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add Adjoint Functor
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hooyuser committed Jan 11, 2024
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*.synctex(busy)
*.toc
*.thm
*.auxlock

.history/*
175 changes: 164 additions & 11 deletions algebraic_construction.tex
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\documentclass{report}

\input{preamble}

%\tikzexternalize % activate!

\usepackage{chemfig}
\usepackage{relsize}
\usepackage{multirow}
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}

\newcolumntype{P}[1]{>{\centering\arraybackslash}p{#1}}
\newcommand{\bld}[1]{\mbox{\boldmath $#1$}}
\newcommand{\spec}{\operatorname{Spec}}
\newcommand{\midv}{\,\middle\vert\,}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Start document
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\begin{titlepage}
\begin{center}
~\\
\vspace{6em}
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\vspace{5in}
{\large Latest Update: \today}
\end{center}
\end{titlepage}

\makeatletter
\MHInternalSyntaxOn
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\kern-\wd0 \lower.55ex\box0}}
\MHInternalSyntaxOff
\makeatother
\newcommand{\spec}{\operatorname{Spec}}
\newcommand{\midv}{\,\middle\vert\,}

\newpage
% table of contents
\tableofcontents

\thispagestyle{empty}
% Your document content here

\chapter*{Notation Conventions}
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\chapter{Set Theory}
\thispagestyle{empty}
\setcounter{page}{1}
\section{Set}
\subsection{Basic Operations}
\dfn{Family of Sets}{
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S\left(A^{\prime}\right) \arrow[r, "h^{\prime}"'] \& T\left(B^{\prime}\right)
\end{tikzcd}
\]


Morphisms are composed by taking $\left(f^{\prime}, g^{\prime}\right) \circ(f, g)$ to be $\left(f^{\prime} \circ f, g^{\prime} \circ g\right)$, whenever the latter expression is defined. The identity morphism on an object $(A, B, h)$ is $\left(\operatorname{id}_A, \operatorname{id}_B\right)$.
\end{itemize}
}
\dfn{Comma Category with Constant Functor from $\mathsf{1}$}{
Suupose $\mathsf{A}, \mathsf{B}$, and $\mathsf{C}$ are categories and $X\in\mathrm{Ob}\left(\mathsf{C}\right)$. The comma category $(\Delta X\downarrow \mathsf{B})$ for the following pair of functors
$$
\mathsf{1} \xlongrightarrow{\Delta X} \mathsf{C} \xlongleftarrow{T} \mathsf{B}
$$
can be written as $\left(X\downarrow \mathsf{B}\right)$ for short. And morphisms in $\left(X\downarrow \mathsf{B}\right)$ can be simplified to
commutative triangles
\[
\begin{tikzcd}[ampersand replacement=\&, row sep=9pt]
\& T(B) \arrow[dd, "T(g)"]\\
X\arrow[ru, "h"]\arrow[rd, "h^{\prime}"']\& \\
\& T\left(B^{\prime} \right)
\end{tikzcd}
\]
Similarly, the comma category $(\mathsf{A}\downarrow \Delta X)$ for the following pair of functors
$$
\mathsf{A} \xlongrightarrow{S} \mathsf{C} \xlongleftarrow{\Delta X} \mathsf{1}
$$
can be written as $\left(\mathsf{A}\downarrow X\right)$ for short. And morphisms in $\left(\mathsf{A}\downarrow X\right)$ can be simplified to
commutative triangles
\[
\begin{tikzcd}[ampersand replacement=\&, row sep=9pt]
S(A) \arrow[rd, "h"] \arrow[dd, "S(f)"'] \& \\
\& X\\
S\left(A^{\prime}\right) \arrow[ru, "h^{\prime}"'] \&
\end{tikzcd}
\]
}
\dfn{Slice Category}{
Suppose $\mathsf{C}$ is a category and $X\in\mathrm{Ob}(\mathsf{C})$. The \textbf{slice category} $\left(\mathsf{C} / X\right)$ is the comma category $(\mathrm{id}_{\mathsf{C}} \downarrow\Delta X)$, where functors are illustrated as follows
Suppose $\mathsf{C}$ is a category and $X\in\mathrm{Ob}(\mathsf{C})$. The \textbf{slice category} $\left(\mathsf{C} / X\right)$ is the comma category $(\mathrm{id}_{\mathsf{C}} \downarrow X)$, where functors are illustrated as follows
\[
\begin{tikzcd}[ampersand replacement=\&]
\mathsf{C} \arrow[r, "\mathrm{id}_{\mathsf{C}}"] \& \mathsf{C} \& \mathsf{1} \arrow[l, "\Delta X"']
Expand All @@ -545,7 +584,7 @@ \section{Category}
\]
}
\dfn{Coslice Category}{
Suppose $\mathsf{C}$ is a category and $X\in\mathrm{Ob}(\mathsf{C})$. The \textbf{coslice category} $\left(X / \mathsf{C}\right)$ is the comma category $(\Delta X \downarrow\mathrm{id}_{\mathsf{C}})$, where functors are illustrated as follows
Suppose $\mathsf{C}$ is a category and $X\in\mathrm{Ob}(\mathsf{C})$. The \textbf{coslice category} $\left(X / \mathsf{C}\right)$ is the comma category $(X \downarrow\mathrm{id}_{\mathsf{C}})$, where functors are illustrated as follows
\[
\begin{tikzcd}[ampersand replacement=\&]
\mathsf{1} \arrow[r, "\Delta X"] \& \mathsf{C} \& \mathsf{C} \arrow[l, "\mathrm{id}_{\mathsf{C}}"']
Expand All @@ -555,7 +594,7 @@ \section{Category}
\[
\begin{tikzcd}[ampersand replacement=\&, row sep=9pt]
\& C \arrow[dd, "f"]\\
X\arrow[rd, "h"']\arrow[ru, "h^{\prime}"]\& \\
X\arrow[ru, "h"]\arrow[rd, "h^{\prime}"']\& \\
\& C^{\prime}
\end{tikzcd}
\]
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}

