diff --git a/algebraic_construction.tex b/algebraic_construction.tex index 70dadfd..9a210aa 100644 --- a/algebraic_construction.tex +++ b/algebraic_construction.tex @@ -11,6 +11,7 @@ \usepackage{multirow} \usepackage{colortbl} \usepackage{mathrsfs} +\usepackage{stmaryrd} \usepackage{bbm} diff --git a/category_theory.tex b/category_theory.tex index 6e3c452..2d4284d 100644 --- a/category_theory.tex +++ b/category_theory.tex @@ -487,7 +487,6 @@ \subsubsection{Composition of Functors} \draw (7,2) node {$=$}; \begin{scope}[shift={(8,0)}] - % Draw background % Draw background \fill[leftcolor] (0,0) rectangle (2,4); \fill[rightcolor] (2,0) rectangle (4,4); @@ -522,7 +521,6 @@ \subsubsection{Composition of Functors} \draw (22,2) node {$=$}; \begin{scope}[shift={(23,0)}] - % Draw background % Draw background \fill[leftcolor] (0,0) rectangle (2,4); \fill[rightcolor] (2,0) rectangle (4,4); @@ -538,7 +536,7 @@ \subsubsection{Composition of Functors} \end{center} \subsubsection{Vertical Composition} -The vertical composition of natural transformations $\varphi:F\Rightarrow G$ and $\psi:G\Rightarrow H$ collapses two adjacent points in a string into one single point and ``eats" the intermediate segment, which is illustrated as follows +The vertical composition of natural transformations $\varphi:F\Rightarrow H$ and $\psi:G\Rightarrow I$ collapses two adjacent points in a string into one single point and ``eats" the intermediate segment, which is illustrated as follows \begin{center} \begin{tikzpicture}[x=0.6cm,y=0.6cm, baseline=(current bounding box.center)] @@ -581,8 +579,7 @@ \subsubsection{Vertical Composition} \end{tikzpicture} \end{center} \subsubsection{Horizontal Composition} -The horizontal composition of natural transformations $\varphi:F\Rightarrow G$ and $\theta:H\Rightarrow I$ stick two natural transformations in a horizontal line together, which is illustrated as follows - +To do horizontal composition for natural transformations $\varphi:F\Rightarrow H$ and $\theta:G\Rightarrow I$ on a horizontal line. First we can stick these two points together, which is illustrated as follows \begin{center} \begin{tikzpicture}[x=0.6cm,y=0.6cm, baseline=(current bounding box.center)] @@ -600,23 +597,25 @@ \subsubsection{Horizontal Composition} \draw[line width=0.7pt] (2,0) -- (2,4); \filldraw [black] (2,2) circle (2pt) node[anchor=east] {$\varphi$}; \draw[line width=0.7pt] (4,0) -- (4,4); - \filldraw [black] (4,2) circle (2pt) node[anchor=east] {$\theta$}; + \filldraw [black] (4,2) circle (2pt) node[anchor=west] {$\theta$}; + + % Draw dashed line + \draw[dashed, color=black!40, line width=0.7pt] (0,0) -- (6,0); + \draw[dashed, color=black!40, line width=0.7pt] (0,4) -- (6,4); % Label axes - \draw (2,0) node[below] {$G$}; + \draw (2,0) node[below] {$H$}; \draw (2,4) node[above] {$F$}; \draw (4,0) node[below] {$I$}; - \draw (4,4) node[above] {$H$}; + \draw (4,4) node[above] {$G$}; \end{scope} - \draw (7.5,2) node {$=$}; + \draw (8,2) node {$=$}; - \begin{scope}[shift={(9,0)}] - % Draw background - % Draw background - + \begin{scope}[shift={(10,0)}] \begin{scope} \clip (0,0) rectangle (6,4); + % Draw background \fill[leftcolor] (0,0) rectangle (3,4); \fill[rightcolor] (3,0) rectangle (6,4); \filldraw[fill=blue!20, line width=0.7pt] (2,5)-- (2,4) .. controls (2, 2.5) and (2.5, 2) .. (3,2).. controls (3.5,2) and (4, 2.5) .. (4,4)--(4,5)--cycle; @@ -625,17 +624,71 @@ \subsubsection{Horizontal Composition} % Draw vertical line and point \filldraw [black] (3,2) circle (2pt) node[left=9pt] {$\theta\circ\varphi$}; - + + % Draw dashed line + \draw[dashed, color=black!40, line width=0.7pt] (0,0) -- (6,0); + \draw[dashed, color=black!40, line width=0.7pt] (0,4) -- (6,4); + % Label axes - \draw (2,0) node[below] {$G$}; + \draw (2,0) node[below] {$H$}; \draw (2,4) node[above] {$F$}; \draw (4,0) node[below] {$I$}; - \draw (4,4) node[above] {$H$}; + \draw (4,4) node[above] {$G$}; \end{scope} \end{tikzpicture} \end{center} +Then we can composite the functors $F$ and $G$ to get a new functor $G\circ F$. Also we can composite the functors $H$ and $I$ to get a new functor $I\circ H$. By gluing these strings together, we get a new natural transformation $\theta\circ\varphi:G\circ F\Rightarrow I\circ H$. This is illustrated as follows + + +\begin{center} + \begin{tikzpicture}[x=0.6cm,y=0.6cm] + \definecolor{leftcolor}{RGB}{255,255,204} % light yellow + \definecolor{midcolor}{RGB}{204,204,255} % light purple + \definecolor{rightcolor}{RGB}{204,255,204} + + \begin{scope}[shift={(0,0)}] + \begin{scope} + \clip (0,0) rectangle (6,4); + % Draw background + \fill[leftcolor] (0,0) rectangle (3,4); + \fill[rightcolor] (3,0) rectangle (6,4); + \filldraw[fill=blue!20, line width=0.7pt] (2,5)-- (2,2) .. controls (2,1) and (2.5, 0.6) .. (3,0.6).. controls (3.5,0.6) and (4, 1) .. (4,2)--(4,5)--cycle; + \end{scope} + + % Draw vertical line and point + \draw[line width=0.7pt] (3,0) -- (3,0.6); + \filldraw [black] (3,0.6) circle (2pt); + \filldraw [black] (2,2) circle (2pt) node[anchor=east] {$\varphi$}; + \filldraw [black] (4,2) circle (2pt) node[anchor=west] {$\theta$}; + + % Label axes + \draw (2,4) node[above] {$F$}; + \draw (4,4) node[above] {$G$}; + \draw (1.75,1) node {$H$}; + \draw (4.1,1) node {$I$}; + \end{scope} + \draw (8,2) node {$=$}; + + \begin{scope}[shift={(10,0)}] + % Draw background + % Draw background + \fill[leftcolor] (0,0) rectangle (2,4); + \fill[rightcolor] (2,0) rectangle (4,4); + + % Draw vertical line and point + \draw[line width=0.7pt] (2,0) -- (2,4); + \filldraw [black] (2,0.6) circle (2pt); + \filldraw [black] (2,2) circle (2pt) node[left=1pt] {$\theta\circ \varphi$}; + \draw (2, 1.3) node[anchor=west] {$I\circ H$}; + + % Label axes + \draw (2,4) node[above] {$G\circ F$}; + \end{scope} + + \end{tikzpicture} +\end{center} \subsection{Morphism as Natural Transformation} If $f:X\to Y$ is a morphism in $\mathsf{C}$, then the following string diagram