diff --git a/algebraic_construction.tex b/algebraic_construction.tex index 8954a43..c4c3fa2 100644 --- a/algebraic_construction.tex +++ b/algebraic_construction.tex @@ -104,15 +104,16 @@ \chapter*{Notation Conventions} \end{itemize} We use sans-serif font for categories. Some common categories are \begin{itemize} + \item $\mathsf{FinSet}$: the category of finite sets. \item $\mathsf{Set}$: the category of sets. \item $\mathsf{Mon}$: the category of monoids. \item $\mathsf{Grp}$: the category of groups. \item $\mathsf{Ab}$: the category of abelian groups. \item $\mathsf{Ring}$: the category of rings. \item $\mathsf{CRing}$: the category of commutative rings. - \item $\mathsf{Fld}$: the category of fields. + \item $\mathsf{Field}$: the category of fields. \item $R\text{-}\mathsf{Mod}$: the category of left $R$-modules, where $R\in \mathrm{Ob}\left(\mathsf{Ring}\right)$. - \item $K\text{-}\mathsf{Vect}$: the category of $K$-vector spaces, where $K\in \mathrm{Ob}\left(\mathsf{Fld}\right)$. + \item $K\text{-}\mathsf{Vect}$: the category of $K$-vector spaces, where $K\in \mathrm{Ob}\left(\mathsf{Field}\right)$. \item $R\text{-}\mathsf{Alg}$: the category of associative $R$-algebras, where $R\in \mathrm{Ob}\left(\mathsf{CRing}\right)$. \item $R\text{-}\mathsf{CAlg}$: the category of commutative $R$-algebras, where $R\in \mathrm{Ob}\left(\mathsf{CRing}\right)$. \item $\mathsf{Top}$: the category of topological spaces. diff --git a/category_theory.tex b/category_theory.tex index 2d4284d..158b181 100644 --- a/category_theory.tex +++ b/category_theory.tex @@ -7,7 +7,7 @@ \section{Category} \item $\mathsf{C}$ is \textbf{locally $\mathscr{U}$-small} if $\mathrm{Hom}_{\mathsf{C}}(X,Y)$ is a $\mathscr{U}$-small set for all $X,Y\in \mathrm{Ob}(\mathsf{C})$. \item $\mathsf{C}$ is \textbf{$\mathscr{U}$-small} if $\mathsf{C}$ is locally $\mathscr{U}$-small and $\mathrm{Ob}(\mathsf{C})$ is a \hyperref[th:small_set]{$\mathscr{U}$-small set}, \end{itemize} - We can simply say $\mathsf{C}$ is \textbf{small} or \textbf{locally small} if the choice of $\mathscr{U}$ can be understood from the context. + We can simply say $\mathsf{C}$ is \textbf{small} or \textbf{locally small} if the choice of $\mathscr{U}$ can be inferred from the context. } \dfn{Isomorphism of Categories}{ Suppose $\mathsf{C}$ and $\mathsf{D}$ are two categories. A \textbf{functor} $F:\mathsf{C}\to \mathsf{D}$ is called an \textbf{isomorphism} if there exists a functor $G:\mathsf{D}\to \mathsf{C}$ such that $F\circ G=\mathrm{id}_{\mathsf{D}}$ and $G\circ F=\mathrm{id}_{\mathsf{C}}$. In this case, we say $\mathsf{C}$ and $\mathsf{D}$ are \textbf{isomorphic} and write $\mathsf{C}\cong \mathsf{D}$. @@ -44,6 +44,28 @@ \section{Category} \end{tabular} \end{table} +\dfn{Subcategory}{ + Suppose $\mathsf{C}$ is a category. A \textbf{subcategory} $\mathsf{D}$ of $\mathsf{C}$ is a category such that + \begin{itemize} + \item $\mathrm{Ob}(\mathsf{D})\subseteq \mathrm{Ob}(\mathsf{C})$. + \item For any $X,Y\in \mathrm{Ob}(\mathsf{D})$, $\mathrm{Hom}_{\mathsf{D}}(X,Y)\subseteq \mathrm{Hom}_{\mathsf{C}}(X,Y)$. + \item For any $X\in \mathrm{Ob}(\mathsf{D})$, the identity morphism $\mathrm{id}_X$ of $\mathsf{D}$ is the same as the identity morphism $\mathrm{id}_X$ of $\mathsf{C}$. + \item For any $X,Y,Z\in \mathrm{Ob}(\mathsf{D})$ and $g\in\mathrm{Hom}_{\mathsf{D}}\left(X,Y\right)$, $f\in\mathrm{Hom}_{\mathsf{D}}\left(Y,Z\right)$, $f\circ_{\mathsf{D}}g=f\circ_{\mathsf{C}}g$. + \end{itemize} + A subcategory $\mathsf{D}$ of $\mathsf{C}$ is called a \textbf{full subcategory} if for any $X,Y\in \mathrm{Ob}(\mathsf{D})$, $\mathrm{Hom}_{\mathsf{D}}(X,Y)= \mathrm{Hom}_{\mathsf{C}}(X,Y)$. +} +There is a looser way to think about a subcategory: $\mathsf{D}$ is a subcategory of $\mathsf{C}$ if there exists a fully faithful functor $F:\mathsf{D}\to \mathsf{C}$ that is injective on objects.\\ + +\ex{Examples of Full Subcategories}{ + Here are some examples of full subcategories: + \begin{itemize} + \item $\mathsf{Ab}\subseteq\mathsf{Grp}\subseteq\mathsf{Mon}$ + \item $\mathsf{Field}\subseteq\mathsf{CRing}\subseteq\mathsf{Ring}$ + \item $K\text{-}\mathsf{Vect}\subseteq K\text{-}\mathsf{Mod}$ + \item $R\text{-}\mathsf{CAlg}\subseteq R\text{-}\mathsf{Alg}$ + \end{itemize} +} + \dfn{Opposite Category}{ Suppose $\mathsf{C}$ is a category. The \textbf{opposite category} $\mathsf{C}^{\mathrm{op}}$ is defined as follows \begin{itemize} @@ -228,13 +250,12 @@ \section{Category} 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 +\dfn[universal_morphism]{Universal Morphism}{ + Suupose $\mathsf{A}, \mathsf{B}$, and $\mathsf{C}$ are categories and $X\in\mathrm{Ob}\left(\mathsf{C}\right)$. The comma category $(\Delta X\downarrow T)$ 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 +can be written as $\left(X\downarrow T\right)$ for short. Morphisms in $\left(X\downarrow \mathsf{B}\right)$ can be simplified to commutative triangles \[ \begin{tikzcd}[ampersand replacement=\&, row sep=9pt] \& T(B) \arrow[dd, "T(g)"]\\ @@ -242,12 +263,13 @@ \section{Category} \& T\left(B^{\prime} \right) \end{tikzcd} \] +We say $\left(Y,X\xrightarrow{u} T(Y)\right)$ is a \textbf{universal morphism from $X$ to $T$} if it is initial in $\left(X\downarrow T\right)$. + 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 +can be written as $\left(\mathsf{A}\downarrow X\right)$ for short. Morphisms in $\left(\mathsf{A}\downarrow X\right)$ can be simplified to commutative triangles \[ \begin{tikzcd}[ampersand replacement=\&, row sep=9pt] S(A) \arrow[rd, "h"] \arrow[dd, "S(f)"'] \& \\ @@ -255,7 +277,11 @@ \section{Category} S\left(A^{\prime}\right) \arrow[ru, "h^{\prime}"'] \& \end{tikzcd} \] +We say $\left(Y,S(Y)\xrightarrow{u} X\right)$ is a \textbf{universal morphism from $S$ to $X$} if it is terminal in $\left(\mathsf{A}\downarrow X\right)$. } + + + \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 $(\mathrm{id}_{\mathsf{C}} \downarrow X)$, where functors are illustrated as follows \[ @@ -289,6 +315,10 @@ \section{Category} \] } +\dfn{Category of Pointed Objects}{ + Suppose $\mathsf{C}$ is a category with terminal object $\bullet$. The coslice category $\left(\bullet / \mathsf{C}\right)$ is called the \textbf{category of pointed objects} of $\mathsf{C}$ and is denoted by $\mathsf{C}_\bullet$. +} + \ex{$\mathsf{Disc}$ Functor}{ Given a set $X$, we can define a functor $\mathsf{Disc}:\mathsf{Set}\to \mathsf{Cat}$ as follows \[ @@ -312,6 +342,19 @@ \section{Category} It is easy to check that the functor $\mathsf{Disc}$ is fully faithful. } +\dfn{Grothendieck Construction}{ + Let $\mathsf{C}$ be a category and let $F: \mathsf{C} \rightarrow \mathsf{Cat}$ be a functor from any small category to the category of small categories. The \textbf{Grothendieck construction} of $F$ is a category $\int_{\mathsf{C}} F$ (also written $\Gamma(F)$) defined as follows: + \begin{itemize} + \item Objects are pairs $(A, a)$ where $A \in \mathrm{Ob}(\mathsf{C})$ and $a \in F(A)$. + \item Morphisms $(A, a) \rightarrow(B, b)$ are pairs $(f, g)$ where $f: A \rightarrow B$ is a morphism in $\mathsf{C}$ and $g: F(f)(a) \rightarrow b$ is a morphism in $F(B)$. + \end{itemize} + The composition of morphisms is defined as follows: + \[ + (f', g') \circ(f, g)=\left(f' \circ f, g'\circ \left(F(f')(g)\right)\right) + \] + where $F(f')(g): F(f')(F(f)(a)) \rightarrow F(f')(b)$ is the morphism induced by $g$ under the functor $F(f')$. +} + \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} $\left(\{*\} \downarrow F\right)$ where functors are illustrated as follows @@ -338,6 +381,8 @@ \section{Category} \end{tikzcd} \] } +By viewing sets in $\mathsf{Set}$ as discrete categories, we have an inclusion $\mathsf{Set}\hookrightarrow\mathsf{Cat}$. Hence the category of elements of $F$ is a special case of the Grothendieck construction when the image of $F$ on $\mathrm{Ob}(\mathsf{C})$ on + \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. @@ -429,7 +474,7 @@ \subsubsection{Composition of Functors} \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[fill=midcolor, rounded corners=0.6cm, line width=0.7pt] (1,1) rectangle (3, 3); % Draw vertical line and point \draw[line width=0.7pt] (2,0) -- (2,1); @@ -556,7 +601,7 @@ \subsubsection{Vertical Composition} % Label axes \draw (2,-2) node[below] {$H$}; - \draw (2,0) node[left] {$G$}; + \draw (2,0) node[right] {$G$}; \draw (2,2) node[above] {$F$}; \end{scope} @@ -720,80 +765,112 @@ \subsection{Morphism as Natural Transformation} 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; + \begin{tikzpicture}[x=0.8cm,y=0.8cm, 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} [shift={(-3.5,0)}] + % Draw background + \fill[leftcolor] (-0.5,0) rectangle (1,3); + \fill[rightcolor] (1,0) rectangle (2.5,3); + + % Draw vertical line and point + \draw[line width=0.7pt] (1,0) -- (1,3); + \filldraw [black] (1,1) circle (2pt) node[left=2pt] {$G(f)$}; + \filldraw [black] (1,2) circle (2pt) node[left=2pt] {$\varphi_X$}; + % Label axes + \draw (1,0) node[below] {$G(Y)$}; + \draw (1,3) node[above] {$F(X)$}; + \draw (3,1.5) node {$=$}; \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; + % 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=0.8cm, line width=0.7pt] (0,2) rectangle (2, 5); + \draw[fill=blue!20, rounded corners=0.8cm, line width=0.7pt] (0,2) rectangle (2, -1); + \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$}; + \draw (3.5,1.5) node {$=$}; \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} -\] + + \begin{scope} [shift={(4,0)}] + % 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] {$F$}; + \draw (3.5,1.5) node {$=$}; + \end{scope} + + \begin{scope}[shift={(8,0)}] + % 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=0.8cm, line width=0.7pt] (0,1) rectangle (2, 4); + \draw[fill=blue!20, rounded corners=0.8cm, 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] {$F$}; + \draw (3.5,1.5) node {$=$}; + \end{scope} + + \begin{scope} [shift={(12.5,0)}] + % Draw background + \fill[leftcolor] (-0.5,0) rectangle (1,3); + \fill[rightcolor] (1,0) rectangle (2.5,3); + + % Draw vertical line and point + \draw[line width=0.7pt] (1,0) -- (1,3); + \filldraw [black] (1,2) circle (2pt) node[left=2pt] {$F(f)$}; + \filldraw [black] (1,1) circle (2pt) node[left=2pt] {$\varphi_Y$}; + % Label axes + \draw (1,0) node[below] {$G(Y)$}; + \draw (1,3) node[above] {$F(X)$}; + \end{scope} + \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 @@ -805,193 +882,408 @@ \subsection{Morphism as Natural Transformation} % Draw vertical line and point \draw[line width=0.7pt] (2,0) -- (2,2); - \filldraw [black] (2,1) circle (2pt) node[anchor=east] {$f$}; + \filldraw [black] (2,1) circle (1.2pt) 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 $f$. +\end{tikzpicture} to the left of the string diagram of $\varphi$ is equivalent to evaluating $\varphi$ at $f:X\to Y$. -\section{Limit and Colimit} -\dfn{Cone}{ - Let $\mathsf{J},\mathsf{C}$ be categories and $F:\mathsf{J}\to\mathsf{C}$ be a functor. Consider functors +\section{Representable Functor} +\dfn{Presheaf}{ + Let $\mathsf{C}$ be a category. A \textbf{presheaf} on $\mathsf{C}$ is a functor $F:\mathsf{C}^{\mathrm{op}}\to \mathsf{Set}$. +} + +\dfn[yoneda_embedding_functor]{Yoneda Embedding Functor}{ + Let $\mathsf{C}$ be a category. The \textbf{Yoneda embedding functor} is the functor $Y_\mathsf{C}:\mathsf{C}\to \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right]$ defined as follows \[ - \begin{tikzcd}[ampersand replacement=\&] - \mathsf{C} \arrow[r, "\Delta"] \& {[\mathsf{J},\mathsf{C}]} \& \mathsf{1} \arrow[l, "\Delta F"'] - \end{tikzcd} - \] - The comma category $\left(\Delta \downarrow \Delta F\right)$ is called the \textbf{cone category from $\textsf{C}$ to $F$}, denoted by $\mathsf{Cone}(\textsf{C},F)$. - \begin{itemize} - \item Objects: The objects in $\mathsf{Cone}(\textsf{C},F)$ are all natural transformations - \[ - \begin{tikzcd}[ampersand replacement=\&] - \mathsf{J} \arrow[r, "\Delta C"{name=A, above}, bend left] \arrow[r, "F"'{name=B, below}, bend right] \&[+30pt] \mathsf{C} - \arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "h"] - \end{tikzcd} - \] - where $C\in \mathrm{Ob}(\mathsf{C})$. Such $h:\Delta C\Rightarrow F$ is called a \textbf{cone from $C$ to $F$} because it can be viewed as a family of morphisms $\left(h_i:C\to F(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ in $\mathsf{C}$ such that the following diagram commutes for each morphism $\lambda:i\to j$ in $\mathsf{J}$ - \[ - \begin{tikzcd}[ampersand replacement=\&] - \& C \arrow[ld, "h_i"'] \arrow[rd, "h_j"] \& \\ - F(i) \arrow[rr, "F(\lambda)"'] \& \& F(j) + \begin{tikzcd}[ampersand replacement=\&] + \mathsf{C} % column 1 + \&[-25pt] % column 2 + \&[+30pt] % column 3 + \&[-30pt] \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right] % column 4 + \&[+25pt] % column 5 + \&[-25pt] % column 6 + \& % column 7 + \\[-15pt] + Y_1 \arrow[dd, "g"{name=L, left}] \&\& \& \mathrm{Hom}_{\mathsf{C}}\left(-,Y_1\right) \arrow[dd, Rightarrow, "g_\star"{name=R}] \& \mathrm{Hom}_{\mathsf{C}}(X_1,Y_1) \arrow[dd, "{g_*}"'] \arrow[rr, "{f^*}"] \& \& \mathrm{Hom}_{\mathsf{C}}(X_2,Y_1) \arrow[dd, "g_*"] \\[-5pt] + \& \phantom{.}\arrow[r, "Y_\mathsf{C}", squigarrow]\&\phantom{.} \&\phantom{.} \&\phantom{.}\&\\[-5pt] + Y_2 \& \& \& \mathrm{Hom}_{\mathsf{C}}\left(-,Y_2\right)\& \mathrm{Hom}_{\mathsf{C}}(X_1,Y_2) \arrow[rr, "{f^*}"'] \& \& \mathrm{Hom}_{\mathsf{C}}(X_2,Y_2) + \arrow[from=4-5, to=2-5, dash, start anchor={[xshift=-8ex,yshift=-3ex]}, end anchor={[xshift=-8ex,yshift=3ex]},decorate,decoration={brace,amplitude=10pt,raise=4pt}] + \arrow[from=3-4, to=3-5, dashed, start anchor={[xshift=9pt, yshift=-2pt]}, end anchor={[xshift=-11ex, yshift=-2pt]},decorate] \end{tikzcd} - \] - \item Morphisms: The morphisms in $\mathsf{Cone}(\textsf{C},F)$ are commutative triangles shown as follows - \[ - \begin{tikzcd}[ampersand replacement=\&, row sep=9pt] - \Delta C \arrow[rd, Rightarrow, "h"] \arrow[dd, Rightarrow, "f_\bullet"'] \& \\ - \& F\\ - \Delta C' \arrow[ru, Rightarrow, "h^{\prime}"'] \& - \end{tikzcd} - \] - Equivalently, a morphism from cone $\left(h_i:C\to F(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ to cone $\left(h_i':C'\to F(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ is a morphism $f:C\to C'$ in $\mathsf{C}$ such that the following diagram commutes for each $i\in \mathrm{Ob}(\mathsf{J})$ - \[ - \begin{tikzcd}[ampersand replacement=\&, row sep=9pt] - C \arrow[rd, "h_i"] \arrow[dd, "f"'] \& \\ - \& F(i)\\ - C' \arrow[ru, "h^{\prime}_i"'] \& - \end{tikzcd} \] - \end{itemize} -} -\dfn{Limit}{ - Let $\mathsf{J},\mathsf{C}$ be categories and $F:\mathsf{J}\to\mathsf{C}$ be a functor. The \textbf{limit} of $F$, denoted by $\varprojlim F$, is the terminal object of $\mathsf{Cone}(\textsf{C},F)$. For each morphism $\lambda:i\to j$ in $\mathsf{J}$, we have the following commutative diagram + where the natural transformation $g_\star$ is defined pointwise as follows: $\left(g_\star\right)_X=g_*$ for any $X\in \mathrm{Ob}(\mathsf{C}^{\mathrm{op}})$.\\ + The contravariant version is $Y_{\mathsf{C}^{\mathrm{op}}}:\mathsf{C}^{\mathrm{op}}\to \left[\mathsf{C},\mathsf{Set}\right]$, which is defined as follows \[ - \begin{tikzcd}[ampersand replacement=\&, every arrow/.append style={-latex, line width=1.1pt}] - \& [-1.5em] \&[-1em] C \arrow[rdd, draw=cyan, "h_j"] \arrow[ldd, draw=cyan, "h_i"'] \arrow[d, dash pattern=on 4pt off 2pt, draw=arrowRed] \& [-1em] \\[+0.2cm] - \& \& \varprojlim F\arrow[ld, draw=arrowBlue, shorten <=-3pt, "\phi_i" yshift=4.5pt] \arrow[rd, draw=arrowBlue, shorten <=-4.5pt, "\phi_j"' yshift=4pt] \& \\[-0.1cm] - \& F(i) \arrow[rr, "F(\lambda)"']\& \& F(j) - \end{tikzcd} + \begin{tikzcd}[ampersand replacement=\&] + \mathsf{C}^{\mathrm{op}} % column 1 + \&[-25pt] % column 2 + \&[+30pt] % column 3 + \&[-30pt] \left[\mathsf{C},\mathsf{Set}\right] % column 4 + \&[+25pt] % column 5 + \&[-25pt] % column 6 + \& % column 7 + \\[-15pt] + X_1 \arrow[dd, "f"{name=L, left}] \&\& \& \mathrm{Hom}_{\mathsf{C}}\left(X_1,-\right) \arrow[dd, Rightarrow, "f^\star"{name=R}] \& \mathrm{Hom}_{\mathsf{C}}(X_1,Y_1) \arrow[dd, "{f^*}"'] \arrow[rr, "{g_*}"] \& \& \mathrm{Hom}_{\mathsf{C}}(X_1,Y_2) \arrow[dd, "f^*"] \\[-5pt] + \& \phantom{.}\arrow[r, "Y_{\mathsf{C}^{\mathrm{op}}}", squigarrow]\&\phantom{.} \&\phantom{.} \&\phantom{.}\&\\[-5pt] + X_2 \& \& \& \mathrm{Hom}_{\mathsf{C}}\left(X_2,-\right)\& \mathrm{Hom}_{\mathsf{C}}(X_2,Y_1) \arrow[rr, "{g_*}"'] \& \& \mathrm{Hom}_{\mathsf{C}}(X_2,Y_2) + \arrow[from=4-5, to=2-5, dash, start anchor={[xshift=-8ex,yshift=-3ex]}, end anchor={[xshift=-8ex,yshift=3ex]},decorate,decoration={brace,amplitude=10pt,raise=4pt}] + \arrow[from=3-4, to=3-5, dashed, start anchor={[xshift=9pt, yshift=-2pt]}, end anchor={[xshift=-11ex, yshift=-2pt]},decorate] + \end{tikzcd} \] + Here we use the fact $\mathrm{Hom}_{\mathsf{C}^{\mathrm{op}}}\left(-,X\right)=\mathrm{Hom}_{\mathsf{C}}\left(X,-\right)$. + } -\dfn{Cocone}{ - Let $\mathsf{J},\mathsf{C}$ be categories and $F:\mathsf{J}\to\mathsf{C}$ be a functor. Consider functors +\thm[yoneda_lemma]{Yoneda Lemma}{ + Let $\mathsf{C}$ be a locally small category. For any functor $F:\mathsf{C}^{\mathrm{op}}\to \mathsf{Set}$ and any $A\in \mathrm{Ob}(\mathsf{C})$, there is a natural bijection + \begin{align*} + q_{A,F}:\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(\operatorname{Hom}_{\mathsf{C}}\left(-,A\right), F\right) & \overset{\sim}{\longrightarrow} F(A) \\ + { \left[ \begin{tikzcd}[ampersand replacement=\&] + \mathsf{C}^{\mathrm{op}} \arrow[r, "\operatorname{Hom}_{\mathsf{C}}\left(-{,}A\right)"{name=A, above}, bend left] \arrow[r, "F"'{name=B, below}, bend right] \&[+30pt] \mathsf{Set} + \arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "\phi"] + \end{tikzcd}\right] } & \longmapsto \phi_A\left(\operatorname{id}_A\right) + \end{align*} + The naturality of $q_{A,F}$ means that \[ \begin{tikzcd}[ampersand replacement=\&] - \mathsf{1} \arrow[r, "\Delta F"] \& {[\mathsf{J},\mathsf{C}]} \&\mathsf{C} \arrow[l, "\Delta "'] + \mathsf{C}^{\mathrm{op}}\times \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right]\&[-34pt]\&[+62pt]\&[-25pt] \mathsf{Set}\&[-25pt]\&[-25pt] \\ [-15pt] + (A_1,F_1) \arrow[dd, "\left(g{,} \eta\right)"{name=L, left}] + \&[-25pt] \& [+10pt] + \& [-30pt]\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(\mathrm{Hom}_{\mathsf{C}}(-,A_1),F_1)\arrow[dd, ""{name=R}] \& \ni \& \phi \arrow[dd,mapsto]\\ [-8pt] + \& \phantom{.}\arrow[r, "\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(Y_\mathsf{C}(-){,}-\right)", squigarrow]\&\phantom{.} \& \\[-8pt] + (A_2,F_2) \& \& \& \operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(\mathrm{Hom}_{\mathsf{C}}(-,A_2),F_2)\& \ni \& \eta\circ \phi\circ g_\star \end{tikzcd} \] - The comma category $\left(\Delta F \downarrow \Delta\right)$ is called the \textbf{cocone category from $F$ to $\textsf{C}$}, denoted by $\mathsf{Cocone}(F,\textsf{C})$. - \begin{itemize} - \item Objects: The objects in $\mathsf{Cocone}(F,\textsf{C})$ are all natural transformations - \[ - \begin{tikzcd}[ampersand replacement=\&] - \mathsf{J} \arrow[r, "F"{name=A, above}, bend left] \arrow[r, "\Delta C"'{name=B, below}, bend right] \&[+30pt] \mathsf{C} - \arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "h"] - \end{tikzcd} - \] - where $C\in \mathrm{Ob}(\mathsf{C})$. Such $h:F\Rightarrow \Delta C$ is called a \textbf{cocone from $F$ to $C$} because it can be viewed as a family of morphisms $\left(h_i:F(i)\to C\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ in $\mathsf{C}$ such that the following diagram commutes for each morphism $\lambda:i\to j$ in $\mathsf{J}$ + is a functor isomorphic to the \hyperref[th:evaluation_functor]{evaluation functor} + \begin{align*} + \mathrm{ev}:\mathsf{C}^{\mathrm{op}}\times \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right]&\longrightarrow \mathsf{Set}\\ + \left(A,F\right)&\longmapsto F(A) + \end{align*} + \textbf{Covariant version}\\ + For any functor $F:\mathsf{C}\to \mathsf{Set}$ and any $A\in \mathrm{Ob}(\mathsf{C})$, there is a natural bijection + \begin{align*} + \operatorname{Hom}_{\left[\mathsf{C},\mathsf{Set}\right]}\left(\operatorname{Hom}_{\mathsf{C}}\left(A,-\right), F\right) & \overset{\sim}{\longrightarrow} F(A) \\ + { \left[ \begin{tikzcd}[ampersand replacement=\&] + \mathsf{C} \arrow[r, "\operatorname{Hom}_{\mathsf{C}}\left(A{,}-\right)"{name=A, above}, bend left] \arrow[r, "F"'{name=B, below}, bend right] \&[+30pt] \mathsf{Set} + \arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "\phi"] + \end{tikzcd}\right] } & \longmapsto \phi_A\left(\operatorname{id}_A\right) + \end{align*} +} +\pf{ + We break the proof into following steps. +\begin{itemize}[leftmargin=*] + \item \textbf{$\phi$ is determined by $\phi_A\left(\operatorname{id}_A\right)$}.