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add string diagram for representable functors
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add string diagram for representable functors ; add Limit Characterized by Representability
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hooyuser committed Jan 17, 2024
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155 changes: 142 additions & 13 deletions category_theory.tex
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Expand Up @@ -1119,8 +1119,99 @@ \section{Representable Functor}
A \textbf{representation of $F$} is a pair $(A,\phi)$, where $A\in \mathrm{Ob}(\mathsf{C})$ and $\phi:F\stackrel{\sim\;}{\Rightarrow} \operatorname{Hom}_{\mathsf{C}}\left(A,-\right)$ is a natural isomorphism. \\
According to \hyperref[th:yoneda_lemma]{Yoneda lemma}, $\phi:\operatorname{Hom}_{\mathsf{C}}\left(A,-\right)\Rightarrow F$ is 1-1 correspondence with an element $\phi_A(\mathrm{id}_A)\in F(A)$. We define an \textbf{universal element of $F$} is a pair $(A,u)$ where $A\in \mathrm{Ob}(\mathsf{C})$ and $u\in F(A)$ such that $u$ corresponds to a natural isomorphism $\phi: \operatorname{Hom}_{\mathsf{C}}\left(A,-\right)\stackrel{\sim\;}{\Rightarrow}F$. Specifying a universal element of $F$ is equivalent to specifying a representation of $F$.
}
\prop[universal_element_characterization]{Uniqueness of Universal Element}{
Suppose $F:\mathsf{C}\to \mathsf{Set}$ is a representable functor. Then $(A,u)$ is a universal element of $F$ if and only if $(A,u)$ is initial in the category \hyperref[th:category_of_elements]{$\int_{\mathsf{C}}F$}. That is, if $(A,u)$ is a universal element of $F$, then for any $(X,x)\in \mathrm{Ob}(\int_{\mathsf{C}}F)$, there is a unique morphism $(A,u)\to (X,x)$ in $\int_{\mathsf{C}}F$ (which is a morphism $f:A\to X$ in $\mathsf{C}$ such that $F(f)(u)=x$).
\[
\begin{tikzpicture}[x=0.6cm,y=0.6cm, baseline=(current bounding box.center), line width=0.7pt]
\definecolor{leftcolor}{RGB}{255,255,204}
\definecolor{midcolor}{HTML}{BBE6FB}
\definecolor{rightcolor}{HTML}{F5D4BC}
\node at (0, 6) {$\mathrm{Hom}_{\mathsf{C}}\left(A,X\right)$};
\node at (4.75, 6) {$\xlongrightarrow{\sim}$};
\node at (4.75, -4.75) {$\longmapsfrom$};
\node at (4.75, 1.8) {$\longmapsto$};
\node at (9.5, 6) {$F(X)$};
\begin{scope}[line width=0.7pt]
\begin{scope}
\clip (-3,0.4) rectangle (3,3.2);
\fill[fill=leftcolor] (-3,0) rectangle (0, 3.2);
\fill[fill=rightcolor] (0,0) rectangle (3, 3.2);
%\draw[fill=midcolor, rounded corners=0.6cm, line width=0.7pt] (-1, -2) rectangle (1, 2);
\end{scope}
\node[below] at (0, 0.4) {$X$};
\node[left=1pt] at (0, 1.8) {$f$};
\draw[fill=black] (0, 1.8) circle (0.1);
\draw (0, 3.2) -- (0, 0.4);
\node[above] at (0, 3.2) {$A$};
\end{scope}
\begin{scope}[shift={(9.5,0)}]
\fill[fill=leftcolor] (-3,-0.4) rectangle (0,4);
\fill[fill=midcolor] (0, -0.4) rectangle (3,4);
\fill[fill=rightcolor] (-1, -0.4) rectangle (1,2);
\begin{scope}
\clip (-2.5,0.4) rectangle (2.5,3.2);
\fill[fill=leftcolor] (-2.5,0) rectangle (0, 3.2);
\fill[fill=rightcolor] (0,0) rectangle (2.5, 3.2);
%\draw[fill=midcolor, rounded corners=0.6cm, line width=0.7pt] (-1, -2) rectangle (1, 2);
\end{scope}
\draw[line width=1.7pt, color=black!80, rounded corners=2.5pt] (-2.5, 0.4) rectangle (2.5, 3.2); % inner rectangle