should be understood as follows @@ -1289,37 +1342,37 @@ \section{Representable Functor} \section{Adjoint Functor} \dfn{Adjoint Pair of Functors}{ - An \textbf{adjoint pair of functors} is a tuple $\left(L,R,\varphi\right)$ consisting of a pair of functors $\begin{tikzcd}[ampersand replacement=\&] + An \textbf{adjoint pair of functors} is a tuple $\left(L,R,\phi\right)$ consisting of a pair of functors $\begin{tikzcd}[ampersand replacement=\&] \mathsf{C} \arrow[r, "L", bend left] \& \mathsf{D} \arrow[l, "R", bend left] \end{tikzcd}$ and a natural isomorphism \[ \begin{tikzcd}[ampersand replacement=\&] - \mathsf{C}^{\mathrm{op}}\times\mathsf{D} \arrow[r, "\mathrm{Hom}_{\mathsf{D}}\left(L(-){,}-\right)"{name=A, above}, bend left] \arrow[r, "\mathrm{Hom}_{\mathsf{C}}\left(-{,}R(-)\right)"'{name=B, below}, bend right] \&[+50pt] \mathsf{Set} - \arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "\varphi", "\sim\hspace{1.5pt}"'] + \mathsf{C}^{\mathrm{op}}\times\mathsf{D} \arrow[r, "\scalebox{1.2}{$\mathrm{Hom}_{\mathsf{D}}\left(L(-){,}-\right)$}"{name=A, above}, bend left] \arrow[r, "\scalebox{1.2}{$\mathrm{Hom}_{\mathsf{C}}\left(-{,}R(-)\right)$}"'{name=B, below}, bend right] \&[+60pt] \mathsf{Set} + \arrow[Rightarrow, shorten <=5.5pt, shorten >=5.5pt, from=A.south-|B, to=B, "\phi", "\sim\hspace{1.5pt}"'] \end{tikzcd} \] which means for any $X\in \mathrm{Ob}(\mathsf{C})$ and $Y\in \mathrm{Ob}(\mathsf{D})$, there is a bijection \begin{align*} - \varphi_{X,Y}:\mathrm{Hom}_{\mathsf{D}}\left(L(X),Y\right)&\xlongrightarrow{\sim} \mathrm{Hom}_{\mathsf{C}}\left(X,R(Y)\right)\\ + \phi_{X,Y}:\mathrm{Hom}_{\mathsf{D}}\left(L(X),Y\right)&\xlongrightarrow{\sim} \mathrm{Hom}_{\mathsf{C}}\left(X,R(Y)\right)\\ \Big(L(X)\xlongrightarrow{f}Y\Big)&\longmapsto \Big(X\xlongrightarrow{\tilde{f}}R(Y)\Big) \end{align*} natural in $X$ and $Y$. $L$ is called the \textbf{left adjoint} of $R$, and $R$ is called the \textbf{right adjoint} of $L$. We write $L\dashv R$ to denote that $L$ is left adjoint to $R$. } -The naturality square of $\varphi$ means that for any morphism $g:X_2\to X_1$ in $\mathsf{C}$ and $h:Y_1\to Y_2$ in $\mathsf{D}$, the following diagram commutes +The naturality square of $\phi$ means that for any morphism $g:X_2\to X_1$ in $\mathsf{C}$ and $h:Y_1\to Y_2$ in $\mathsf{D}$, the following diagram commutes \[ \begin{tikzcd}[ampersand replacement=\&] - \mathrm{Hom}_{\mathsf{D}}\left(L(X_1),Y_1\right) \arrow[r, "{\mathrm{Hom}_{\mathsf{D}}\left(L(g){,}h\right)}"] \arrow[d, "\varphi_{X_1,Y_1}"', "\sim"] \&[+50pt] \mathrm{Hom}_{\mathsf{D}}\left(L(X_2),Y_2\right) \arrow[d, "\varphi_{X_2,Y_2}", "\sim"'] \\[+20pt] + \mathrm{Hom}_{\mathsf{D}}\left(L(X_1),Y_1\right) \arrow[r, "{\mathrm{Hom}_{\mathsf{D}}\left(L(g){,}h\right)}"] \arrow[d, "\phi_{X_1,Y_1}"', "\sim"] \&[+50pt] \mathrm{Hom}_{\mathsf{D}}\left(L(X_2),Y_2\right) \arrow[d, "\phi_{X_2,Y_2}", "\sim"'] \\[+20pt] \mathrm{Hom}_{\mathsf{C}}\left(X_1,R(Y_1)\right) \arrow[r, "{\mathrm{Hom}_{\mathsf{C}}\left(g{,}R(h)\right)}"'] \& \mathrm{Hom}_{\mathsf{C}}\left(X_2,R(Y_2)\right) \end{tikzcd} \] which