\\ + Suppose $\phi:\mathrm{Hom}_{\mathsf{C}}\left(-,A\right)\Rightarrow F$ is a natural transformation. Given any morphism $f:X\to A$ in $\mathsf{C}$, the naturality of $\phi$ gives the following commutative diagram \[ - \begin{tikzcd}[ampersand replacement=\&] - \& C\& \\ -F(i)\arrow[ru, "h_i"] \arrow[rr, "F(\lambda)"'] \& \& F(j) \arrow[lu, "h_j"'] -\end{tikzcd} -\] - \item Morphisms: The morphisms in $\mathsf{Cocone}(F,\textsf{C})$ are commutative triangles shown as follows + \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}}(X,A)} \arrow[d, "\phi_X"]\&[-25pt]\ni\&[-25pt]f \arrow[d, mapsto]\\ + \phi_A(\mathrm{id}_A)\&\in\&{F(A)}\arrow[rr, "{F(f)}"'] \& \& {F(X)} \&\ni\& \phi_X(f) + \end{tikzcd} + \] + Thus we see $\phi$ is determined by $\phi_A(\mathrm{id}_A)$ as follows \[ - \begin{tikzcd}[ampersand replacement=\&, row sep=9pt] - \&\Delta C \arrow[dd, Rightarrow, "f_\bullet"] \\ - F \arrow[ru, Rightarrow, "h"]\arrow[rd, Rightarrow, "h^{\prime}"']\& \\ - \&\Delta C' - \end{tikzcd} + \phi_X(f)=F(f)(\phi_A(\mathrm{id}_A)),\quad\forall X\in \mathrm{Ob}(\mathsf{C}),\forall f\in \mathrm{Hom}_{\mathsf{C}}(X,A). \] - \end{itemize} - Equivalently, a morphism from cocone $\left(h_i:F(i)\to C\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ to cocone $\left(h_i':F(i)\to C'\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ is a morphism $f:C\to C'$ in $\mathsf{C}$ such that the following diagram commutes for each $i\in \mathrm{Ob}(\mathsf{J})$ + This implies that the injectivity of $q_{A,F}$. + \item \textbf{Construct the inverse of $q_{A,F}$}.\\ + Define + \begin{align*} + r_{A,F}: F(A)&\longrightarrow\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(\operatorname{Hom}_{\mathsf{C}}\left(-,A\right), F\right) \\ + u & \longmapsto \phi^u:\left[\begin{aligned}\phi_X^u: \mathrm{Hom}_{\mathsf{C}}(X,A)&\longrightarrow F(X)\\ + f&\longmapsto F(f)(u)\end{aligned} \right] + \end{align*} + To ensure $r_{A,F}$ is well-defined, we need to check that $r_{A,F}(u)=\phi^u$ is always a natural transformation. In fact, given any morphism $h:X_1\to X_2$ in $\mathsf{C}$, it is easy to check the following diagram commutes \[ - \begin{tikzcd}[ampersand replacement=\&, row sep=9pt] - \&\Delta C \arrow[dd, "f"] \\ - F(i) \arrow[ru, "h_I"]\arrow[rd, "h^{\prime}_i"']\& \\ - \&\Delta C' + \begin{tikzcd}[ampersand replacement=\&, row sep = 3.5em] + g\arrow[mapsto]{d}\&[-25pt]\in\&[-25pt]\mathrm{Hom}_{\mathsf{C}}(X_2,A) \arrow[d, "{\phi_{X_2}^u}"'] \arrow[rr, "{h^*}"] \&[-20pt] \& {\mathrm{Hom}_{\mathsf{C}}(X_1,A)} \arrow[d, "\phi_{X_1}^u"]\&[-25pt]\ni\&[-25pt]g\circ h \arrow[mapsto]{d}\\ + F(g)(u)\&\in\&{F(X_2)}\arrow[rr, "{F(h)}"'] \& \& {F(X_1)} \&\ni\& F(g\circ h)(u) \end{tikzcd} \] -} + because the functoriality of $F$ gives the identity $F(g\circ h)=F(h)\circ F(g)$. -\dfn{Colimit}{ - Let $\mathsf{J},\mathsf{C}$ be categories and $F:\mathsf{J}\to\mathsf{C}$ be a functor. - The \textbf{colimit} of $F$, denoted by $\varinjlim F$, is the initial object of $\mathsf{Cocone}(F,\textsf{C})$. For each morphism $\lambda:i\to j$ in $\mathsf{J}$, we have the following commutative diagram + For any $u\in F(A)$, we have + \begin{align*} + \left(q_{A,F}\circ r_{A,F}\right)(u) + &=q_{A,F}(\phi^u)\\ + &=\phi^u_A(\mathrm{id}_A)\\ + &=F(\mathrm{id}_A)(u)\\ + &=\mathrm{id}_{F(A)}(u)\\ + &=u, + \end{align*} + which implies that $q_{A,F}$ is surjective. Therefore, $q_{A,F}$ is a bijection and $r_{A,F}$ is the inverse of $q_{A,F}$. + + We can also manually check that $\left(r_{A,F}\circ q_{A,F}\right)(\phi)=\phi$ for any natural transformation $\phi:\mathrm{Hom}_{\mathsf{C}}\left(-,A\right)\Rightarrow F$, by evaluating the natural transformations at $\mathrm{id}_A$, + \begin{align*} + \left(\left(r_{A,F}\circ q_{A,F}\right)(\phi)\right)_A(\mathrm{id}_A)&=(r_{A,F}(\phi_A(\mathrm{id}_A)))_A(\mathrm{id}_A)\\ + &=F(\mathrm{id}_A)(\phi_A(\mathrm{id}_A))\\ + &=\mathrm{id}_{F(A)}(\phi_A(\mathrm{id}_A))\\ + &=\phi_A(\mathrm{id}_A). + \end{align*} + \item \textbf{$q_{A,F}$ is natural in $A$ and $F$}.\\ + $\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(Y_\mathsf{C}(-){,}-\right)$ is a functor obtained by the composition \[ - \begin{tikzcd}[ampersand replacement=\&, every arrow/.append style={-latex, line width=1.1pt}] - \& [-1.5em] \&[-1em] C \& [-1em] \\[+0.2cm] - \& \& \varinjlim F \arrow[u, dash pattern=on 4pt off 2pt, draw=arrowRed] \& \\[-0.1cm] - \& F(i) \arrow[ruu, draw=cyan, "h_i"]\arrow[ru, draw=arrowBlue, shorten >=-3.5pt, "\phi_i"' yshift=4.5pt] \arrow[rr, "F(\lambda)"']\& \& F(j)\arrow[luu, draw=cyan, "h_j"']\arrow[lu, draw=arrowBlue, shorten >=-3.7pt, "\phi_j" yshift=4.5pt] + \mathsf{C}^{\mathrm{op}}\times \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right]\xrightarrow{\left(Y^{\mathrm{op}}_{\mathsf{C}},\mathrm{id}\right)}\left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right]^{\mathrm{op}}\times \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right]\xrightarrow{\mathrm{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(-,-\right)}\mathsf{Set} + \] + Given any $g:A_2\to A_1$ and $\eta:F_1\to F_2$, we should check the following diagram commutes + \[ + \begin{tikzcd}[row sep=3.5em, column sep=2.5em, ampersand replacement=\&] + {\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(Y_\mathsf{C}(A_1),F_1)} \arrow[d, "{q_{A_1,F_1}}"'] \arrow[rr, "\left(g_\star\right)^*\circ \eta_*"] \& \& {\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(Y_\mathsf{C}(A_2),F_2)} \arrow[d, "q_{A_2,F_2}"] \\ + {F_1(A_1)} \arrow[rr, "F_2(g)\circ \eta_{A_1}"'] \& \& {F_2(A_2)} + \end{tikzcd} + \] + It suffices to check that the following diagram commutes + \[ + \begin{tikzcd}[ampersand replacement=\&, row sep = 3.5em] + \operatorname{Hom}(Y_\mathsf{C}(A_1),F_1) \arrow[rr,"\left(g_\star\right)^*"] \arrow[dr,swap,"\eta_*"] \arrow[dd,swap,"q_{A_1,F_1}"] \&\& + \operatorname{Hom}(Y_\mathsf{C}(A_2),F_1) \arrow[dd,swap,"q_{A_2,F_1}" near start] \arrow[dr,"\eta_*"] \\ + \& \operatorname{Hom}(Y_\mathsf{C}(A_1),F_2) \arrow[rr,crossing over,"\left(g_\star\right)^*" near start] \&\& + \operatorname{Hom}(Y_\mathsf{C}(A_2),F_2) \arrow[dd,"q_{A_2,F_2}"] \\ + F_1(A_1) \arrow[rr,"F_1(g)" near end] \arrow[dr,swap,"\eta_{A_1}"] \&\& F_1(A_2) \arrow[dr,swap,"\eta_{A_2}"] \\ + \& F_2(A_1) \arrow[rr,"F_2(g)"] \arrow[uu,<-,crossing over,"q_{A_1,F_2}" near end]\&\& F_2(A_2) \end{tikzcd} \] -} - -\ex{Limit in $\mathsf{Set}$}{ - Let $F:\mathsf{J}\to \mathsf{Set}$ be a functor. Then +\end{itemize} + To check the commutativity of the left square + \[ + \begin{tikzcd}[row sep=3.5em, column sep=2.5em, ampersand replacement=\&] + {\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(Y_\mathsf{C}(A_1),F_1)} \arrow[d, "{q_{A_1,F_1}}"'] \arrow[rr, "\eta_*"] \& \& {\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(Y_\mathsf{C}(A_1),F_2)} \arrow[d, "q_{A_1,F_2}"] \\ + {F_1(A_1)} \arrow[rr, "\eta_{A_1}"'] \& \& {F_2(A_1)} + \end{tikzcd} + \] + we can verify that for any $\phi\in \operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(Y_\mathsf{C}(A_1),F_1\right)$, \begin{align*} - \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\} + q_{A_1,F_2}\circ \eta_*\left(\phi\right)&=q_{A_1,F_2}\left(\eta\circ \phi\right)\\ + &=\left(\eta\circ \phi\right)_{A_1}\left(\mathrm{id}_{A_1}\right)\\ + &=\eta_{A_1}(\phi_{A_1}(\mathrm{id}_{A_1}))\\ + &=\eta_{A_1}\circ q_{A_1,F_1}(\phi). \end{align*} - where $\alpha$ and $\beta$ are defined by - and the map $\phi_i:\varprojlim F\to F(i)$ is given by the composition + To check the commutativity of the front square \[ - \varprojlim F\xrightarrow{\quad\iota\quad}\prod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i)\xrightarrow{\quad\pi_i\quad}F(i). + \begin{tikzcd}[row sep=3.5em, column sep=2.5em, ampersand replacement=\&] + {\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(Y_\mathsf{C}(A_1),F_2)} \arrow[d, "{q_{A_1,F_2}}"'] \arrow[rr, "\left(g_\star\right)^*"] \& \& {\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(Y_\mathsf{C}(A_2),F_2)} \arrow[d, "q_{A_2,F_2}"] \\ + {F_2(A_1)} \arrow[rr, "F_2(g)"'] \& \& {F_2(A_2)} + \end{tikzcd} \] - $\varprojlim F$ also can be constructed from product and equalizer + we can verify that for any $\psi\in \operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(Y_\mathsf{C}(A_1),F_2\right)$, we have \begin{align*} - \varprojlim F\cong \ker\left[ - \begin{tikzcd}[ampersand replacement=\&] - \prod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i) \arrow[r, "\alpha", shift left] \arrow[r, "\beta"', shift right] \& \prod\limits_{\lambda \in \mathrm{Mor}(\mathsf{J})}F(\mathrm{codom}\left(\lambda\right)) - \end{tikzcd} - \right] + q_{A_2,F_2}\circ \left(g_\star\right)^*(\psi)&=q_{A_2,F_2}\left(\psi\circ g_\star\right)\\ + &=\left(\psi\circ g_\star\right)_{A_2}\left(\mathrm{id}_{A_2}\right)\\ + &=\psi_{A_2}\circ g_*\left(\mathrm{id}_{A_2}\right)\\ + &=\psi_{A_2}(g)\\ + &=F_2(g)(\psi_{A_1}(\mathrm{id}_{A_1}))\\ + &=F_2(g)\circ q_{A_1,F_2}(\psi). \end{align*} - where $\alpha$ and $\beta$ are induced by the following diagrams +} +\cor{Yoneda Embedding}{ + Let $\mathsf{C}$ be a locally small category. The \hyperref[th:yoneda_embedding_functor]{Yoneda embedding functor} $Y_\mathsf{C}:\mathsf{C}\to \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right]$ is fully faithful. That is, for any $A_1,A_2\in\mathrm{Ob}\left(\mathsf{C}\right)$, the morphism set map + \begin{align*} + \left.Y_{\mathsf{C}}\right|_{\operatorname{Hom}_{\mathsf{C}}\left(A_1,A_2\right)}:\operatorname{Hom}_{\mathsf{C}}\left(A_1,A_2\right) &\longrightarrow\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(\operatorname{Hom}_{\mathsf{C}}\left(-,A_1\right), \operatorname{Hom}_{\mathsf{C}}\left(-,A_2\right)\right) \\ + f & \longmapsto f_\star + \end{align*} + is a bijection. That justifies the name ``embedding'' because $\mathsf{C}$ is embedded into $\mathsf{Psh}(\mathsf{C})$ via $Y_\mathsf{C}$ as a full subcategory. +} +\pf{ + By \hyperref[th:yoneda_lemma]{Yoneda lemma}, there is a natural bijection \[ - \begin{tikzcd}[ampersand replacement=\&] - F\left(\mathrm{dom}\left(\lambda\right)\right) \arrow[d, "F(\lambda)"'] \&[+25pt] \prod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i) \arrow[l, "\pi_{\mathrm{dom}\left(\lambda\right)}"'] \arrow[d, "\alpha", dashed] \&[-15pt]\ni \& [-15pt] (x_i)_{i\in \mathrm{Ob}(\mathsf{J})} \arrow[d, "\alpha", mapsto] \\ [+10pt] - F\left(\mathrm{codom}\left(\lambda\right)\right) \& {\prod\limits_{\lambda \in \mathrm{Mor}(\mathsf{J})}F(\mathrm{codom}\left(\lambda\right))} \arrow[l, "{\pi_{\lambda}}"]\&\ni\& (F(\lambda)(x_i))_{\lambda:i\to j\in \mathrm{Mor}(\mathsf{J})} - \end{tikzcd} + \operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(\operatorname{Hom}_{\mathsf{C}}\left(-,A_1\right), \operatorname{Hom}_{\mathsf{C}}\left(-,A_2\right)\right)\cong \operatorname{Hom}_{\mathsf{C}}\left(A_1,A_2\right), \] + which is given by + \begin{align*} + r_{A_1,Y_{\mathsf{C}}(A_2)}: \operatorname{Hom}_{\mathsf{C}}\left(A_1,A_2\right) &\longrightarrow\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(\operatorname{Hom}_{\mathsf{C}}\left(-,A_1\right), \operatorname{Hom}_{\mathsf{C}}\left(-,A_2\right)\right) \\ + f & \longmapsto f_\star:\left[\begin{aligned}\left(f_\star\right)_X: \mathrm{Hom}_{\mathsf{C}}(X,A_1)&\longrightarrow \mathrm{Hom}_{\mathsf{C}}(X,A_2)\\ + g&\longmapsto f_*(g)=f\circ g\end{aligned} \right] + \end{align*} + and + \begin{align*} + q_{A_1,Y_{\mathsf{C}}(A_2)}:\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(\operatorname{Hom}_{\mathsf{C}}\left(-,A_1\right), \operatorname{Hom}_{\mathsf{C}}\left(-,A_2\right)\right) & \overset{\sim}{\longrightarrow} \operatorname{Hom}_{\mathsf{C}}\left(A_1,A_2\right) \\ + { \left[ \begin{tikzcd}[ampersand replacement=\&] + \mathsf{C}^{\mathrm{op}} \arrow[r, "\operatorname{Hom}_{\mathsf{C}}\left(-{,}A_1\right)"{name=A, above}, bend left] \arrow[r, "\operatorname{Hom}_{\mathsf{C}}\left(-{,}A_2\right)"'{name=B, below}, bend right] \&[+30pt] \mathsf{Set} + \arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "\phi"] + \end{tikzcd}\right] } & \longmapsto \phi_{A_1}\left(\operatorname{id}_{A_1}\right) + \end{align*} + Note that $r_{A_1,Y_{\mathsf{C}}(A_2)}$ is exactly the morphism set map $\left.Y_{\mathsf{C}}\right|_{\operatorname{Hom}_{\mathsf{C}}\left(A_1,A_2\right)}$. +} +Yoneda embedding allows us to regard any category $\mathsf{C}$ as a full subcategory of $\mathsf{Psh}(\mathsf{C})$. So in the sense of categorical isomorphism, we can identify any object $A\in \mathrm{Ob}(\mathsf{C})$ with a presheaf $\operatorname{Hom}_{\mathsf{C}}(-, A)$. + +An object in a category is equivalent to the collection of all arrows pointing to that object. In other words, the information we care about is which arrows from an object itself and other objects would point to this object, while the object's internals are completely regarded as a black box. + +The arrows between objects in a category are equivalent to the transformations between sets of arrows pointing to objects. Specifically, suppose $f$ is an arrow from object $A$ to object $B$, then any arrow pointing to $A$ can be transformed into an arrow pointing to $B$ by appending an $f$, which is exactly what $f_\star$ does. Conversely, if there is a way to naturally transform all arrows pointing to $A$ into arrows pointing to $B$, then this transformation must be realized by appending an arrow between $A$ and $B$. + +\dfn{Representable Functor}{ + Let $\mathsf{C}$ be a locally small category. + \begin{itemize} + \item A functor $F:\mathsf{C}\to \mathsf{Set}$ is called \textbf{representable} if it is naturally isomorphic to $\operatorname{Hom}_{\mathsf{C}}\left(A,-\right)$ for some $A\in \mathrm{Ob}(\mathsf{C})$. + \item A functor $F:\mathsf{C}^{\mathrm{op}}\to \mathsf{Set}$ is called \textbf{representable} if it is naturally isomorphic to $\operatorname{Hom}_{\mathsf{C}}\left(-,A\right)$ for some $A\in \mathrm{Ob}(\mathsf{C})$. + \end{itemize} + 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 $F(f)(u)=x$). +} +\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 \[ - \begin{tikzcd}[ampersand replacement=\&] - \&[+25pt] \prod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i) \arrow[ld, "\pi_{\mathrm{codom}\left(\lambda\right)}"'] \arrow[d, "\beta", dashed] \&[-15pt]\ni \&[-15pt] (x_i)_{i\in \mathrm{Ob}(\mathsf{J})} \arrow[d, "\beta", mapsto] \\ [+10pt] - F\left(\mathrm{codom}\left(\lambda\right)\right) \& {\prod\limits_{\lambda \in \mathrm{Mor}(\mathsf{J})}F(\mathrm{codom}\left(\lambda\right))} \arrow[l, "{\pi_{\lambda}}"]\&\ni\& (x_j)_{\lambda:i\to j\in \mathrm{Mor}(\mathsf{J})} - \end{tikzcd} + \begin{tikzcd}[ampersand replacement=\&, row sep = 3.5em] + \mathrm{id}_A\arrow[d, mapsto]\&[-25pt]\in\&[-25pt]\mathrm{Hom}_{\mathsf{C}}(A,A) \arrow[d, "{\phi_A}"'] \arrow[rr, "{f_*}"] \&[-20pt] \& {\mathrm{Hom}_{\mathsf{C}}(A,X)} \arrow[d, "\phi_X"]\&[-25pt]\ni\&[-25pt]f \arrow[d, mapsto]\\ + u=\phi_A(\mathrm{id}_A)\&\in\&{F(A)}\arrow[rr, "{F(f)}"'] \& \& {F(X)} \&\ni\& \phi_X(f) + \end{tikzcd} + \] + which implies + \[ + F(f)(u)=\phi_X(f),\quad\forall f\in \mathrm{Hom}_{\mathsf{C}}(A,X). \] + Thus we have + \begin{align*} + (A,u)\text{ is initial in }\int_{\mathsf{C}}F&\iff \forall(X,x)\in \mathrm{Ob}\left(\int_{\mathsf{C}}F\right),\;\exists! f\in \mathrm{Hom}_{\mathsf{C}}(A,X),\;F(f)(u)=x\\ + &\iff\forall X\in \mathrm{Ob}\left(\mathsf{C}\right),\; \forall x\in F(X),\;\exists! f\in \mathrm{Hom}_{\mathsf{C}}(A,X),\;\phi_X(f)=x\\ + &\iff \forall X\in \mathrm{Ob}\left(\mathsf{C}\right),\;\phi_X \text{ is bijective}\\ + &\iff \phi \text{ is a natural isomorphism}\\ + &\iff (A,u) \text{ is a universal element of }F. + \end{align*} +} +\cor{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 $\Delta\{*\}$. } -\ex{Colimit in $\mathsf{Set}$}{ - Let $F:\mathsf{J}\to \mathsf{Set}$ be a functor. Define a relation $\sim^*$ on $\coprod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i)$: for any $(i,x),(j,y)\in \coprod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i)$, +\section{Limit and Colimit} +\dfn{Cone}{ + Let $\mathsf{J},\mathsf{C}$ be categories and $F:\mathsf{J}\to\mathsf{C}$ be a functor. Consider functors \[ - (i,x)\sim^* (j,y)\iff \text{there exists }\lambda:i\to j\text{ in }\mathrm{Hom}_\mathsf{J}(i,j)\text{ such that }F(\lambda)(x)=y. + \begin{tikzcd}[ampersand replacement=\&] + \mathsf{C} \arrow[r, "\Delta"] \& {[\mathsf{J},\mathsf{C}]} \& \mathsf{1} \arrow[l, "\Delta F"'] + \end{tikzcd} \] - Let $\sim$ denote the equivalence relation generated by $\sim^*$. Then + The comma category $\left(\Delta \downarrow \Delta F\right)$ is called the \textbf{cone category from $\textsf{C}$ to $F$}, denoted by $\mathsf{Cone}(\textsf{C},F)$. + \begin{itemize} + \item Objects: The objects in $\mathsf{Cone}(\textsf{C},F)$ are all natural transformations + \[ + \begin{tikzcd}[ampersand replacement=\&] + \mathsf{J} \arrow[r, "\Delta C"{name=A, above}, bend left] \arrow[r, "F"'{name=B, below}, bend right] \&[+30pt] \mathsf{C} + \arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "h"] + \end{tikzcd} + \] + where $C\in \mathrm{Ob}(\mathsf{C})$. Such $h:\Delta C\Rightarrow F$ is called a \textbf{cone from $C$ to $F$} because it can be viewed as a family of morphisms $\left(h_i:C\to F(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ in $\mathsf{C}$ such that the following diagram commutes for each morphism $\lambda:i\to j$ in $\mathsf{J}$ + \[ + \begin{tikzcd}[ampersand replacement=\&] + \& C \arrow[ld, "h_i"'] \arrow[rd, "h_j"] \& \\ + F(i) \arrow[rr, "F(\lambda)"'] \& \& F(j) + \end{tikzcd} + \] + \item Morphisms: The morphisms in $\mathsf{Cone}(\textsf{C},F)$ are commutative triangles shown as follows \[ - \varinjlim F\cong\coprod_{i\in \mathrm{Ob}(\mathsf{J})}F(i)/\sim - \] - and the map $\phi_i:F(i)\to \varinjlim F$ is given by the composition + \begin{tikzcd}[ampersand replacement=\&, row sep=9pt] + \Delta C \arrow[rd, Rightarrow, "h"] \arrow[dd, Rightarrow, "f_\bullet"'] \& \\ + \& F\\ + \Delta C' \arrow[ru, Rightarrow, "h^{\prime}"'] \& + \end{tikzcd} + \] + Equivalently, a morphism from cone $\left(h_i:C\to F(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ to cone $\left(h_i':C'\to F(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ is a morphism $f:C\to C'$ in $\mathsf{C}$ such that the following diagram commutes for each $i\in \mathrm{Ob}(\mathsf{J})$ \[ - F(i)\xrightarrow{\quad\iota_i\quad}\coprod_{i\in \mathrm{Ob}(\mathsf{J})}F(i)\xrightarrow{\quad\pi\quad}\coprod_{i\in \mathrm{Ob}(\mathsf{J})}F(i)/\sim. + \begin{tikzcd}[ampersand replacement=\&, row sep=9pt] + C \arrow[rd, "h_i"] \arrow[dd, "f"'] \& \\ + \& F(i)\\ + C' \arrow[ru, "h^{\prime}_i"'] \& + \end{tikzcd} \] - If $\mathsf{J}$ is a filtered category, then the equivalence relation $\sim$ has a explicit description: for any $(i,x),(j,y)\in \coprod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i)$, + \end{itemize} +} +\dfn{Limit}{ + Let $\mathsf{J},\mathsf{C}$ be categories and $F:\mathsf{J}\to\mathsf{C}$ be a functor. The \textbf{limit} of $F$, denoted by $\varprojlim F$, is the terminal object of $\mathsf{Cone}(\textsf{C},F)$. For each morphism $\lambda:i\to j$ in $\mathsf{J}$, we have the following commutative diagram \[ - (i,x)\sim (j,y)\iff \text{there exists }\lambda_1:i\to k\text{ and }\lambda_2:j\to k\text{ such that }F(\lambda_1)(x)=F(\lambda_2)(y). + \begin{tikzcd}[ampersand replacement=\&, every arrow/.append style={-latex, line width=1.1pt}] + \& [-1.5em] \&[-1em] C \arrow[rdd, draw=cyan, "h_j"] \arrow[ldd, draw=cyan, "h_i"'] \arrow[d, dash pattern=on 4pt off 2pt, draw=arrowRed] \& [-1em] \\[+0.2cm] + \& \& \varprojlim F\arrow[ld, draw=arrowBlue, shorten <=-3pt, "\phi_i" yshift=4.5pt] \arrow[rd, draw=arrowBlue, shorten <=-4.5pt, "\phi_j"' yshift=4pt] \& \\[-0.1cm] + \& F(i) \arrow[rr, "F(\lambda)"']\& \& F(j) + \end{tikzcd} \] } -\pf{ - It is easy to show +\dfn{Cocone}{ + Let $\mathsf{J},\mathsf{C}$ be categories and $F:\mathsf{J}\to\mathsf{C}$ be a functor. Consider functors \[ - \varinjlim F\cong\coprod_{i\in \mathrm{Ob}(\mathsf{J})}F(i)/\sim + \begin{tikzcd}[ampersand