\node[above,shift={(0.35,0.1)}] at (0, 0.4) {$X$};
\node[left=1pt] at (0, 1.8) {$f$};
\draw[fill=black] (0, 1.8) circle (0.1);
\draw (0, 4) -- (0, 0.4);
\node[below,shift={(0.35,-0.1)}] at (0, 3.2) {$A$};
\draw (1, 0.4) -- (1, -0.4);
\draw (-1, 0.4) -- (-1, -0.4);
\node[above] at (0, 4) {$\{*\}$};
\node[below] at (-1, -0.4) {$X$};
\node[below] at (1, -0.4) {$F$};
\end{scope}
\begin{scope}[shift={(0, -6.5)}, line width=0.7pt]
\fill[fill=leftcolor] (-3,-0.4) rectangle (0,4);
\fill[fill=midcolor] (0, -0.4) rectangle (3,4);
\begin{scope}
\clip (-2.5,0.4) rectangle (2.5, 3.2);
\fill[fill=leftcolor] (-2.5,0) rectangle (0, 3.2);
\fill[fill=rightcolor] (0,0) rectangle (2.5, 3.2);
\draw[fill=midcolor, rounded corners=0.6cm] (-1, -2) rectangle (1, 2);
\end{scope}
\node[shift={(-0.35,0.35)}] at (-1, 0.4) {$X$};
\node[shift={(0.35,0.35)}] at (1, 0.4) {$F$};
\node[below=2pt] at (0, 2) {$s$};
\draw[fill=black] (0, 2) circle (0.1);
\draw (0,4) -- (0,2);
\draw (0, -0.4) -- (0,0.4);
\node[below,shift={(-0.5,0.1)}] at (0, 3) {\scalebox{.8}{$\{*\}$}};
\draw[line width=1.7pt, color=black!80, rounded corners=2.5pt] (-2.5, 0.4) rectangle (2.5, 3.2); % inner rectangle
\node[above] at (0, 4){$A$};
\node[below] at (0, -0.6){$X$};
\end{scope}
\begin{scope}[shift={(9.5, -6.5)}, line width=0.7pt]
\begin{scope}
\clip (-3,0.4) rectangle (3,3.2);
\fill[fill=leftcolor] (-3,0) rectangle (0, 3.2);
\fill[fill=rightcolor] (0,0) rectangle (3, 3.2);
\draw[fill=midcolor, rounded corners=0.6cm] (-1, -2) rectangle (1, 2);
\end{scope}
\node[below] at (-1, 0.4) {$X$};
\node[below] at (1, 0.4) {$F$};
\node[below=2pt] at (0, 2) {$s$};
\draw[fill=black] (0, 2) circle (0.1);
\draw (0,3.2) -- (0,2);
\node[above] at (0, 3.2) {$\{*\}$};
\end{scope}
\end{tikzpicture}
\]
\prop[universal_element_characterization]{Equivalent Characterizations of Representable Functor}{
Suppose $F:\mathsf{C}\to \mathsf{Set}$ is a functor. Then the following statements are equivalent:
\begin{enumerate}[(i)]
\item $F$ is representable by universal element $(A,u)$
\item $(A,u)$ is initial in the category \hyperref[th:category_of_elements]{$\int_{\mathsf{C}}F$}.
\item $\left(A,\Delta u:\{*\}\to A\right)$ is a universal morphism from $\{*\}$ to $F$.
\item For any $(X,x)\in \mathrm{Ob}(\int_{\mathsf{C}}F)$, there is a unique morphism $(A,u)\to (X,x)$ in $\int_{\mathsf{C}}F$ (which is a morphism $f:A\to X$ in $\mathsf{C}$ such that $F(f)(u)=x$).
\end{enumerate}
}
\pf{
Suppose $(A,u)$ is an object of $\int_{\mathsf{C}}F$ and $\phi:\operatorname{Hom}_{\mathsf{C}}\left(A,-\right)\Rightarrow F$ is the natural isomorphism corresponding to $u\in F(A)$. For any $(X,x)\in \mathrm{Ob}(\int_{\mathsf{C}}F)$, we have the following commutative diagram
Expand All @@ -1143,18 +1234,22 @@ \section{Representable Functor}
&\iff (A,u) \text{ is a universal element of }F.
\end{align*}
}
\cor{Initial Object Characterized by Representable Functor}{
\cor[initial_object_representable_functor]{Initial Object Characterized by Representable Functor}{
Suppose $\mathsf{C}$ is a locally small category.
\begin{itemize}
\item $A\in\mathrm{Ob}(\mathsf{C})$ is initial in $\mathsf{C}$ if and only if the functor $\Delta \{*\}:\mathsf{C}\to \mathsf{Set}$ is naturally isomorphic to $\mathrm{Hom}_\mathsf{C}\left(A,-\right)$.