\dfn[category_of_elements]{Category of Elements}{
Let $\mathsf{C}$ be a category and let $F: \mathsf{C} \rightarrow \mathsf{Set}$ be a functor. The \textbf{category of elements of $F$} is the the \hyperref[th:comma_category]{comma category} $\left(\Delta\{*\} \downarrow F\right)$
Let $\mathsf{C}$ be a category and let $F: \mathsf{C} \rightarrow \mathsf{Set}$ be a functor. The \textbf{category of elements of $F$} is the the \hyperref[th:comma_category]{comma category} $\left(\{*\} \downarrow F\right)$
where functors are illustrated as follows
\[
\begin{tikzcd}[ampersand replacement=\&]
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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 $\Delta\{*\}$.
}
\section{Adjoint Functor}
\dfn{Adjoint Pair of Functors}{
An \textbf{adjoint pair of functors} is a tuple $\left(L,R,\varphi\right)$ consisting of a pair of functors $\begin{tikzcd}[ampersand replacement=\&]
\mathsf{C} \arrow[r, "L", bend left] \& \mathsf{D} \arrow[l, "R", bend left]
\end{tikzcd}$
and a natural isomorphism
\[
\begin{tikzcd}[ampersand replacement=\&]
\mathsf{C}^{\mathrm{op}}\times\mathsf{D} \arrow[r, "\mathrm{Hom}_{\mathsf{D}}\left(L(-){,}-\right)"{name=A, above}, bend left] \arrow[r, "\mathrm{Hom}_{\mathsf{C}}\left(-{,}R(-)\right)"'{name=B, below}, bend right] \&[+50pt] \mathsf{Set}
\arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "\varphi", "\sim\hspace{1.5pt}"']
\end{tikzcd}
\]
which means for any $X\in \mathrm{Ob}(\mathsf{C})$ and $Y\in \mathrm{Ob}(\mathsf{D})$, there is a bijection
\begin{align*}
\varphi_{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)
\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 $R$.
}
The naturality square of $\varphi$ means that for any morphism $g:X_2\to X_1$ in $\mathsf{C}$ and $h:Y_1\to Y_2$ in $\mathsf{D}$, the following diagram commutes
\[
\begin{tikzcd}[ampersand replacement=\&]
\mathrm{Hom}_{\mathsf{D}}\left(L(X_1),Y_1\right) \arrow[r, "{\mathrm{Hom}_{\mathsf{D}}\left(L(g){,}h\right)}"] \arrow[d, "\varphi_{X_1,Y_1}"', "\sim"] \&[+50pt] \mathrm{Hom}_{\mathsf{D}}\left(L(X_2),Y_2\right) \arrow[d, "\varphi_{X_2,Y_2}", "\sim"'] \\[+20pt]
\mathrm{Hom}_{\mathsf{C}}\left(X_1,R(Y_1)\right) \arrow[r, "{\mathrm{Hom}_{\mathsf{C}}\left(g{,}R(h)\right)}"'] \& \mathrm{Hom}_{\mathsf{C}}\left(X_2,R(Y_2)\right)
\end{tikzcd}
\]
which in turn can be explicitly written as for any $f:L(X_1)\to Y_1$ in $\mathsf{D}$,
\[
\varphi_{X_2,Y_2}\left(h\circ f\circ L(g)\right)=R(h)\circ\varphi_{X_1,Y_1}(f)\circ g.
\]