replacement=\&] + \mathsf{1} \arrow[r, "\Delta F"] \& {[\mathsf{J},\mathsf{C}]} \&\mathsf{C} \arrow[l, "\Delta "'] + \end{tikzcd} \] - by checking the universal property of $\varinjlim F$. If $\mathsf{J}$ is a filtered category, first we can check + The comma category $\left(\Delta F \downarrow \Delta\right)$ is called the \textbf{cocone category from $F$ to $\textsf{C}$}, denoted by $\mathsf{Cocone}(F,\textsf{C})$. + \begin{itemize} + \item Objects: The objects in $\mathsf{Cocone}(F,\textsf{C})$ are all natural transformations + \[ + \begin{tikzcd}[ampersand replacement=\&] + \mathsf{J} \arrow[r, "F"{name=A, above}, bend left] \arrow[r, "\Delta C"'{name=B, below}, bend right] \&[+30pt] \mathsf{C} + \arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "h"] + \end{tikzcd} + \] + where $C\in \mathrm{Ob}(\mathsf{C})$. Such $h:F\Rightarrow \Delta C$ is called a \textbf{cocone from $F$ to $C$} because it can be viewed as a family of morphisms $\left(h_i:F(i)\to C\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ in $\mathsf{C}$ such that the following diagram commutes for each morphism $\lambda:i\to j$ in $\mathsf{J}$ \[ - (i,x)\approx (j,y)\iff \text{there exists }\lambda_1:i\to k\text{ and }\lambda_2:j\to k\text{ such that }F(\lambda_1)(x)=F(\lambda_2)(y). + \begin{tikzcd}[ampersand replacement=\&] + \& C\& \\ +F(i)\arrow[ru, "h_i"] \arrow[rr, "F(\lambda)"'] \& \& F(j) \arrow[lu, "h_j"'] +\end{tikzcd} +\] + \item Morphisms: The morphisms in $\mathsf{Cocone}(F,\textsf{C})$ are commutative triangles shown as follows + \[ + \begin{tikzcd}[ampersand replacement=\&, row sep=9pt] + \&\Delta C \arrow[dd, Rightarrow, "f_\bullet"] \\ + F \arrow[ru, Rightarrow, "h"]\arrow[rd, Rightarrow, "h^{\prime}"']\& \\ + \&\Delta C' + \end{tikzcd} \] - is an equivalence relation. To show $\approx\;=\;\sim$, let's assume $\backsimeq$ is any equivalence relation containing $\sim^*$. If $(i,x)\approx(j,y)$, then there exists $\lambda_1:i\to k$ and $\lambda_2:j\to k$ such that $F(\lambda_1)(x)=F(\lambda_2)(y)=z$. Hence + \end{itemize} + Equivalently, a morphism from cocone $\left(h_i:F(i)\to C\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ to cocone $\left(h_i':F(i)\to C'\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ is a morphism $f:C\to C'$ in $\mathsf{C}$ such that the following diagram commutes for each $i\in \mathrm{Ob}(\mathsf{J})$ \[ - (i,x)\sim^* (k,z)\text{ and }(j,y)\sim^* (k,z)\implies(i,x)\backsimeq (k,z)\text{ and }(j,y)\backsimeq (k,z)\implies (i,x)\backsimeq (j,y). + \begin{tikzcd}[ampersand replacement=\&, row sep=9pt] + \&\Delta C \arrow[dd, "f"] \\ + F(i) \arrow[ru, "h_I"]\arrow[rd, "h^{\prime}_i"']\& \\ + \&\Delta C' + \end{tikzcd} + \] +} + +\dfn{Colimit}{ + Let $\mathsf{J},\mathsf{C}$ be categories and $F:\mathsf{J}\to\mathsf{C}$ be a functor. + The \textbf{colimit} of $F$, denoted by $\varinjlim F$, is the initial object of $\mathsf{Cocone}(F,\textsf{C})$. For each morphism $\lambda:i\to j$ in $\mathsf{J}$, we have the following commutative diagram + \[ + \begin{tikzcd}[ampersand replacement=\&, every arrow/.append style={-latex, line width=1.1pt}] + \& [-1.5em] \&[-1em] C \& [-1em] \\[+0.2cm] + \& \& \varinjlim F \arrow[u, dash pattern=on 4pt off 2pt, draw=arrowRed] \& \\[-0.1cm] + \& F(i) \arrow[ruu, draw=cyan, "h_i"]\arrow[ru, draw=arrowBlue, shorten >=-3.5pt, "\phi_i"' yshift=4.5pt] \arrow[rr, "F(\lambda)"']\& \& F(j)\arrow[luu, draw=cyan, "h_j"']\arrow[lu, draw=arrowBlue, shorten >=-3.7pt, "\phi_j" yshift=4.5pt] + \end{tikzcd} \] - This implies $\backsimeq$ contains $\approx$. Therefore, $\approx$ is the smallest equivalence relation containing $\sim^*$, which means $\approx$ coincides with $\sim$. } + \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. } \dfn{Cocomplete Category}{ A category $\mathsf{C}$ is \textbf{cocomplete} if it has all small colimits. That is, for any functor $F:\mathsf{J}\to \mathsf{C}$ with $\mathsf{J}$ small, $\varinjlim F$ exists. } +\dfn{Bicomplete Category}{ + A category $\mathsf{C}$ is \textbf{bicomplete} if it is both complete and cocomplete. +} +\ex{Examples of Bicomplete Categories}{ + The following categories are bicomplete: $\mathsf{Set}$, $\mathsf{Grp}$, $\mathsf{Ab}$, $\mathsf{Ring}$, + $R\text{-}\mathsf{Mod}$, $K\text{-}\mathsf{Vect}$, $\mathsf{Top}$. +} +\dfn{Finitely Complete category}{ + A category $\mathsf{C}$ is \textbf{finitely complete} if it has all finite limits. That is, for any functor $F:\mathsf{J}\to \mathsf{C}$ with $\mathsf{J}$ a finite category, $\varprojlim F$ exists. +} +\dfn{Finitely Cocomplete category}{ + A category $\mathsf{C}$ is \textbf{finitely cocomplete} if it has all finite colimits. That is, for any functor $F:\mathsf{J}\to \mathsf{C}$ with $\mathsf{J}$ a finite category, $\varinjlim F$ exists. +} \thm{Existance Theorem for Limits}{ Let $\mathsf{C}$ be a category and $F:\mathsf{J}\to\mathsf{C}$ be a functor. If a category $\mathsf{C}$ has equalizers and all products indexed by the classes $\mathrm{Ob}(\mathsf{J})$ and $\mathrm{Hom}(\mathsf{J})$, then $\mathsf{C}$ has all limits of shape $\mathsf{J}$. @@ -1055,288 +1347,88 @@ \section{Limit and Colimit} \ex{Filtered Set}{ A filtered set can be regarded as a filtered (0,1)-category with objects being elements of the set and morphisms being \begin{align*} - \mathrm{Hom}(x,y)=\begin{cases} - \{*\} & \text{if }x\leq y\\ - \varnothing & \text{otherwise} - \end{cases} - \end{align*} -} - -\dfn{}{ - For any class of diagrams $K: \mathsf{J} \rightarrow \mathsf{C}$ valued in $\mathsf{C}$, a functor $F: \mathsf{C} \rightarrow \mathsf{D}$ - \begin{itemize} - \item \textbf{preserves} those limits if for any diagram $K: \mathsf{J} \rightarrow \mathsf{C}$ and limit cone over $K$, the image of this cone defines a limit cone over the composite diagram $F\circ K: \mathsf{J} \rightarrow \mathsf{D}$; - \item \textbf{reflects} those limits if any cone over a diagram $K: \mathsf{J} \rightarrow \mathsf{C}$, whose image upon applying $F$ is a limit cone for the diagram $F\circ K: \mathsf{J} \rightarrow \mathsf{D}$, is a limit cone over $K$; - \item \textbf{creates} those limits if whenever $F\circ K: \mathsf{J} \rightarrow \mathsf{D}$ has a limit in $\mathsf{D}$, there is some limit cone over $F\circ K$ that can be lifted to a limit cone over $K$, and moreover $F$ reflects the limits in the class of diagrams. - \end{itemize} -} - -\section{Representable Functor} -\dfn{Presheaf}{ - Let $\mathsf{C}$ be a category. A \textbf{presheaf} on $\mathsf{C}$ is a functor $F:\mathsf{C}^{\mathrm{op}}\to \mathsf{Set}$. -} - -\dfn[yoneda_embedding_functor]{Yoneda Embedding Functor}{ - Let $\mathsf{C}$ be a category. The \textbf{Yoneda embedding functor} is the functor $Y_\mathsf{C}:\mathsf{C}\to \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right]$ defined as follows - \[ - \begin{tikzcd}[ampersand replacement=\&] - \mathsf{C} % column 1 - \&[-25pt] % column 2 - \&[+30pt] % column 3 - \&[-30pt] \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right] % column 4 - \&[+25pt] % column 5 - \&[-25pt] % column 6 - \& % column 7 - \\[-15pt] - Y_1 \arrow[dd, "g"{name=L, left}] \&\& \& \mathrm{Hom}_{\mathsf{C}}\left(-,Y_1\right) \arrow[dd, Rightarrow, "g_\star"{name=R}] \& \mathrm{Hom}_{\mathsf{C}}(X_1,Y_1) \arrow[dd, "{g_*}"'] \arrow[rr, "{f^*}"] \& \& \mathrm{Hom}_{\mathsf{C}}(X_2,Y_1) \arrow[dd, "g_*"] \\[-5pt] - \& \phantom{.}\arrow[r, "Y_\mathsf{C}", squigarrow]\&\phantom{.} \&\phantom{.} \&\phantom{.}\&\\[-5pt] - Y_2 \& \& \& \mathrm{Hom}_{\mathsf{C}}\left(-,Y_2\right)\& \mathrm{Hom}_{\mathsf{C}}(X_1,Y_2) \arrow[rr, "{f^*}"'] \& \& \mathrm{Hom}_{\mathsf{C}}(X_2,Y_2) - \arrow[from=4-5, to=2-5, dash, start anchor={[xshift=-8ex,yshift=-3ex]}, end anchor={[xshift=-8ex,yshift=3ex]},decorate,decoration={brace,amplitude=10pt,raise=4pt}] - \arrow[from=3-4, to=3-5, dashed, start anchor={[xshift=9pt, yshift=-2pt]}, end anchor={[xshift=-11ex, yshift=-2pt]},decorate] - \end{tikzcd} - \] - where the natural transformation $g_\star$ is defined pointwise as follows: $\left(g_\star\right)_X=g_*$ for any $X\in \mathrm{Ob}(\mathsf{C}^{\mathrm{op}})$.\\ - The contravariant version is $Y_{\mathsf{C}^{\mathrm{op}}}:\mathsf{C}^{\mathrm{op}}\to \left[\mathsf{C},\mathsf{Set}\right]$, which is defined as follows - \[ - \begin{tikzcd}[ampersand replacement=\&] - \mathsf{C}^{\mathrm{op}} % column 1 - \&[-25pt] % column 2 - \&[+30pt] % column 3 - \&[-30pt] \left[\mathsf{C},\mathsf{Set}\right] % column 4 - \&[+25pt] % column 5 - \&[-25pt] % column 6 - \& % column 7 - \\[-15pt] - X_1 \arrow[dd, "f"{name=L, left}] \&\& \& \mathrm{Hom}_{\mathsf{C}}\left(X_1,-\right) \arrow[dd, Rightarrow, "f^\star"{name=R}] \& \mathrm{Hom}_{\mathsf{C}}(X_1,Y_1) \arrow[dd, "{f^*}"'] \arrow[rr, "{g_*}"] \& \& \mathrm{Hom}_{\mathsf{C}}(X_1,Y_2) \arrow[dd, "f^*"] \\[-5pt] - \& \phantom{.}\arrow[r, "Y_{\mathsf{C}^{\mathrm{op}}}", squigarrow]\&\phantom{.} \&\phantom{.} \&\phantom{.}\&\\[-5pt] - X_2 \& \& \& \mathrm{Hom}_{\mathsf{C}}\left(X_2,-\right)\& \mathrm{Hom}_{\mathsf{C}}(X_2,Y_1) \arrow[rr, "{g_*}"'] \& \& \mathrm{Hom}_{\mathsf{C}}(X_2,Y_2) - \arrow[from=4-5, to=2-5, dash, start anchor={[xshift=-8ex,yshift=-3ex]}, end anchor={[xshift=-8ex,yshift=3ex]},decorate,decoration={brace,amplitude=10pt,raise=4pt}] - \arrow[from=3-4, to=3-5, dashed, start anchor={[xshift=9pt, yshift=-2pt]}, end anchor={[xshift=-11ex, yshift=-2pt]},decorate] - \end{tikzcd} - \] - Here we use the fact $\mathrm{Hom}_{\mathsf{C}^{\mathrm{op}}}\left(-,X\right)=\mathrm{Hom}_{\mathsf{C}}\left(X,-\right)$. - -} -\thm[yoneda_lemma]{Yoneda Lemma}{ - Let $\mathsf{C}$ be a locally small category. For any functor $F:\mathsf{C}^{\mathrm{op}}\to \mathsf{Set}$ and any $A\in \mathrm{Ob}(\mathsf{C})$, there is a natural bijection - \begin{align*} - q_{A,F}:\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(\operatorname{Hom}_{\mathsf{C}}\left(-,A\right), F\right) & \overset{\sim}{\longrightarrow} F(A) \\ - { \left[ \begin{tikzcd}[ampersand replacement=\&] - \mathsf{C}^{\mathrm{op}} \arrow[r, "\operatorname{Hom}_{\mathsf{C}}\left(-{,}A\right)"{name=A, above}, bend left] \arrow[r, "F"'{name=B, below}, bend right] \&[+30pt] \mathsf{Set} - \arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "\phi"] - \end{tikzcd}\right] } & \longmapsto \phi_A\left(\operatorname{id}_A\right) - \end{align*} - The naturality of $q_{A,F}$ means that - \[ - \begin{tikzcd}[ampersand replacement=\&] - \mathsf{C}^{\mathrm{op}}\times \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right]\&[-34pt]\&[+62pt]\&[-25pt] \mathsf{Set}\&[-25pt]\&[-25pt] \\ [-15pt] - (A_1,F_1) \arrow[dd, "\left(g{,} \eta\right)"{name=L, left}] - \&[-25pt] \& [+10pt] - \& [-30pt]\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(\mathrm{Hom}_{\mathsf{C}}(-,A_1),F_1)\arrow[dd, ""{name=R}] \& \ni \& \phi \arrow[dd,mapsto]\\ [-8pt] - \& \phantom{.