\item $A\in\mathrm{Ob}(\mathsf{C})$ is terminal in $\mathsf{C}$ if and only if the functor $\Delta \{*\}:\mathsf{C}^{\mathrm{op}}\to \mathsf{Set}$ is naturally isomorphic to $\mathrm{Hom}_\mathsf{C}\left(-,A\right)$.
\end{itemize}
}
\pf{
Let $\Delta \{*\}:\mathsf{C}\to \mathsf{Set}$ be a constant functor. It is easy to see that the category $\int_\mathsf{C}\Delta \{*\}$ is isomorphic to $\mathsf{C}$.
According to \Cref{th:universal_element_characterization},
$A\in\mathrm{Ob}(\mathsf{C})$ is initial in $\int_\mathsf{C}\Delta \{*\}$ if and only if $\Delta$ is a representable functor with a universal element $(A,*)$.\\
If $\Delta\{*\}$ is naturally isomorphic to $\mathrm{Hom}_\mathsf{C}\left(A,-\right)$ through $\theta:\mathrm{Hom}_\mathsf{C}\left(A,-\right)\stackrel{\sim\;}{\Rightarrow}\Delta \{*\}$, we have no choice but to define $\theta_X(\mathrm{id}_X)=*$. Note that $\Delta \{*\}(A)=\{*\}$. Yoneda lemma also implies that $\theta$ must correspond to $*\in \Delta \{*\}(A)$ and accordingly $\theta$ is the unique natural isomorphism from $\mathrm{Hom}_\mathsf{C}\left(A,-\right)$ to $\Delta\{*\}$.
Let $\Delta \{*\}:\mathsf{C}\to \mathsf{Set}$ be a constant functor. It is easy to see that the category $\int_\mathsf{C}\Delta \{*\}$ is isomorphic to $\mathsf{C}$ through the functor
\begin{align*}
p:\int_\mathsf{C}\Delta \{*\}&\longrightarrow \mathsf{C}\\
(C,*) &\longmapsto C
\end{align*}
As established in \Cref{th:universal_element_characterization},
$(A,*)\in\mathrm{Ob}(\mathsf{C})$ is initial in $\int_\mathsf{C}\Delta \{*\}$ if and only if $\Delta\{*\}$ is a representable functor with a universal element $(A,*)$, which proves the first statement. The second statement can be obtained by applying the first statement to $\mathsf{C}^{\mathrm{op}}$.\\
In addition, an alternative ad-hoc proof is conceivable. If $\Delta\{*\}$ is naturally isomorphic to $\mathrm{Hom}_\mathsf{C}\left(A,-\right)$ through $\theta:\mathrm{Hom}_\mathsf{C}\left(A,-\right)\stackrel{\sim\;}{\Rightarrow}\Delta \{*\}$, we have no choice but to define $\theta_X(\mathrm{id}_X)=*$. Note that $\Delta \{*\}(A)=\{*\}$. Yoneda lemma also implies that $\theta$ must correspond to $*\in \Delta \{*\}(A)$ and accordingly $\theta$ is the unique natural isomorphism from $\mathrm{Hom}_\mathsf{C}\left(A,-\right)$ to $\Delta\{*\}$.
