shift right=1pt,leftharpoondown, "\triangleleft"'] \& {\mathrm{Hom}_{[\mathsf{C}, \mathsf{C'}]}(H, R'KL) } \arrow[d, shift left=1pt, rightharpoonup, "\triangleright"]\arrow[d, shift right=1pt,leftharpoondown, "\triangleleft"'] \\
{ \mathrm{Hom}_{[\mathsf{D}, \mathsf{D'}]}\left(L'HR, K\right)} \arrow[u, shift left=1pt, rightharpoonup, "\triangleright"]\arrow[u, shift right=1pt,leftharpoondown, "\triangleleft"'] \& {\mathrm{Hom}_{[\mathsf{D}, \mathsf{C'}]}(HR, R'K) } \arrow[l, shift left=1pt, rightharpoonup, "\triangleright"]\arrow[l, shift right=1pt,leftharpoondown, "\triangleleft"']
\end{tikzcd}
\]
natural in both $H$ and $K$ such that $\triangleleft=\triangleright^{-1}$ and $\triangleright\triangleright\triangleright\hspace{2.15pt}\triangleright=\mathrm{id}$. $\triangleright$ is called \textbf{wire bending} because its action can be visualized as bending the wires of adjoint pairs in the following string diagrams

\begin{center}
\begin{tikzpicture}[x=0.8cm,y=0.8cm, baseline=(current bounding box.center)]
\draw [|-To] (3,1) -- (4,1) node[midway,above] {$\triangleright$};
\draw [|-To] (7,-1) -- (7,-2) node[midway,right] {$\triangleright$};
\draw [|-To] (4,-4) -- (3,-4) node[midway,above] {$\triangleright$};
\draw [|-To] (0,-2) -- (0,-1) node[midway,right] {$\triangleright$};
\begin{scope}
%\clip (-2,0) rectangle (2,2);
\fill[fill=green!20] (-2,0) rectangle (0,2);
\fill[fill=blue!45] (0,0) rectangle (2,2);
\fill[fill=blue!20] (-1,0) .. controls (-1,0.4) and (-0.8, 1) .. (0,1)--(0,0)--cycle;
\fill[fill=green!40] (1,2) .. controls (1,1.6) and (0.8, 1) .. (0,1)--(0,2)--cycle;
\end{scope}
\draw[line width=0.5pt] (0,0) -- (0,2);
\draw[line width=0.5pt] (-1,0) .. controls (-1, 0.4) and (-0.8, 1) .. (0, 1) .. controls (0.8, 1) and (1, 1.6) .. (1, 2);

\node[above=2pt] at (0, 2) {$H$};
\node[below=2pt] at (0, 0) {$K$};
\node[below=2pt] at (-1, 0) {$L$};
\node[above=2pt] at (1, 2) {$L'$};

\draw[fill=black] (0, 1) circle (0.07);
\node[left=7pt, above=1pt] at (0, 1) {$\alpha$};

\begin{scope}[shift={(7,0)}]
\begin{scope}
%\clip (-2,0) rectangle (2,2);
\fill[fill=green!20] (-2,0) rectangle (0,2);
\fill[fill=green!45] (0,0) rectangle (2,2);
\fill[fill=blue!20] (-1,0) .. controls (-1, 0.4) and (-0.8, 1) .. (0,1)--(0,0)--cycle;
\fill[fill=blue!40] (1,0) .. controls (1, 0.4) and (0.8, 1) .. (0,1)--(0,0)--cycle;
\end{scope}
\draw[line width=0.5pt] (0,0) -- (0,2);
\draw[line width=0.5pt] (-1,0) .. controls (-1, 0.4) and (-0.8, 1) .. (0, 1) .. controls (0.8, 1) and (1, 0.4) .. (1, 0);

\node[above=2pt] at (0, 2) {$H$};
\node[below=2pt] at (0, 0) {$K$};
\node[below=2pt] at (-1, 0) {$L$};
\node[below=2pt] at (1, 0) {$R'$};

\draw[fill=black] (0, 1) circle (0.07);
\node[left=7pt, above=1pt] at (0, 1) {$\beta$};
\end{scope}
\begin{scope}[shift={(0,-5)}]
\begin{scope}
%\clip (-2,0) rectangle (2,2);
\fill[fill=blue!20] (-2,0) rectangle (0,2);
\fill[fill=blue!45] (0,0) rectangle (2,2);
\fill[fill=green!20] (-1,2) .. controls (-1, 1.6) and (-0.8, 1) .. (0,1)--(0,2)--cycle;
\fill[fill=green!40] (1,2) .. controls (1, 1.6) and (0.8, 1) .. (0,1)--(0,2)--cycle;
\end{scope}
\draw[line width=0.5pt] (0,0) -- (0,2);
\draw[line width=0.5pt] (-1,2) .. controls (-1, 1.6) and (-0.8, 1) .. (0, 1) .. controls (0.8, 1) and (1, 1.6) .. (1, 2);

\node[above=2pt] at (0, 2) {$H$};
\node[below=2pt] at (0, 0) {$K$};
\node[above=2pt] at (-1, 2) {$R$};
\node[above=2pt] at (1, 2) {$L'$};

\draw[fill=black] (0, 1) circle (0.07);
\node[left=7pt, below=1pt] at (0, 1) {$\delta$};
\end{scope}
\begin{scope}[shift={(7,-5)}]
\begin{scope}
%\clip (-2,0) rectangle (2,2);
\fill[fill=blue!20] (-2,0) rectangle (0,2);
\fill[fill=green!45] (0,0) rectangle (2,2);
\fill[fill=green!20] (-1,2) .. controls (-1, 1.6) and (-0.8, 1) .. (0,1)--(0,2)--cycle;
\fill[fill=blue!40] (1,0) .. controls (1, 0.4) and (0.8, 1) .. (0,1)--(0,0)--cycle;
\end{scope}
\draw[line width=0.5pt] (0,0) -- (0,2);
\draw[line width=0.5pt] (-1,2) .. controls (-1, 1.6) and (-0.8, 1) .. (0, 1) .. controls (0.8, 1) and (1, 0.4) .. (1, 0);

\node[above=2pt] at (0, 2) {$H$};
\node[below=2pt] at (0, 0) {$K$};
\node[above=2pt] at (-1, 2) {$R$};
\node[below=2pt] at (1, 0) {$R'$};

\draw[fill=black] (0, 1) circle (0.07);
\node[left=7pt, below=1pt] at (0, 1) {$\gamma$};
\end{scope}
\end{tikzpicture}
\end{center}
}
We only explicit define $\alpha^\triangleright$ and $\alpha^\triangleleft$ here. The other wire bendings are similar.


\section{Monoidal Category}
\dfn{Monoidal Category}{
Expand Down

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