diff --git a/.gitignore b/.gitignore index 36701c9..3679c45 100644 --- a/.gitignore +++ b/.gitignore @@ -6,5 +6,6 @@ *.synctex(busy) *.toc *.thm +*.auxlock .history/* \ No newline at end of file diff --git a/algebraic_construction.tex b/algebraic_construction.tex index b3f1477..b6050c4 100644 --- a/algebraic_construction.tex +++ b/algebraic_construction.tex @@ -1,6 +1,9 @@ \documentclass{report} \input{preamble} + +%\tikzexternalize % activate! + \usepackage{chemfig} \usepackage{relsize} \usepackage{multirow} @@ -43,8 +46,16 @@ } \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} @@ -59,6 +70,7 @@ \vspace{5in} {\large Latest Update: \today} \end{center} +\end{titlepage} \makeatletter \MHInternalSyntaxOn @@ -74,12 +86,11 @@ \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} @@ -106,6 +117,8 @@ \chapter*{Notation Conventions} \chapter{Set Theory} +\thispagestyle{empty} +\setcounter{page}{1} \section{Set} \subsection{Basic Operations} \dfn{Family of Sets}{ @@ -523,13 +536,39 @@ \section{Category} 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 \[ \begin{tikzcd}[ampersand replacement=\&] \mathsf{C} \arrow[r, "\mathrm{id}_{\mathsf{C}}"] \& \mathsf{C} \& \mathsf{1} \arrow[l, "\Delta X"'] @@ -545,7 +584,7 @@ \section{Category} \] } \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}}"'] @@ -555,7 +594,7 @@ \section{Category} \[ \begin{tikzcd}[ampersand replacement=\&, row sep=9pt] \& C \arrow[dd, "f"]\\ - X\arrow[rd, "h"']\arrow[ru, "h^{\prime}"]\& \\ + X\arrow[ru, "h"]\arrow[rd, "h^{\prime}"']\& \\ \& C^{\prime} \end{tikzcd} \] @@ -585,7 +624,7 @@ \section{Category} } \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=\&] @@ -1138,6 +1177,94 @@ \section{Representable Functor} 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 @@ -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 $1\in \mathrm{Ob}(\mathsf{V})$ @@ -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 @@ -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 $\sqrt{I^n}=\sqrt{I}$, $\sqrt{I J}=\sqrt{I \cap J}=\sqrt{I} \cap \sqrt{J}$ @@ -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} } @@ -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. diff --git a/preamble.tex b/preamble.tex index de2831d..bf74f37 100644 --- a/preamble.tex +++ b/preamble.tex @@ -68,8 +68,8 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % SELF MADE COLORS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\usetikzlibrary{ shapes.geometric } +\usetikzlibrary{external} +\usetikzlibrary{shapes.geometric} \usetikzlibrary{calc} \usepackage{anyfontsize}