}


Expand Down Expand Up @@ -1205,7 +1300,7 @@ \section{Limit and Colimit}
\[
\begin{tikzcd}[ampersand replacement=\&, every arrow/.append style={-latex, line width=1.1pt}]
\& [-1.5em] \&[-1em] C \arrow[rdd, draw=cyan, "h_j"] \arrow[ldd, draw=cyan, "h_i"'] \arrow[d, dash pattern=on 4pt off 2pt, draw=arrowRed] \& [-1em] \\[+0.2cm]
\& \& \varprojlim F\arrow[ld, draw=arrowBlue, shorten <=-3pt, "\phi_i" yshift=4.5pt] \arrow[rd, draw=arrowBlue, shorten <=-4.5pt, "\phi_j"' yshift=4pt] \& \\[-0.1cm]
\& \& \varprojlim F\arrow[ld, draw=arrowBlue, shorten <=-3pt, "\ell_i" yshift=4.5pt] \arrow[rd, draw=arrowBlue, shorten <=-4.5pt, "\ell_j"' yshift=4pt] \& \\[-0.1cm]
\& F(i) \arrow[rr, "F(\lambda)"']\& \& F(j)
\end{tikzcd}
\]
Expand Down Expand Up @@ -1259,11 +1354,45 @@ \section{Limit and Colimit}
\begin{tikzcd}[ampersand replacement=\&, every arrow/.append style={-latex, line width=1.1pt}]
\& [-1.5em] \&[-1em] C \& [-1em] \\[+0.2cm]
\& \& \varinjlim F \arrow[u, dash pattern=on 4pt off 2pt, draw=arrowRed] \& \\[-0.1cm]
\& F(i) \arrow[ruu, draw=cyan, "h_i"]\arrow[ru, draw=arrowBlue, shorten >=-3.5pt, "\phi_i"' yshift=4.5pt] \arrow[rr, "F(\lambda)"']\& \& F(j)\arrow[luu, draw=cyan, "h_j"']\arrow[lu, draw=arrowBlue, shorten >=-3.7pt, "\phi_j" yshift=4.5pt]
\& F(i) \arrow[ruu, draw=cyan, "h_i"]\arrow[ru, draw=arrowBlue, shorten >=-3.5pt, "\ell_i"' yshift=4.5pt] \arrow[rr, "F(\lambda)"']\& \& F(j)\arrow[luu, draw=cyan, "h_j"']\arrow[lu, draw=arrowBlue, shorten >=-3.7pt, "\ell_j" yshift=4.5pt]
\end{tikzcd}
\]
}

\prop{Limit Characterized by Representability}{
Let $\mathsf{J},\mathsf{C}$ be categories and $F:\mathsf{J}\to\mathsf{C}$ be a functor. The limit of $F$ exists if and only if the functor $\mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\Delta\left(-\right), F\right)$
\begin{align*}
\begin{minipage}{.7\textwidth}
\begin{tikzcd}[ampersand replacement=\&]
\mathsf{C}^{\mathrm{op}}\&[-34pt]\&[+52pt]\&[-30pt] \mathsf{Set}\&[-30pt]\&[-30pt] \\ [-15pt]
C_1 \arrow[dd, "g"{name=L, left}]
\&[-25pt] \& [+10pt]
\& [-30pt]\mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\Delta C_1, F\right)\arrow[dd, "\left(\Delta g\right)^*0.-*+9************9"{name=R}] \& \ni \& \left(h_i:C_1\textcolor{arrowBlue}{\boldsymbol{\longrightarrow}} F(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}\arrow[dd,mapsto,"\left(g^*\right)_{i\in \mathrm{Ob}(\mathsf{J})}"]\\ [-3pt]
\& \phantom{.}\arrow[r, "{\mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\Delta\left(-\right), F\right)}", squigarrow]\&\phantom{.} \& \\[-3pt]
C_2 \& \& \& \mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\Delta C_2, F\right)\& \ni \& \left(h_i\circ g:C_2\textcolor{cyan}{\boldsymbol{\longrightarrow}} F(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}
\end{tikzcd}
\end{minipage}
\begin{minipage}{.2\textwidth}
\begin{tikzcd}[ampersand replacement=\&, every arrow/.append style={-latex, line width=1.1pt},scale=0.5]
\\[-1pt]
\& [-1.5em] \&[-1em] C_2 \arrow[rdd, draw=cyan, "h_j\circ g"] \arrow[ldd, draw=cyan, "h_i\circ g"'] \arrow[d, draw=red!50!yellow!80, "g"'] \& [-1em] \\[+0.1cm]
\& \& C_1\arrow[ld, draw=arrowBlue, shorten <=-3pt, "h_i" yshift=3.7pt] \arrow[rd, draw=arrowBlue, shorten <=-4.5pt, "h_j"' yshift=4pt] \& \\[-0.2cm]
\& F(i) \arrow[rr, "F(\lambda)"']\& \& F(j)
\end{tikzcd}
\end{minipage}
\end{align*}
is representable. In this case, the universal element coincides with $\left(\varprojlim F,\left(\ell_i:\varprojlim F\to F(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}\right)$ and we have
\[
\mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\Delta\left(-\right), F\right)\cong \mathrm{Hom}_{\mathsf{C}}\left(-,\varprojlim F\right).
\]
Dually, the colimit of $F$ exists if and only if the functor $\mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(F,\Delta\left(-\right)\right):\mathsf{C}\to\mathsf{Set}$ is representable. In this case, the universal element coincides with $\left(\varinjlim F,\left(\ell_i:F(i)\to \varinjlim F\right)_{i\in \mathrm{Ob}(\mathsf{J})}\right)$ and we have
\[
\mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(F,\Delta\left(-\right)\right)\cong \mathrm{Hom}_{\mathsf{C}}\left(\varinjlim F,-\right).