}\arrow[r, "\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(Y_\mathsf{C}(-){,}-\right)", squigarrow]\&\phantom{.} \& \\[-8pt] - (A_2,F_2) \& \& \& \operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(\mathrm{Hom}_{\mathsf{C}}(-,A_2),F_2)\& \ni \& \eta\circ \phi\circ g_\star - \end{tikzcd} - \] - is a functor isomorphic to the \hyperref[th:evaluation_functor]{evaluation functor} - \begin{align*} - \mathrm{ev}:\mathsf{C}^{\mathrm{op}}\times \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right]&\longrightarrow \mathsf{Set}\\ - \left(A,F\right)&\longmapsto F(A) - \end{align*} - \textbf{Covariant version}\\ - For any functor $F:\mathsf{C}\to \mathsf{Set}$ and any $A\in \mathrm{Ob}(\mathsf{C})$, there is a natural bijection - \begin{align*} - \operatorname{Hom}_{\left[\mathsf{C},\mathsf{Set}\right]}\left(\operatorname{Hom}_{\mathsf{C}}\left(A,-\right), F\right) & \overset{\sim}{\longrightarrow} F(A) \\ - { \left[ \begin{tikzcd}[ampersand replacement=\&] - \mathsf{C} \arrow[r, "\operatorname{Hom}_{\mathsf{C}}\left(A{,}-\right)"{name=A, above}, bend left] \arrow[r, "F"'{name=B, below}, bend right] \&[+30pt] \mathsf{Set} - \arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "\phi"] - \end{tikzcd}\right] } & \longmapsto \phi_A\left(\operatorname{id}_A\right) + \mathrm{Hom}(x,y)=\begin{cases} + \{*\} & \text{if }x\leq y\\ + \varnothing & \text{otherwise} + \end{cases} \end{align*} } -\pf{ - We break the proof into following steps. -\begin{itemize}[leftmargin=*] - \item \textbf{$\phi$ is determined by $\phi_A\left(\operatorname{id}_A\right)$}.\\ - Suppose $\phi:\mathrm{Hom}_{\mathsf{C}}\left(-,A\right)\Rightarrow F$ is a natural transformation. Given any morphism $f:X\to A$ in $\mathsf{C}$, the naturality of $\phi$ gives the following commutative diagram - \[ - \begin{tikzcd}[ampersand replacement=\&, row sep = 3.5em] - \mathrm{id}_A\arrow[d, mapsto]\&[-25pt]\in\&[-25pt]\mathrm{Hom}_{\mathsf{C}}(A,A) \arrow[d, "{\phi_A}"'] \arrow[rr, "{f^*}"] \&[-20pt] \& {\mathrm{Hom}_{\mathsf{C}}(X,A)} \arrow[d, "\phi_X"]\&[-25pt]\ni\&[-25pt]f \arrow[d, mapsto]\\ - \phi_A(\mathrm{id}_A)\&\in\&{F(A)}\arrow[rr, "{F(f)}"'] \& \& {F(X)} \&\ni\& \phi_X(f) - \end{tikzcd} - \] - Thus we see $\phi$ is determined by $\phi_A(\mathrm{id}_A)$ as follows + +\dfn{Preserve, Reflect, Create Limits}{ + For any class of diagrams $K: \mathsf{J} \rightarrow \mathsf{C}$ valued in $\mathsf{C}$, a functor $F: \mathsf{C} \rightarrow \mathsf{D}$ + \begin{itemize} + \item \textbf{preserves} those limits if for any diagram $K: \mathsf{J} \rightarrow \mathsf{C}$ and limit cone over $K$, the image of this cone defines a limit cone over the composite diagram $F\circ K: \mathsf{J} \rightarrow \mathsf{D}$; + \item \textbf{reflects} those limits if any cone over a diagram $K: \mathsf{J} \rightarrow \mathsf{C}$, whose image upon applying $F$ is a limit cone for the diagram $F\circ K: \mathsf{J} \rightarrow \mathsf{D}$, is a limit cone over $K$; + \item \textbf{creates} those limits if whenever $F\circ K: \mathsf{J} \rightarrow \mathsf{D}$ has a limit in $\mathsf{D}$, there is some limit cone over $F\circ K$ that can be lifted to a limit cone over $K$, and moreover $F$ reflects the limits in the class of diagrams. + \end{itemize} +} + +\ex{Limit in $\mathsf{Set}$}{ + Let $F:\mathsf{J}\to \mathsf{Set}$ be a functor. Then + \begin{align*} + \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 \[ - \phi_X(f)=F(f)(\phi_A(\mathrm{id}_A)),\quad\forall X\in \mathrm{Ob}(\mathsf{C}),\forall f\in \mathrm{Hom}_{\mathsf{C}}(X,A). + \varprojlim F\xrightarrow{\quad\iota\quad}\prod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i)\xrightarrow{\quad\pi_i\quad}F(i). \] - This implies that the injectivity of $q_{A,F}$. - \item \textbf{Construct the inverse of $q_{A,F}$}.\\ - Define + $\varprojlim F$ also can be constructed from product and equalizer \begin{align*} - r_{A,F}: F(A)&\longrightarrow\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(\operatorname{Hom}_{\mathsf{C}}\left(-,A\right), F\right) \\ - u & \longmapsto \phi^u:\left[\begin{aligned}\phi_X^u: \mathrm{Hom}_{\mathsf{C}}(X,A)&\longrightarrow F(X)\\ - f&\longmapsto F(f)(u)\end{aligned} \right] + \varprojlim F\cong \ker\left[ + \begin{tikzcd}[ampersand replacement=\&] + \prod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i) \arrow[r, "\alpha", shift left] \arrow[r, "\beta"', shift right] \& \prod\limits_{\lambda \in \mathrm{Mor}(\mathsf{J})}F(\mathrm{codom}\left(\lambda\right)) + \end{tikzcd} + \right] \end{align*} - To ensure $r_{A,F}$ is well-defined, we need to check that $r_{A,F}(u)=\phi^u$ is always a natural transformation. In fact, given any morphism $h:X_1\to X_2$ in $\mathsf{C}$, it is easy to check the following diagram commutes + where $\alpha$ and $\beta$ are induced by the following diagrams \[ - \begin{tikzcd}[ampersand replacement=\&, row sep = 3.5em] - g\arrow[mapsto]{d}\&[-25pt]\in\&[-25pt]\mathrm{Hom}_{\mathsf{C}}(X_2,A) \arrow[d, "{\phi_{X_2}^u}"'] \arrow[rr, "{h^*}"] \&[-20pt] \& {\mathrm{Hom}_{\mathsf{C}}(X_1,A)} \arrow[d, "\phi_{X_1}^u"]\&[-25pt]\ni\&[-25pt]g\circ h \arrow[mapsto]{d}\\ - F(g)(u)\&\in\&{F(X_2)}\arrow[rr, "{F(h)}"'] \& \& {F(X_1)} \&\ni\& F(g\circ h)(u) + \begin{tikzcd}[ampersand replacement=\&] + F\left(\mathrm{dom}\left(\lambda\right)\right) \arrow[d, "F(\lambda)"'] \&[+25pt] \prod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i) \arrow[l, "\pi_{\mathrm{dom}\left(\lambda\right)}"'] \arrow[d, "\alpha", dashed] \&[-15pt]\ni \& [-15pt] (x_i)_{i\in \mathrm{Ob}(\mathsf{J})} \arrow[d, "\alpha", mapsto] \\ [+10pt] + F\left(\mathrm{codom}\left(\lambda\right)\right) \& {\prod\limits_{\lambda \in \mathrm{Mor}(\mathsf{J})}F(\mathrm{codom}\left(\lambda\right))} \arrow[l, "{\pi_{\lambda}}"]\&\ni\& (F(\lambda)(x_i))_{\lambda:i\to j\in \mathrm{Mor}(\mathsf{J})} \end{tikzcd} \] - because the functoriality of $F$ gives the identity $F(g\circ h)=F(h)\circ F(g)$. - - For any $u\in F(A)$, we have - \begin{align*} - \left(q_{A,F}\circ r_{A,F}\right)(u) - &=q_{A,F}(\phi^u)\\ - &=\phi^u_A(\mathrm{id}_A)\\ - &=F(\mathrm{id}_A)(u)\\ - &=\mathrm{id}_{F(A)}(u)\\ - &=u, - \end{align*} - which implies that $q_{A,F}$ is surjective. Therefore, $q_{A,F}$ is a bijection and $r_{A,F}$ is the inverse of $q_{A,F}$. - - We can also manually check that $\left(r_{A,F}\circ q_{A,F}\right)(\phi)=\phi$ for any natural transformation $\phi:\mathrm{Hom}_{\mathsf{C}}\left(-,A\right)\Rightarrow F$, by evaluating the natural transformations at $\mathrm{id}_A$, - \begin{align*} - \left(\left(r_{A,F}\circ q_{A,F}\right)(\phi)\right)_A(\mathrm{id}_A)&=(r_{A,F}(\phi_A(\mathrm{id}_A)))_A(\mathrm{id}_A)\\ - &=F(\mathrm{id}_A)(\phi_A(\mathrm{id}_A))\\ - &=\mathrm{id}_{F(A)}(\phi_A(\mathrm{id}_A))\\ - &=\phi_A(\mathrm{id}_A). - \end{align*} - \item \textbf{$q_{A,F}$ is natural in $A$ and $F$}.\\ - $\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(Y_\mathsf{C}(-){,}-\right)$ is a functor obtained by the composition \[ - \mathsf{C}^{\mathrm{op}}\times \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right]\xrightarrow{\left(Y^{\mathrm{op}}_{\mathsf{C}},\mathrm{id}\right)}\left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right]^{\mathrm{op}}\times \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right]\xrightarrow{\mathrm{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(-,-\right)}\mathsf{Set} + \begin{tikzcd}[ampersand replacement=\&] + \&[+25pt] \prod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i) \arrow[ld, "\pi_{\mathrm{codom}\left(\lambda\right)}"'] \arrow[d, "\beta", dashed] \&[-15pt]\ni \&[-15pt] (x_i)_{i\in \mathrm{Ob}(\mathsf{J})} \arrow[d, "\beta", mapsto] \\ [+10pt] + F\left(\mathrm{codom}\left(\lambda\right)\right) \& {\prod\limits_{\lambda \in \mathrm{Mor}(\mathsf{J})}F(\mathrm{codom}\left(\lambda\right))} \arrow[l, "{\pi_{\lambda}}"]\&\ni\& (x_j)_{\lambda:i\to j\in \mathrm{Mor}(\mathsf{J})} + \end{tikzcd} \] - Given any $g:A_2\to A_1$ and $\eta:F_1\to F_2$, we should check the following diagram commutes +} + + +\ex{Colimit in $\mathsf{Set}$}{ + Let $F:\mathsf{J}\to \mathsf{Set}$ be a functor. Define a relation $\sim^*$ on $\coprod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i)$: for any $(i,x),(j,y)\in \coprod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i)$, \[ - \begin{tikzcd}[row sep=3.5em, column sep=2.5em, ampersand replacement=\&] - {\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(Y_\mathsf{C}(A_1),F_1)} \arrow[d, "{q_{A_1,F_1}}"'] \arrow[rr, "\left(g_\star\right)^*\circ \eta_*"] \& \& {\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(Y_\mathsf{C}(A_2),F_2)} \arrow[d, "q_{A_2,F_2}"] \\ - {F_1(A_1)} \arrow[rr, "F_2(g)\circ \eta_{A_1}"'] \& \& {F_2(A_2)} - \end{tikzcd} + (i,x)\sim^* (j,y)\iff \text{there exists }\lambda:i\to j\text{ in }\mathrm{Hom}_\mathsf{J}(i,j)\text{ such that }F(\lambda)(x)=y. \] - It suffices to check that the following diagram commutes + Let $\sim$ denote the equivalence relation generated by $\sim^*$. Then \[ - \begin{tikzcd}[ampersand replacement=\&, row sep = 3.5em] - \operatorname{Hom}(Y_\mathsf{C}(A_1),F_1) \arrow[rr,"\left(g_\star\right)^*"] \arrow[dr,swap,"\eta_*"] \arrow[dd,swap,"q_{A_1,F_1}"] \&\& - \operatorname{Hom}(Y_\mathsf{C}(A_2),F_1) \arrow[dd,swap,"q_{A_2,F_1}" near start] \arrow[dr,"\eta_*"] \\ - \& \operatorname{Hom}(Y_\mathsf{C}(A_1),F_2) \arrow[rr,crossing over,"\left(g_\star\right)^*" near start] \&\& - \operatorname{Hom}(Y_\mathsf{C}(A_2),F_2) \arrow[dd,"q_{A_2,F_2}"] \\ - F_1(A_1) \arrow[rr,"F_1(g)" near end] \arrow[dr,swap,"\eta_{A_1}"] \&\& F_1(A_2) \arrow[dr,swap,"\eta_{A_2}"] \\ - \& F_2(A_1) \arrow[rr,"F_2(g)"] \arrow[uu,<-,crossing