\dfn{Adjunction Unit and Counit}{
Let $\left(L,R,\varphi\right)$ be an adjoint pair of functors. The \textbf{adjunction unit} $\eta$ of this adjunction is a natural transformation
\begin{center}
\begin{tikzcd}[ampersand replacement=\&]
\mathsf{C} \arrow[r, "\mathrm{id}_{\mathsf{C}}"{name=A, above}, bend left=40] \arrow[r, "R\circ L"'{name=B, below}, bend right=40]
\&[+25pt] \mathsf{C}
\arrow[Rightarrow, shorten <=3.5pt, shorten >=3.5pt, from=A.south-|B, to=B, "\eta"]
\end{tikzcd}
\hspace{3cm}
\begin{tikzpicture}[x=0.6cm,y=0.6cm, baseline=(current bounding box.center)]
\fill[fill=blue!20] (-2.5,0) rectangle (2.5,2.8);
\filldraw[fill=green!20] (-1,2) arc (180:360:1);
\fill[fill=green!20] (-1,2) rectangle (1,2.8);
\node[above=2pt] at (-1,2.8) {$F$};
\node[above=2pt] at (1,2.8) {$G$};
\node[above=2pt] at (0,1) {$\eta$};
\draw[fill=black] (0, 1) circle (0.07);
\draw (-1,2) -- (-1,2.8);
\draw (1,2) -- (1,2.8);
\end{tikzpicture}
\end{center}
defined by $\eta_X:=\varphi_{X,L(X)}(\mathrm{id}_{L(X)})$ for any $X\in \mathrm{Ob}(\mathsf{C})$, where $\varphi_{X,L(X)}$ is the natural bijection
$$
\begin{aligned}
\varphi_{X,L(X)}:\operatorname{Hom}_{\mathsf{D}}\left(L(X), L(X)\right) & \xlongrightarrow{\sim}\operatorname{Hom}_{\mathsf{C}}(X, RL(X)) \\
\operatorname{id}_{F X} & \longmapsto \eta_X
\end{aligned}
$$
The \textbf{adjunction counit} $\varepsilon$ of this adjunction is a natural transformation
\begin{center}
\begin{tikzcd}[ampersand replacement=\&]
\mathsf{D} \arrow[r, "\scalebox{1.2}{$L\circ R$}"{name=A, above}, bend left=40] \arrow[r, "\scalebox{1.2}{$\mathrm{id}_{\mathsf{D}}$}"'{name=B, below}, bend right=40]
\&[+25pt] \mathsf{D}
\arrow[Rightarrow, shorten <=3.5pt, shorten >=3.5pt, from=A.south-|B, to=B, "\varepsilon"]
\end{tikzcd}
\hspace{3cm}
\begin{tikzpicture}[x=0.6cm,y=0.6cm, baseline=(current bounding box.center)]
\fill[fill=blue!20] (-2.5,0) rectangle (2.5,2.8);
\filldraw[fill=green!20] (-1,2) arc (180:360:1);
\fill[fill=green!20] (-1,2) rectangle (1,2.8);
\node[above=2pt] at (-1,2.8) {$F$};
\node[above=2pt] at (1,2.8) {$G$};
\node[above=2pt] at (0,1) {$\eta$};
\draw[fill=black] (0, 1) circle (0.07);
\draw (-1,2) -- (-1,2.8);
\draw (1,2) -- (1,2.8);
\end{tikzpicture}
\end{center}
}
\pf{
The naturality square of $\eta$ means that for any morphism $g:X_1\to X_2$ in $\mathsf{C}$, the following diagram commutes
\[
\begin{tikzcd}[ampersand replacement=\&]
X_1 \arrow[d, "{\eta_{X_1}}"'] \arrow[r, "g"] \&[+50pt]X_2\arrow[d, "{\eta_{X_2}}"] \\[+20pt]
R(L(X_1))\arrow[r, "R(L(g))"']\& R(L(X_2))
\end{tikzcd}
\]
}
\section{Monoidal Category}
\dfn{Monoidal Category}{
A monoidal category is a category $\mathsf{V}$ equipped with
Expand All @@ -1157,7 +1284,7 @@ \section{Monoidal Category}
\[
\begin{tikzcd}[ampersand replacement=\&]
\mathsf{V}\times\mathsf{V}\times\mathsf{V} \arrow[r, "(-\otimes-)\otimes-"{name=A, above}, bend left] \arrow[r, "-\otimes(-\otimes-)"'{name=B, below}, bend right] \&[+30pt] \mathsf{V}
\arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "a"]
\arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "a", "\sim\hspace{1pt}"']
\end{tikzcd}
\]
\item Unit object: an object $1\in \mathrm{Ob}(\mathsf{V})$
Expand Down Expand Up @@ -1473,6 +1600,30 @@ \subsection{Free Object}
The identity element is the empty word. The inverse of a word is obtained by reversing the order of the letters and replacing each letter by its inverse.
}