\]
}
\pf{
According to \Cref{th:universal_element_characterization}, $\mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\Delta\left(-\right), F\right)$ is representable by the universal element $(A,u)$ if and only if $(A,u)$ is initial in $\int_{\mathsf{C}^{\mathrm{op}}}\mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\Delta\left(-\right), F\right)$, which is equivalent to saying that for any $C\in\mathrm{Ob}\left(\mathsf{C}\right)$ and $(h_i)\in \mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\Delta C, F\right)$, there is a unique morphism $f:C\to A$ in $\mathsf{C}$ such that $h_i=f^*(u_i)=u_i\circ f$ for each $i\in \mathrm{Ob}(\mathsf{J})$. This is exactly the universal property of limit.
}

\dfn{Complete Category}{
A category $\mathsf{C}$ is \textbf{complete} if it has all small limits. That is, for any functor $F:\mathsf{J}\to \mathsf{C}$ with $\mathsf{J}$ small, $\varprojlim F$ exists.
Expand Down Expand Up @@ -1303,7 +1432,7 @@ \section{Limit and Colimit}
\end{enumerate}
}

\prop{Limits commute with limits}{
\prop{Limits Commute with Limits}{
Let $\mathsf{C}$ be a category and $F:\mathsf{I}\times \mathsf{J}\to\mathsf{C}$ be a diagram. If $\varprojlim\limits_{i\in \mathsf{I}}\varprojlim\limits_{j\in \mathsf{J}}F(i,j)$ and $\varprojlim\limits_{j\in \mathsf{J}}\varprojlim\limits_{i\in \mathsf{I}}F(i,j)$ exist, then they are naturally isomorphic.
}

Expand Down Expand Up @@ -1369,7 +1498,7 @@ \section{Limit and Colimit}
\varprojlim F&\cong\left\{(x_i)_{i\in \mathrm{Ob}(\mathsf{J})}\in \prod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i)\;\midv \;\forall \lambda:i\to j\text{ in }\mathrm{Hom}_\mathsf{J}(i,j),\,F(\lambda)(x_i)=x_j\right\}
\end{align*}
where $\alpha$ and $\beta$ are defined by
and the map $\phi_i:\varprojlim F\to F(i)$ is given by the composition
and the map $\ell_i:\varprojlim F\to F(i)$ is given by the composition
\[
\varprojlim F\xrightarrow{\quad\iota\quad}\prod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i)\xrightarrow{\quad\pi_i\quad}F(i).
\]
Expand Down Expand Up @@ -1406,7 +1535,7 @@ \section{Limit and Colimit}
\[
\varinjlim F\cong\coprod_{i\in \mathrm{Ob}(\mathsf{J})}F(i)/\sim
\]
and the map $\phi_i:F(i)\to \varinjlim F$ is given by the composition
and the map $\ell_i:F(i)\to \varinjlim F$ is given by the composition
\[
F(i)\xrightarrow{\quad\iota_i\quad}\coprod_{i\in \mathrm{Ob}(\mathsf{J})}F(i)\xrightarrow{\quad\pi\quad}\coprod_{i\in \mathrm{Ob}(\mathsf{J})}F(i)/\sim.
\]
Expand Down Expand Up @@ -1447,7 +1576,7 @@ \section{Adjoint Functor}
which means for any $X\in \mathrm{Ob}(\mathsf{C})$ and $Y\in \mathrm{Ob}(\mathsf{D})$, there is a bijection
\begin{align*}
\phi_{X,Y}:\mathrm{Hom}_{\mathsf{D}}\left(L(X),Y\right)&\xlongrightarrow{\sim} \mathrm{Hom}_{\mathsf{C}}\left(X,R(Y)\right)\\
\Big(L(X)\xlongrightarrow{f}Y\Big)&\longmapsto \Big(X\xlongrightarrow{\tilde{f}}R(Y)\Big)
\Big(L(X)\xlongrightarrow{f}Y\Big)&\longmapsto \Big(X\xlongrightarrow{f^{\triangleright}}R(Y)\Big)
\end{align*}
natural in $X$ and $Y$. $L$ is called the \textbf{left adjoint} of $R$, and $R$ is called the \textbf{right adjoint} of $L$. We write $L\dashv R$ to denote that $L$ is left adjoint to $R$.
}
Expand Down

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