over,"q_{A_1,F_2}" near end]\&\& F_2(A_2) - \end{tikzcd} + \varinjlim F\cong\coprod_{i\in \mathrm{Ob}(\mathsf{J})}F(i)/\sim \] -\end{itemize} - To check the commutativity of the left square + and the map $\phi_i:F(i)\to \varinjlim F$ is given by the composition \[ - \begin{tikzcd}[row sep=3.5em, column sep=2.5em, ampersand replacement=\&] - {\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(Y_\mathsf{C}(A_1),F_1)} \arrow[d, "{q_{A_1,F_1}}"'] \arrow[rr, "\eta_*"] \& \& {\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(Y_\mathsf{C}(A_1),F_2)} \arrow[d, "q_{A_1,F_2}"] \\ - {F_1(A_1)} \arrow[rr, "\eta_{A_1}"'] \& \& {F_2(A_1)} - \end{tikzcd} + F(i)\xrightarrow{\quad\iota_i\quad}\coprod_{i\in \mathrm{Ob}(\mathsf{J})}F(i)\xrightarrow{\quad\pi\quad}\coprod_{i\in \mathrm{Ob}(\mathsf{J})}F(i)/\sim. \] - we can verify that for any $\phi\in \operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(Y_\mathsf{C}(A_1),F_1\right)$, - \begin{align*} - q_{A_1,F_2}\circ \eta_*\left(\phi\right)&=q_{A_1,F_2}\left(\eta\circ \phi\right)\\ - &=\left(\eta\circ \phi\right)_{A_1}\left(\mathrm{id}_{A_1}\right)\\ - &=\eta_{A_1}(\phi_{A_1}(\mathrm{id}_{A_1}))\\ - &=\eta_{A_1}\circ q_{A_1,F_1}(\phi). - \end{align*} - To check the commutativity of the front square + If $\mathsf{J}$ is a filtered category, then the equivalence relation $\sim$ has a explicit description: for any $(i,x),(j,y)\in \coprod\limits_{i\in \mathrm{Ob}(\mathsf{J})}F(i)$, \[ - \begin{tikzcd}[row sep=3.5em, column sep=2.5em, ampersand replacement=\&] - {\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(Y_\mathsf{C}(A_1),F_2)} \arrow[d, "{q_{A_1,F_2}}"'] \arrow[rr, "\left(g_\star\right)^*"] \& \& {\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}(Y_\mathsf{C}(A_2),F_2)} \arrow[d, "q_{A_2,F_2}"] \\ - {F_2(A_1)} \arrow[rr, "F_2(g)"'] \& \& {F_2(A_2)} - \end{tikzcd} + (i,x)\sim (j,y)\iff \text{there exists }\lambda_1:i\to k\text{ and }\lambda_2:j\to k\text{ such that }F(\lambda_1)(x)=F(\lambda_2)(y). \] - we can verify that for any $\psi\in \operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(Y_\mathsf{C}(A_1),F_2\right)$, we have - \begin{align*} - q_{A_2,F_2}\circ \left(g_\star\right)^*(\psi)&=q_{A_2,F_2}\left(\psi\circ g_\star\right)\\ - &=\left(\psi\circ g_\star\right)_{A_2}\left(\mathrm{id}_{A_2}\right)\\ - &=\psi_{A_2}\circ g_*\left(\mathrm{id}_{A_2}\right)\\ - &=\psi_{A_2}(g)\\ - &=F_2(g)(\psi_{A_1}(\mathrm{id}_{A_1}))\\ - &=F_2(g)\circ q_{A_1,F_2}(\psi). - \end{align*} -} -\cor{Yoneda Embedding}{ - Let $\mathsf{C}$ be a locally small category. The \hyperref[th:yoneda_embedding_functor]{Yoneda embedding functor} $Y_\mathsf{C}:\mathsf{C}\to \left[\mathsf{C}^{\mathrm{op}},\mathsf{Set}\right]$ is fully faithful. That is, for any $A_1,A_2\in\mathrm{Ob}\left(\mathsf{C}\right)$, the morphism set map - \begin{align*} - \left.Y_{\mathsf{C}}\right|_{\operatorname{Hom}_{\mathsf{C}}\left(A_1,A_2\right)}:\operatorname{Hom}_{\mathsf{C}}\left(A_1,A_2\right) &\longrightarrow\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(\operatorname{Hom}_{\mathsf{C}}\left(-,A_1\right), \operatorname{Hom}_{\mathsf{C}}\left(-,A_2\right)\right) \\ - f & \longmapsto f_\star - \end{align*} - is a bijection. That justifies the name ``embedding'' because $\mathsf{C}$ is embedded into $\mathsf{Psh}(\mathsf{C})$ via $Y_\mathsf{C}$ as a full subcategory. } -\pf{ - By \hyperref[th:yoneda_lemma]{Yoneda lemma}, there is a natural bijection +\pf{ + It is easy to show \[ - \operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(\operatorname{Hom}_{\mathsf{C}}\left(-,A_1\right), \operatorname{Hom}_{\mathsf{C}}\left(-,A_2\right)\right)\cong \operatorname{Hom}_{\mathsf{C}}\left(A_1,A_2\right), + \varinjlim F\cong\coprod_{i\in \mathrm{Ob}(\mathsf{J})}F(i)/\sim \] - which is given by - \begin{align*} - r_{A_1,Y_{\mathsf{C}}(A_2)}: \operatorname{Hom}_{\mathsf{C}}\left(A_1,A_2\right) &\longrightarrow\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(\operatorname{Hom}_{\mathsf{C}}\left(-,A_1\right), \operatorname{Hom}_{\mathsf{C}}\left(-,A_2\right)\right) \\ - f & \longmapsto f_\star:\left[\begin{aligned}\left(f_\star\right)_X: \mathrm{Hom}_{\mathsf{C}}(X,A_1)&\longrightarrow \mathrm{Hom}_{\mathsf{C}}(X,A_2)\\ - g&\longmapsto f_*(g)=f\circ g\end{aligned} \right] - \end{align*} - and - \begin{align*} - q_{A_1,Y_{\mathsf{C}}(A_2)}:\operatorname{Hom}_{\mathsf{Psh}(\mathsf{C})}\left(\operatorname{Hom}_{\mathsf{C}}\left(-,A_1\right), \operatorname{Hom}_{\mathsf{C}}\left(-,A_2\right)\right) & \overset{\sim}{\longrightarrow} \operatorname{Hom}_{\mathsf{C}}\left(A_1,A_2\right) \\ - { \left[ \begin{tikzcd}[ampersand replacement=\&] - \mathsf{C}^{\mathrm{op}} \arrow[r, "\operatorname{Hom}_{\mathsf{C}}\left(-{,}A_1\right)"{name=A, above}, bend left] \arrow[r, "\operatorname{Hom}_{\mathsf{C}}\left(-{,}A_2\right)"'{name=B, below}, bend right] \&[+30pt] \mathsf{Set} - \arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "\phi"] - \end{tikzcd}\right] } & \longmapsto \phi_{A_1}\left(\operatorname{id}_{A_1}\right) - \end{align*} - Note that $r_{A_1,Y_{\mathsf{C}}(A_2)}$ is exactly the morphism set map $\left.Y_{\mathsf{C}}\right|_{\operatorname{Hom}_{\mathsf{C}}\left(A_1,A_2\right)}$. -} -Yoneda embedding allows us to regard any category $\mathsf{C}$ as a full subcategory of $\mathsf{Psh}(\mathsf{C})$. So in the sense of categorical isomorphism, we can identify any object $A\in \mathrm{Ob}(\mathsf{C})$ with a presheaf $\operatorname{Hom}_{\mathsf{C}}(-, A)$. - -An object in a category is equivalent to the collection of all arrows pointing to that object. In other words, the information we care about is which arrows from an object itself and other objects would point to this object, while the object's internals are completely regarded as a black box. - -The arrows between objects in a category are equivalent to the transformations between sets of arrows pointing to objects. Specifically, suppose $f$ is an arrow from object $A$ to object $B$, then any arrow pointing to $A$ can be transformed into an arrow pointing to $B$ by appending an $f$, which is exactly what $f_\star$ does. Conversely, if there is a way to naturally transform all arrows pointing to $A$ into arrows pointing to $B$, then this transformation must be realized by appending an arrow between $A$ and $B$. - -\dfn{Representable Functor}{ - Let $\mathsf{C}$ be a locally small category. - \begin{itemize} - \item A functor $F:\mathsf{C}\to \mathsf{Set}$ is called \textbf{representable} if it is naturally isomorphic to $\operatorname{Hom}_{\mathsf{C}}\left(A,-\right)$ for some $A\in \mathrm{Ob}(\mathsf{C})$. - \item A functor $F:\mathsf{C}^{\mathrm{op}}\to \mathsf{Set}$ is called \textbf{representable} if it is naturally isomorphic to $\operatorname{Hom}_{\mathsf{C}}\left(-,A\right)$ for some $A\in \mathrm{Ob}(\mathsf{C})$. - \end{itemize} - 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]{}{ - 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 $F(f)(u)=x$). -} -\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 + by checking the universal property of $\varinjlim F$. If $\mathsf{J}$ is a filtered category, first we can check \[ - \begin{tikzcd}[ampersand replacement=\&, row sep = 3.5em] - \mathrm{id}_A\arrow[d, mapsto]\&[-25pt]\in\&[-25pt]\mathrm{Hom}_{\mathsf{C}}(A,A) \arrow[d, "{\phi_A}"'] \arrow[rr, "{f_*}"] \&[-20pt] \& {\mathrm{Hom}_{\mathsf{C}}(A,X)} \arrow[d, "\phi_X"]\&[-25pt]\ni\&[-25pt]f \arrow[d, mapsto]\\ - u=\phi_A(\mathrm{id}_A)\&\in\&{F(A)}\arrow[rr, "{F(f)}"'] \& \& {F(X)} \&\ni\& \phi_X(f) - \end{tikzcd} - \] - which implies + (i,x)\approx (j,y)\iff \text{there exists }\lambda_1:i\to k\text{ and }\lambda_2:j\to k\text{ such that }F(\lambda_1)(x)=F(\lambda_2)(y). + \] + is an equivalence relation. To show $\approx\;=\;\sim$, let's assume $\backsimeq$ is any equivalence relation containing $\sim^*$. If $(i,x)\approx(j,y)$, then there exists $\lambda_1:i\to k$ and $\lambda_2:j\to k$ such that $F(\lambda_1)(x)=F(\lambda_2)(y)=z$. Hence \[ - F(f)(u)=\phi_X(f),\quad\forall f\in \mathrm{Hom}_{\mathsf{C}}(A,X). + (i,x)\sim^* (k,z)\text{ and }(j,y)\sim^* (k,z)\implies(i,x)\backsimeq (k,z)\text{ and }(j,y)\backsimeq (k,z)\implies (i,x)\backsimeq (j,y). \] - Thus we have - \begin{align*} - (A,u)\text{ is initial in }\int_{\mathsf{C}}F&\iff \forall(X,x)\in \mathrm{Ob}\left(\int_{\mathsf{C}}F\right),\;\exists! f\in \mathrm{Hom}_{\mathsf{C}}(A,X),\;F(f)(u)=x\\ - &\iff\forall X\in \mathrm{Ob}\left(\mathsf{C}\right),\; \forall x\in F(X),\;\exists! f\in \mathrm{Hom}_{\mathsf{C}}(A,X),\;\phi_X(f)=x\\ - &\iff \forall X\in \mathrm{Ob}\left(\mathsf{C}\right),\;\phi_X \text{ is bijective}\\ - &\iff \phi \text{ is a natural isomorphism}\\ - &\iff (A,u) \text{ is a universal element of }F. - \end{align*} -} -\cor{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 $\Delta\{*\}$. + This implies $\backsimeq$ contains $\approx$. Therefore, $\approx$ is the smallest equivalence relation containing $\sim^*$, which means $\approx$ coincides with $\sim$. } @@ -1384,7 +1476,7 @@ \section{Adjoint Functor} \begin{scope} \clip (-2.5,0.2) rectangle (2.5,3); \fill[fill=green!20] (-2.5,0.2) rectangle (2.5,3); - \draw[fill=blue!20, rounded corners=0.6cm, line width=0.5pt] (-1,-1) rectangle (1, 2); + \draw[fill=blue!20, rounded corners=0.6cm, line width=0.7pt] (-1,-1) rectangle (1, 2); \end{scope} \node[below=2pt] at (-1,0.2) {$L$}; \node[below=2pt] at (1,0.2) {$R$}; @@ -1411,7 +1503,7 @@ \section{Adjoint Functor} \begin{scope} \clip (-2.5,0) rectangle (2.5,2.8); \fill[fill=blue!20] (-2.5,0) rectangle (2.5,2.8); - \draw[fill=green!20, rounded corners=0.6cm, line width=0.5pt] (-1,1) rectangle (1, 4); + \draw[fill=green!20, rounded corners=0.6cm, line width=0.7pt] (-1,1) rectangle (1, 4); \end{scope} \node[above=2pt] at (-1,2.8) {$R$}; \node[above=2pt] at (1,2.8) {$L$}; @@ -1489,7 +1581,7 @@ \section{Adjoint Functor} \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) + \filldraw[fill=blue!20, rounded corners=0.48cm, line width=0.7pt] (-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} @@ -1507,7 +1599,7 @@ \section{Adjoint Functor} \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); + \filldraw[fill=blue!20, line width=0.7pt] (5.2,-1) rectangle (7,4); \end{scope} \node[below] at (5.2,0) {$L$}; @@ -1576,8 +1668,8 @@ \section{Adjoint Functor} \fill[fill=blue!20] (-1,0) .. controls (-1,0.4) and (-0.8, 1) .. (0,1)--(0,0)--cycle; \fill[fill=green!45] (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); + \draw[line width=0.7pt] (0,0) -- (0,2); + \draw[line width=0.7pt] (-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) {$K$}; @@ -1594,8 +1686,8 @@ \section{Adjoint Functor} \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); + \draw[line width=0.7pt] (0,0) -- (0,2); + \draw[line width=0.7pt] (-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) {$K$}; @@ -1612,8 +1704,8 @@ \section{Adjoint Functor} \fill[fill=green!20] (-1,2) .. controls (-1, 1.6) and (-0.8, 1) .. (0,1)--(0,2)--cycle; \fill[fill=green!45] (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); + \draw[line width=0.7pt] (0,0) -- (0,2); + \draw[line width=0.7pt] (-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) {$K$}; @@ -1630,8 +1722,8 @@ \section{Adjoint Functor} \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); + \draw[line width=0.7pt] (0,0) -- (0,2); + \draw[line width=0.7pt] (-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) {$K$}; @@ -1652,8 +1744,8 @@ \section{Adjoint Functor} \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; - \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); + \draw[line width=0.7pt] (0,0) -- (0,2); + \draw[line width=0.7pt] (-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) {$K$}; @@ -1672,13 +1764,13 @@ \section{Adjoint Functor} \fill[fill=green!20] (-2,0) rectangle (0,4); \fill[fill=green!45] (0,0) rectangle (4,4); \filldraw[fill=blue!20] (-1,-1) -- (-1,0) .. controls (-1,0.4) and (-0.8, 1) .. (0,1)--(0,-1)--cycle; - \filldraw[fill=blue!45, line width=0.5pt] (0, -1) -- + \filldraw[fill=blue!45, line width=0.7pt] (0, -1) -- (0, 1) .. controls (0.8, 1) and (1, 1.6) .. (1, 2) .. controls (1, 2.4) and (1.2, 3) .. (2, 3) .. controls (2.8, 3) and (3, 2.4) .. (3, 2)--(3, -1)--cycle; \end{scope} - \draw[line width=0.5pt] (0,0) -- (0,4); + \draw[line width=0.7pt] (0,0) -- (0,4); \node[above=2pt] at (0, 4) {$H$}; \node[below=2pt] at (0, 0) {$K$}; @@ -1731,9 +1823,9 @@ \section{Adjoint Functor} \fill[fill=blue!20] (0,0) rectangle (2,2); \fill[fill=green!20] (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[dashed, color=black!50, line width=0.5pt] (-1,0) .. controls (-1, 0.4) and (-0.8, 1) .. (0, 1) ; - \draw[line width=0.5pt] (0, 1) .. controls (0.8, 1) and (1, 1.6) .. (1, 2); + \draw[line width=0.7pt] (0,0) -- (0,2); + \draw[dashed, color=black!50, line width=0.7pt] (-1,0) .. controls (-1, 0.4) and (-0.8, 1) .. (0, 1) ; + \draw[line width=0.7pt] (0, 1) .. controls (0.8, 1) and (1, 1.6) .. (1, 2); \node[above=2pt] at (0, 2) {$X$}; \node[below=2pt] at (0, 0) {$Y$}; @@ -1750,9 +1842,9 @@ \section{Adjoint Functor} \fill[fill=blue!20] (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[dashed, color=black!40, line width=0.5pt] (-1,0) .. controls (-1, 0.4) and (-0.8, 1) .. (0, 1); - \draw[line width=0.5pt] (0, 1) .. controls (0.8, 1) and (1, 0.4) .. (1, 0); + \draw[line width=0.7pt] (0,0) -- (0,2); + \draw[dashed, color=black!40, line width=0.7pt] (-1,0) .. controls (-1, 0.4) and (-0.8, 1) .. (0, 1); + \draw[line width=0.7pt] (0, 1) .. controls (0.8, 1) and (1, 0.4) .. (1, 0); \node[above=2pt] at (0, 2) {$X$}; \node[below=2pt] at (0, 0) {$Y$}; \node[below=2pt] at (1, 0) {$R$}; @@ -1767,8 +1859,8 @@ \section{Adjoint Functor} \fill[fill=green!20] (0,0) rectangle (2,2); \filldraw[fill=blue!20] (1.5,-1)--(1.5,0) .. controls (1.5, 0.5) and (0.7, 2.3) .. (0,1)--(0,-1)--cycle; \end{scope} - \draw[line width=0.5pt] (0,0) -- (0,2); - \draw[dashed, color=black!40, line width=0.5pt] (-0.5,0) .. controls (-0.5, 0.2) and (-0.3, 0.6) .. (0, 1); + \draw[line width=0.7pt] (0,0) -- (0,2); + \draw[dashed, color=black!40, line width=0.7pt] (-0.5,0) .. controls (-0.5, 0.2) and (-0.3, 0.6) .. (0, 1); \node[above=2pt] at (0, 2) {$X$}; \node[below=2pt] at (0, 0) {$Y$}; \node[below=2pt] at (1.5, 0) {$R$}; @@ -1785,9 +1877,9 @@ \section{Adjoint Functor} \fill[fill=blue!20] (1.4,0) .. controls (1.4,1) and (1, 1.4) .. (0,1.4)--(0,0)--cycle; \end{scope} - \draw[line width=0.5pt] (0,0) -- (0,2); + \draw[line width=0.7pt] (0,0) -- (0,2); - \draw[line width=0.5pt] (0, 1.4) .. controls (1, 1.4) and (1.4, 1) .. (1.4, 0); + \draw[line width=0.7pt] (0, 1.4) .. controls (1, 1.4) and (1.4, 1) .. (1.4, 0); \node[above=2pt] at (0, 2) {$X$}; \node[below=2pt] at (0, 0) {$Y$}; \node[below=2pt] at (1.4, 0) {$R$}; @@ -1805,6 +1897,30 @@ \section{Adjoint Functor} which is natural in both $X$ and $Y$. Hence $\phi$ is a natural isomorphism. It is straightforward to check that $T$ is a bijection. } +\prop{Equivalent Definition of Adjoint Functor Using Universal Morphism}{ + Given pair of functors $\begin{tikzcd}[ampersand replacement=\&] + \mathsf{C} \arrow[r, "L", bend left] \& \mathsf{D} \arrow[l, "R", bend left] + \end{tikzcd}$, then the following are equivalent + \begin{enumerate}[(i)] + \item $L \dashv R$. + \item For every object $X\in\mathrm{Ob}(\mathsf{C})$, there exists $\left(L(X), X\xrightarrow{\eta_X} R(L(X))\right)$ initial in $\left(X\downarrow R\right)$, i.e. there exists a \hyperref[th:universal_morphism]{universal morphism} from $X$ to $R$. + \item For every object $Y\in\mathrm{Ob}(\mathsf{D})$, there exists $\left(R(Y), L(R(Y))\xrightarrow{\varepsilon_Y} Y\right)$ terminal in $\left(L\downarrow Y\right)$, i.e. there exists a \hyperref[th:universal_morphism]{universal morphism} from $L$ to $Y$. + \end{enumerate} +} +\pf{ + If $L \dashv R$, then for every $X\in\mathrm{Ob}(\mathsf{C})$, we can let $X\xrightarrow{\eta_X} R(L(X))$ be the unit of the adjunction. Then for any $X\xrightarrow{g} R(Y)$, we have the following commutative diagram + \[ + \begin{tikzcd}[ampersand replacement=\&] + X \arrow[rd, "f"'] \arrow[r, "\eta_X"] \& R(L(X))\arrow[d, "{R\left(\phi_{X,Y}^{-1}(g)\right)}"] \\ + \& R(Y) + \end{tikzcd} + \] + because + \[ + R\left(\phi_{X,Y}^{-1}(g)\right)\circ \eta_X=R\left(\eta_Y\circ L(g)\right)\circ R\left(\eta_X\right)=R\left(\eta_Y\circ L(g)\circ \eta_X\right) + \] +} + \section{Monoidal Category} \dfn{Monoidal Category}{ A monoidal category is a category $\mathsf{V}$ equipped with @@ -1848,10 +1964,11 @@ \section{Monoidal Category} \dfn{Cartesian/Cocartesian Monoidal Category}{ \begin{itemize}[leftmargin=8pt] - \item A \textbf{cartesian monoidal category} is a monoidal category where the tensor product is the categorical product and the unit object is the terminal object. - \item A \textbf{cocartesian monoidal category} is a monoidal category where the tensor product is the categorical coproduct and the unit object is the initial object. + \item A \textbf{cartesian monoidal category} is a monoidal category with finite products endowed with a where the tensor product is the categorical product and the unit object is the terminal object. If a category has all finite products, then we say it is \textbf{cartesian monoidal}. + \item A \textbf{cocartesian monoidal category} is a monoidal category where the tensor product is the categorical coproduct and the unit object is the initial object. If a category has all finite coproducts, then we say it is \textbf{cocartesian monoidal}. \end{itemize} } + \ex{Category of Endofunctors is a Monoidal Category}{ Let $\mathsf{C}$ be a category. The \textbf{category of endofunctors} $\left[\mathsf{C},\mathsf{C}\right]$ is a monoidal category with the following structure \begin{enumerate}[(i)] @@ -1901,6 +2018,50 @@ \section{Enriched Category} \end{enumerate} } +\section{2-Category} + +\dfn{Strict 2-Category}{ + Let $\mathsf{Cat}$ be the Cartesian monoidal category consisting of all small categories. A \textbf{strict 2-category} is a $\mathsf{Cat}$-enriched category. We define the following sets + \begin{itemize} + \item 0-morphism set: $\mathrm{Ob}(\mathsf{C})$ + \item 1-morphism set: $\mathrm{Hom}_{\mathsf{C}}(X,Y)$ for any $X,Y\in \mathrm{Ob}(\mathsf{C})$ + \item 2-morphism set: $\mathrm{Hom}_{\mathsf{C}}(f,g)$ for any $X,Y\in \mathrm{Ob}(\mathsf{C})$ and $f,g\in \mathrm{Hom}_{\mathsf{C}}(X,Y)$ + \end{itemize} + + For any $X,Y,Z\in\mathrm{Ob}(\mathsf{C})$, we have the following composition bifunctor + \begin{align*} + \begin{tikzcd}[ampersand replacement=\&] + \mathrm{Hom}_{\mathsf{C}}(Y,Z)\times \mathrm{Hom}_{\mathsf{C}}(X,Y)\&[-25pt]\&[+10pt]\&[-30pt] \mathrm{Hom}_{\mathsf{C}}(X,Z)\&[-30pt]\&[-30pt] \\ [-15pt] + (f',f) \arrow[dd, "{(\theta', \theta)}"{name=L, left},Rightarrow] + \&[-25pt] \& [+10pt] + \& [-30pt]f'\circ f\arrow[dd, "\theta'\circ \theta"{name=R},Rightarrow] \\ [-10pt] + \& \phantom{.}\arrow[r, "\Theta", squigarrow]\&\phantom{.} \& \\[-10pt] + (g',g) \& \& \& g'\circ g + \end{tikzcd} + \end{align*} + which are called the horizontal composition of 2-morphisms. The functoriality of $\Theta$ means that given any vertical composition of 2-morphisms + \[ + \begin{tikzcd}[ampersand replacement=\&] + { (f',f)} \arrow[d, "{(\theta', \theta)}"', Rightarrow] \\ + { (g',g)} \arrow[d, "{(\psi', \psi)}"', Rightarrow] \\ + { (h',h)} + \end{tikzcd} + \] + we have the following equality + \begin{align*} + \Theta\left(\left(\psi'\circ\theta'\right),\left(\psi\circ\theta\right)\right)=\left(\psi'\circ\theta'\right)\circ\left(\psi\circ\theta\right)=\Theta\left(\psi',\psi\right)\circ\Theta\left(\theta',\theta\right)=\left(\psi'\circ\psi\right)\circ\left(\theta'\circ\theta\right) + \end{align*} + which is called the \textbf{interchange law}. +} + +\ex{$\mathsf{Cat}$ as 2-category}{ + The category $\mathsf{Cat}$ is a strict 2-category with the following structure + \begin{itemize} + \item 0-morphism set: $\mathrm{Ob}(\mathsf{Cat})$ + \item 1-morphism set: $\mathrm{Hom}_{\mathsf{Cat}}(\mathsf{C},\mathsf{D}):=[\mathsf{C},\mathsf{D}]$ for any $\mathsf{C},\mathsf{D}\in \mathrm{Ob}(\mathsf{Cat})$ + \item 2-morphism set: $\mathrm{Hom}_{\mathsf{Cat}}(F,G):=\mathrm{Hom}_{[\mathsf{C},\mathsf{D}]}\left(F,G\right)$ for any $\mathsf{C},\mathsf{D}\in \mathrm{Ob}(\mathsf{Cat})$ and $F,G\in [\mathsf{C},\mathsf{D}]$ + \end{itemize} +} \section{Abelian Category} Some literature refers to $\mathsf{Ab}$-categories as preadditive categories. We will not use this term in this note.