in turn can be explicitly written as for any $f:L(X_1)\to Y_1$ in $\mathsf{D}$, \[ - \varphi_{X_2,Y_2}\left(h\circ f\circ L(g)\right)=R(h)\circ\varphi_{X_1,Y_1}(f)\circ g. + \phi_{X_2,Y_2}\left(h\circ f\circ L(g)\right)=R(h)\circ\phi_{X_1,Y_1}(f)\circ g. \] \dfn{Adjunction Unit and Counit}{ -Let $\left(L,R,\varphi\right)$ be an adjoint pair of functors. The \textbf{adjunction unit} $\eta$ of this adjunction is a natural transformation +Let $\left(L,R,\phi\right)$ be an adjoint pair of functors. The \textbf{adjunction unit} $\eta$ of this adjunction is a natural transformation \begin{center} \begin{tikzcd}[ampersand replacement=\&] \mathsf{C} \arrow[r, "\scalebox{1.2}{$\mathrm{id}_{\mathsf{C}}$}"{name=A, above}, bend left=40] \arrow[r, "\scalebox{1.2}{$R\circ L$}"'{name=B, below}, bend right=40] @@ -1339,10 +1392,10 @@ \section{Adjoint Functor} \draw[fill=black] (0, 2) circle (0.07); \end{tikzpicture} \end{center} -defined by $\eta_X:=\varphi_{X,L(X)}\left(\mathrm{id}_{L(X)}\right)$ for any $X\in \mathrm{Ob}(\mathsf{C})$, where $\varphi_{X,L(X)}$ is the natural bijection +defined by $\eta_X:=\phi_{X,L(X)}\left(\mathrm{id}_{L(X)}\right)$ for any $X\in \mathrm{Ob}(\mathsf{C})$, where $\phi_{X,L(X)}$ is the natural bijection $$ \begin{aligned} - \varphi_{X,L(X)}:\operatorname{Hom}_{\mathsf{D}}\left(L(X), L(X)\right) & \xlongrightarrow{\sim}\operatorname{Hom}_{\mathsf{C}}(X, RL(X)) \\ + \phi_{X,L(X)}:\operatorname{Hom}_{\mathsf{D}}\left(L(X), L(X)\right) & \xlongrightarrow{\sim}\operatorname{Hom}_{\mathsf{C}}(X, RL(X)) \\ \operatorname{id}_{F(X)} & \longmapsto \eta_X \end{aligned} $$ @@ -1366,19 +1419,19 @@ \section{Adjoint Functor} \draw[fill=black] (0, 1) circle (0.07); \end{tikzpicture} \end{center} -defined by $\varepsilon_Y:=\varphi_{R(Y),Y}^{-1}\left(\mathrm{id}_{R(Y)}\right)$ for any $Y\in \mathrm{Ob}(\mathsf{D})$, where $\varphi_{R(Y),Y}^{-1}$ is the natural bijection +defined by $\varepsilon_Y:=\phi_{R(Y),Y}^{-1}\left(\mathrm{id}_{R(Y)}\right)$ for any $Y\in \mathrm{Ob}(\mathsf{D})$, where $\phi_{R(Y),Y}^{-1}$ is the natural bijection $$ \begin{aligned} - \varphi_{R(Y),Y}^{-1}:\operatorname{Hom}_{\mathsf{D}}\left(R(Y), R(Y)\right) & \xlongrightarrow{\sim}\operatorname{Hom}_{\mathsf{C}}(LR(Y), R(Y)) \\ + \phi_{R(Y),Y}^{-1}:\operatorname{Hom}_{\mathsf{D}}\left(R(Y), R(Y)\right) & \xlongrightarrow{\sim}\operatorname{Hom}_{\mathsf{C}}(LR(Y), R(Y)) \\ \mathrm{id}_{R(Y)} & \longmapsto \varepsilon_Y \end{aligned} $$ } \pf{ - By naturality of $\varphi$, for any morphism $g:X_2\to X_1$ in $\mathsf{C}$, we have the following commutative diagram + By naturality of $\phi$, for any morphism $g:X_2\to X_1$ in $\mathsf{C}$, we have the following commutative diagram \[ \begin{tikzcd}[ampersand