\prop{Free-Forgetful Adjunction}{
The free group functor $\mathrm{Free}_{\mathsf{Grp}}$ is left adjoint to the forgetful functor $U:\mathsf{Grp}\to \mathsf{Set}$
$$
\begin{tikzcd}[ampersand replacement=\&]
\mathsf{Set} \arrow[rr, "\mathrm{Free}_{\mathsf{Grp}}", bend left] \&[-10pt]\bot\&[-10pt] \mathsf{Grp} \arrow[ll, "U", bend left]
\end{tikzcd}
$$
The adjunction isomorphism is given by
\begin{align*}
\varphi_{S,G}:\mathrm{Hom}_{\mathsf{Grp}}(\mathrm{Free}_{\mathsf{Grp}}(S),G)&\xlongrightarrow{\sim} \mathrm{Hom}_{\mathsf{Set}}(S,U(G))\\
g&\longmapsto g\circ \iota
\end{align*}
}
\pf{
First we show that $\varphi_{S,G}$ is injective. Suppose $g_1,g_2:\mathrm{Free}_{\mathsf{Grp}}(S)\to G$ are two group homomorphisms such that $g_1\circ \iota=g_2\circ \iota$. By the universal property of free group, we have $g_1=g_2$. Then we show that $\varphi_{S,G}$ is surjective. Suppose $f:S\to U(G)$ is a function. By the universal property there exists a group homomorphism $\widetilde{f}:\mathrm{Free}_{\mathsf{Grp}}(S)\to G$ such that $\varphi_{S,G}(\widetilde{f})=\widetilde{f}\circ \iota=f$. Finally, we show that $\varphi_{S,G}$ is natural in $S$ and $G$. Suppose $h:S_1\to S_2$ is a function and $q:G_1\to G_2$ is a group homomorphism. Then we can check that for any $g\in \mathrm{Hom}_{\mathsf{Grp}}(\mathrm{Free}_{\mathsf{Grp}}(S_2),G_2)$,
\begin{align*}
\varphi_{S_1,G_1}(q\circ g\circ \iota_{S_1})&=(q\circ g\circ \iota_{S_1})\circ \iota_{S_1}\\
&=q\circ g\circ (\iota_{S_1}\circ \iota_{S_1})\\
&=q\circ g\circ \iota_{S_2}\\
&=\varphi_{S_2,G_2}(g\circ \iota_{S_2}).
\end{align*}