replacement=\&] - \mathrm{id}_{L(X_1)}\arrow[d,mapsto]\&[-30pt]\ni\&[-29pt]{\operatorname{Hom}_{\mathsf{D}}\left(L(X_1), L(X_1)\right) } \arrow[d, "{\varphi_{X_1,L(X_1)}}"'] \arrow[r,"{\left(L(g)\right)^*}"] \& [+12pt]{\operatorname{Hom}_{\mathsf{D}}\left(L(X_2), L(X_1)\right) } \arrow[d, "{\varphi_{X_2,L(X_1)}}"] \&[+12pt] {\operatorname{Hom}_{\mathsf{D}}\left(L(X_2), L(X_2)\right) } \arrow[d, "{\varphi_{X_2,L(X_2)}}"] \arrow[l,"{\left(L(g)\right)_*}"'] \&[-28pt]\in\&[-28pt]\mathrm{id}_{L(X_2)}\arrow[d,mapsto]\\[+20pt] + \mathrm{id}_{L(X_1)}\arrow[d,mapsto]\&[-30pt]\ni\&[-29pt]{\operatorname{Hom}_{\mathsf{D}}\left(L(X_1), L(X_1)\right) } \arrow[d, "{\phi_{X_1,L(X_1)}}"'] \arrow[r,"{\left(L(g)\right)^*}"] \& [+12pt]{\operatorname{Hom}_{\mathsf{D}}\left(L(X_2), L(X_1)\right) } \arrow[d, "{\phi_{X_2,L(X_1)}}"] \&[+12pt] {\operatorname{Hom}_{\mathsf{D}}\left(L(X_2), L(X_2)\right) } \arrow[d, "{\phi_{X_2,L(X_2)}}"] \arrow[l,"{\left(L(g)\right)_*}"'] \&[-28pt]\in\&[-28pt]\mathrm{id}_{L(X_2)}\arrow[d,mapsto]\\[+20pt] \eta_{X_1}\&\ni\&{\operatorname{Hom}_{\mathsf{C}}(X_1, RL(X_1))} \arrow[r,"g^*"] \& {\operatorname{Hom}_{\mathsf{C}}(X_2, RL(X_1))} \& {\operatorname{Hom}_{\mathsf{C}}(X_2, RL(X_2))} \arrow[l, "\left(RL(g)\right)_*"'] \&[-30pt]\in\&[-30pt]\eta_{X_2} \end{tikzcd} \] @@ -1388,7 +1441,7 @@ \section{Adjoint Functor} \] we have \[ - g^*(\eta_{X_1})=\varphi_{X_2,L(X_1)}(L(g))=\left(RL(g)\right)_*(\eta_{X_2}), + g^*(\eta_{X_1})=\phi_{X_2,L(X_1)}(L(g))=\left(RL(g)\right)_*(\eta_{X_2}), \] which implies the naturality square of $\eta$ commutes \[ @@ -1406,22 +1459,30 @@ \section{Adjoint Functor} \] } \prop{Adjunction Isomorphism Determined by Unit or Counit}{ - Let $\left(L,R,\varphi\right)$ be an adjoint pair of functors and $\eta$ and $\varepsilon$ be the adjunction unit and counit respectively. Then + Let $\left(L,R,\phi\right)$ be an adjoint pair of functors and $\eta$ and $\varepsilon$ be the adjunction unit and counit respectively. Then \begin{align*} -\varphi_{X,Y}(f) & =G f \circ \eta_X: X \rightarrow G Y, \quad \forall f: F X \rightarrow Y , \\ -\varphi_{X,Y}^{-1}(g) & =\varepsilon_Y \circ F g: F X \rightarrow Y, \quad \forall g: X \rightarrow G Y . +\phi_{X,Y}(f) & =R(f) \circ \eta_X: X \rightarrow R(Y), \quad \forall f: L(X) \rightarrow Y , \\ +\phi_{X,Y}^{-1}(g) & =\varepsilon_Y \circ L(g): L(X) \rightarrow Y, \quad \forall g: X \rightarrow R(Y) . \end{align*} } \pf{ + For any object $X\in \mathrm{Ob}(\mathsf{C})$ and any morphism $f:L(X)\to Y$ in $\mathsf{D}$, we have the following commutative diagram \[ \begin{tikzcd}[ampersand replacement=\&, row sep = 3.5em] - \mathrm{id}_A\arrow[d, mapsto]\&[-25pt]\in\&[-25pt]\mathrm{Hom}_{\mathsf{C}}(A,A) \arrow[d, "{\phi_A}"'] \arrow[rr, "{f_*}"] \&[-20pt] \& {\mathrm{Hom}_{\mathsf{C}}(A,X)} \arrow[d, "\phi_X"]\&[-25pt]\ni\&[-25pt]f \arrow[d, mapsto]\\ - u=\phi_A(\mathrm{id}_A)\&\in\&{F(A)}\arrow[rr, "{F(f)}"'] \& \& {F(X)} \&\ni\& \phi_X(f) + \mathrm{id}_{L(X)}\arrow[d, mapsto]\&[-25pt]\in\&[-25pt]\mathrm{Hom}_{\mathsf{C}}(L(X),L(X)) \arrow[d, "{\phi_{X,L(X)}}"'] \arrow[rr, "{f_*}"] \&[-15pt] \& {\mathrm{Hom}_{\mathsf{C}}(L(X),Y)}\arrow[d, "\phi_{X,Y}"]\&[-25pt]\ni\&[-25pt]f \arrow[d, mapsto]\\ + \eta_X\&\in\&{\mathrm{Hom}_{\mathsf{C}}(X,RL(X))} \arrow[rr, "{\left(R(f)\right)_*}"'] \& \& {\mathrm{Hom}_{\mathsf{C}}(X,R(Y))} \&\ni\& \phi_{X,Y}(f) \end{tikzcd} \] + For any object $Y\in \mathrm{Ob}(\mathsf{D})$ and any morphism $g:X\to R(Y)$ in $\mathsf{C}$, we have the following commutative diagram + \[ + \begin{tikzcd}[ampersand replacement=\&, row sep = 3.5em] + \mathrm{id}_{R(Y)}\arrow[d, mapsto]\&[-25pt]\in\&[-25pt]\mathrm{Hom}_{\mathsf{D}}(R(Y),R(Y)) \arrow[d, "{\phi_{R(Y),Y}^{-1}}"'] \arrow[rr, "{g^*}"] \&[-15pt] \& {\mathrm{Hom}_{\mathsf{D}}(X,R(Y))}\arrow[d, "\phi_{R(Y),Y}^{-1}"]\&[-25pt]\ni\&[-25pt]g \arrow[d, mapsto]\\ + \varepsilon_Y\&\in\&{\mathrm{Hom}_{\mathsf{C}}(LR(Y),Y)} \arrow[rr, "{\left(L(g)\right)^*}"'] \& \& {\mathrm{Hom}_{\mathsf{C}}(L(X),Y)} \&\ni\& \phi_{X,Y}^{-1}(g) + \end{tikzcd} + \] } \lemm{Snake Equations}{ - Let $\left(L,R,\varphi\right)$ be an adjoint pair of functors and $\eta$ and $\varepsilon$ be the adjunction unit and counit respectively. Then we have the following equalities of natural transformations + Let $\left(L,R,\phi\right)$ be an adjoint pair of functors and $\eta$ and $\varepsilon$ be the adjunction unit and counit respectively. Then we have the following equalities of natural transformations \begin{center} \begin{tikzpicture}[x=0.8cm,y=0.8cm] \node at (-7.4,1.5) {${\left[L \xRightarrow{L \eta}LR L\xRightarrow{\varepsilon L} L\right]=\mathrm{id}_L}$}; @@ -1497,7 +1558,7 @@ \section{Adjoint Functor} There are bijections between the following sets of natural transformations \[ \begin{tikzcd}[ampersand replacement=\&] - {\mathrm{Hom}_{[\mathsf{C}, \mathsf{D'}]}\left(L'H, KL\right) } \arrow[r, shift left=1pt, rightharpoonup, "\triangleright"]\arrow[r, shift right=1pt,leftharpoondown, "\triangleleft"'] \& {\mathrm{Hom}_{[\mathsf{C}, \mathsf{C'}]}(H, R'KL) } \arrow[d, shift left=1pt, rightharpoonup, "\triangleright"]\arrow[d, shift right=1pt,leftharpoondown, "\triangleleft"'] \\ + {\mathrm{Hom}_{[\mathsf{C}, \mathsf{D'}]}\left(L'H, KL\right) } \arrow[r, shift left=1pt, rightharpoonup, "\triangleright"]\arrow[r, shift right=1pt,leftharpoondown, "\triangleleft"'] \& {\mathrm{Hom}_{[\mathsf{C}, \mathsf{C'}]}(H, R'KL) } \arrow[d, shift left=1pt, rightharpoonup, "\triangleright"]\arrow[d, shift right=1pt,leftharpoondown, "\triangleleft"'] \\[+15pt] { \mathrm{Hom}_{[\mathsf{D}, \mathsf{D'}]}\left(L'HR, K\right)} \arrow[u, shift left=1pt, rightharpoonup, "\triangleright"]\arrow[u, shift right=1pt,leftharpoondown, "\triangleleft"'] \& {\mathrm{Hom}_{[\mathsf{D}, \mathsf{C'}]}(HR, R'K) } \arrow[l, shift left=1pt, rightharpoonup, "\triangleright"]\arrow[l, shift right=1pt,leftharpoondown, "\triangleleft"'] \end{tikzcd} \] @@ -1509,12 +1570,11 @@ \section{Adjoint Functor} \draw [|-To] (7,-1) -- (7,-2) node[midway,right] {$\triangleright$}; \draw [|-To] (4,-4) -- (3,-4) node[midway,above] {$\triangleright$}; \draw [|-To] (0,-2) -- (0,-1) node[midway,right] {$\triangleright$}; - \begin{scope} - %\clip (-2,0) rectangle (2,2); + \begin{scope} \fill[fill=green!20] (-2,0) rectangle (0,2); \fill[fill=blue!45] (0,0) rectangle (2,2); \fill[fill=blue!20] (-1,0) .. controls (-1,0.4) and (-0.8, 1) .. (0,1)--(0,0)--cycle; - \fill[fill=green!40] (1,2) .. controls (1,1.6) and (0.8, 1) .. (0,1)--(0,2)--cycle; + \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); @@ -1528,8 +1588,7 @@ \section{Adjoint Functor} \node[left=7pt, above=1pt] at (0, 1) {$\alpha$}; \begin{scope}[shift={(7,0)}] - \begin{scope} - %\clip (-2,0) rectangle (2,2); + \begin{scope} \fill[fill=green!20] (-2,0) rectangle (0,2); \fill[fill=green!45] (0,0) rectangle (2,2); \fill[fill=blue!20] (-1,0) .. controls (-1, 0.4) and (-0.8, 1) .. (0,1)--(0,0)--cycle; @@ -1547,12 +1606,11 @@ \section{Adjoint Functor} \node[left=7pt, above=1pt] at (0, 1) {$\beta$}; \end{scope} \begin{scope}[shift={(0,-5)}] - \begin{scope} - %\clip (-2,0) rectangle (2,2); + \begin{scope} \fill[fill=blue!20] (-2,0) rectangle (0,2); \fill[fill=blue!45] (0,0) rectangle (2,2); \fill[fill=green!20] (-1,2) .. controls (-1, 1.6) and (-0.8, 1) .. (0,1)--(0,2)--cycle; - \fill[fill=green!40] (1,2) .. controls (1, 1.6) and (0.8, 1) .. (0,1)--(0,2)--cycle; + \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); @@ -1566,8 +1624,7 @@ \section{Adjoint Functor} \node[left=7pt, below=1pt] at (0, 1) {$\delta$}; \end{scope} \begin{scope}[shift={(7,-5)}] - \begin{scope} - %\clip (-2,0) rectangle (2,2); + \begin{scope} \fill[fill=blue!20] (-2,0) rectangle (0,2); \fill[fill=green!45] (0,0) rectangle (2,2); \fill[fill=green!20] (-1,2) .. controls (-1, 1.6) and (-0.8, 1) .. (0,1)--(0,2)--cycle; @@ -1587,8 +1644,166 @@ \section{Adjoint Functor} \end{tikzpicture} \end{center} } -We only explicit define $\alpha^\triangleright$ and $\alpha^\triangleleft$ here. The other wire bendings are similar. +We only explicit define $\alpha^\triangleright$ here. The other wire bendings are defined similarly using the unit or counit of the adjunctions. +\begin{center} \begin{tikzpicture}[x=0.8cm,y=0.8cm, baseline=(current bounding box.center)] + \begin{scope} + \fill[fill=green!20] (-2,0) rectangle (0,2); + \fill[fill=green!45] (0,0) rectangle (2,2); + \fill[fill=blue!20] (-1,0) .. controls (-1, 0.4) and (-0.8, 1) .. (0,1)--(0,0)--cycle; + \fill[fill=blue!40] (1,0) .. controls (1, 0.4) and (0.8, 1) .. (0,1)--(0,0)--cycle; + + \draw[line width=0.5pt] (0,0) -- (0,2); + \draw[line width=0.5pt] (-1,0) .. controls (-1, 0.4) and (-0.8, 1) .. (0, 1) .. controls (0.8, 1) and (1, 0.4) .. (1, 0); + + \node[above=2pt] at (0, 2) {$H$}; + \node[below=2pt] at (0, 0) {$K$}; + \node[below=2pt] at (-1, 0) {$L$}; + \node[below=2pt] at (1, 0) {$R'$}; + \draw[fill=black] (0, 1) circle (0.07); + \node[left=8pt, above=1pt] at (0, 1) {$\alpha^{\triangleright}$}; + \end{scope} + + \node at (4, 