}

\subsection{Inverse Limit}
\dfn[inverse_limit_of_groups]{Inverse Limit in $\mathsf{Grp}$}{
Let $\mathsf{I}$ be a \hyperref[th:filtered_category]{filtered} \hyperref[th:thin_category]{thin category} and $F:\mathsf{I}^{\mathrm{op}}\to \mathsf{Grp}$ be a functor. To unpack the information of $F$, denote $I:=\mathrm{Ob}(\mathsf{I})$, $G_i:=F(i)$ and $f_{ij}:=F(i\to j)$. An \textbf{inverse system} is a pair $\left(\left(G_i\right)_{i \in I},\left(f_{i j}\right)_{i \leq j \in I}\right)$ where $f_{i j}: G_{j} \rightarrow G_{i}$ is a group homomorphism for each $i \leq j$ such that
Expand Down Expand Up @@ -2670,6 +2821,7 @@ \subsection{Ideals}
\end{gathered}
$$
\item $I(J K)=(I J) K$
\item $(a)^n=(a^n)$
\item $I^0 \supseteq \sqrt{I} \supseteq I \supseteq I^2 \supseteq I^3 \supseteq \cdots$
\item $\sqrt{\sqrt{I}} = \sqrt{I}$,
\item $\sqrt{I^n}=\sqrt{I}$, $\sqrt{I J}=\sqrt{I \cap J}=\sqrt{I} \cap \sqrt{J}$
Expand All @@ -2678,6 +2830,7 @@ \subsection{Ideals}
\proof{
\begin{enumerate}[(i)]
\item Since $\{ab\mid a\in I,b\in J\}\subseteq I\cap J$, we see $IJ=\left(\{ab\mid a\in I,b\in J\}\right)\subseteq I\cap J$. Also we can check $I \cap J \subseteq I \cup J\subseteq (I \cup J)=I+J$.
\item[(vi)] If $x\in(a)^n$, then $x=r_1(r_2a)^n=r_1r_2^na^n\in(a^n)$. If $y\in(a^n)$, then $y=ra^n\in(a)^n$.
\end{enumerate}
}

Expand Down Expand Up @@ -2711,7 +2864,7 @@ \subsection{Ideals}


\prop{Properties of Radical Ideal}{
\begin{enumerate}
\begin{enumerate}[(i)]
\item For any ideal $I$, $\sqrt{0}\subseteq \sqrt{I}$.
\item $\sqrt{I}$ is the smallest radical ideal containing $I$.
\item $\sqrt{\mathfrak{p}^n}=\sqrt{\mathfrak{p}}=\mathfrak{p}$ for any prime ideal $\mathfrak{p}$, which means prime ideals are radical.
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4 changes: 2 additions & 2 deletions preamble.tex
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SELF MADE COLORS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\usetikzlibrary{ shapes.geometric }
\usetikzlibrary{external}
\usetikzlibrary{shapes.geometric}
\usetikzlibrary{calc}
\usepackage{anyfontsize}

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