1) {$:=$}; + + \begin{scope} [shift={(8,0)}] + \begin{scope} + \clip (-2,0) rectangle (4,4); + \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) -- + (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); + + \node[above=2pt] at (0, 4) {$H$}; + \node[below=2pt] at (0, 0) {$K$}; + \node[below=2pt] at (-1, 0) {$L$}; + \node[left=5pt,above=2pt] at (1, 2) {$L'$}; + \node[below=2pt] at (3, 0) {$R'$}; + + \draw[fill=black] (0, 1) circle (0.07); + \node[left=7pt, above=1pt] at (0, 1) {$\alpha$}; + + \draw[fill=black] (2, 3) circle (0.07); + \node[above=1pt] at (2, 3) {$\eta'$}; + + \draw[dashed, color=black!70, line width=0.7pt] (-2,0) rectangle (2,2); + \end{scope} + \end{tikzpicture} +\end{center} +The fact that $\triangleleft$ is the inverse of $\triangleright$ follows from the snake equations. + +\prop{Equivalent Definition of Adjoint Functor Using Unit and Counit}{ + Given pair of functors $\begin{tikzcd}[ampersand replacement=\&] + \mathsf{C} \arrow[r, "L", bend left] \& \mathsf{D} \arrow[l, "R", bend left] + \end{tikzcd}$ + there is a bijection between the following sets + \begin{align*} + T:\left\{\text{adjoint pair }\left(L,R,\phi\right)\right\}&\xrightleftharpoons{\;\sim\;}\left\{\left(L,R,\eta,\varepsilon\right)\text{ that satisfies snake equations}\right\}\\ + \phi&\longmapsto\left(\eta_X:=\phi\left(\mathrm{id}_{L(X)}\right), \varepsilon_Y:=\phi^{-1}\left(\mathrm{id}_{R(Y)}\right)\right)\\ + \phi_{X,Y}(f):=R(f) \circ \eta_X&\longmapsfrom(\eta, \varepsilon) + \end{align*} +} +\pf{ + Suppose $\left(L,R,\eta,\varepsilon\right)$ satisfies snake equations. Consider the diagram + \[ + \begin{tikzcd}[ampersand replacement=\&] + \mathsf{1} \arrow[dd, "\mathrm{id}"', bend right=38] \arrow[r, "\Delta X"] \&[+36pt] \mathsf{C} \arrow[dd, "L"', bend right=38] \\[-5pt] + \dashv \& \dashv \\[-5pt] + \mathsf{1} \arrow[uu, "\mathrm{id}"', bend right=38] \arrow[r, "\Delta Y"'] \& \mathsf{D} \arrow[uu, "R"', bend right=38] + \end{tikzcd} + \] + + \begin{center} + \begin{tikzpicture}[x=0.8cm,y=0.8cm, baseline=(current bounding box.center)] + \definecolor{leftcolor}{RGB}{255,255,204} + \draw [|-To] (2.5,1) -- (3.5,1) node[midway,above] {$\triangleright$}; + \node at (9, 1) {$:=$}; + \node at (14, 1) {$=$}; + + \begin{scope} + \fill[fill=leftcolor] (-2,0) rectangle (0,2); + \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); + + \node[above=2pt] at (0, 2) {$X$}; + \node[below=2pt] at (0, 0) {$Y$}; + \node[above=2pt] at (1, 2) {$L$}; + + \draw[fill=black] (0, 1) circle (0.07); + \node[left=7pt, above=-2pt] at (0, 1) {$f$}; + + \begin{scope}[shift={(6,0)}] + \begin{scope} + + \fill[fill=leftcolor] (-2,0) rectangle (0,2); + \fill[fill=green!20] (0,0) rectangle (2,2); + + \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); + \node[above=2pt] at (0, 2) {$X$}; + \node[below=2pt] at (0, 0) {$Y$}; + \node[below=2pt] at (1, 0) {$R$}; + + \draw[fill=black] (0, 1) circle (0.07); + \node[left=7pt, above=-2pt] at (0, 1) {$f^{\hspace{0.2pt}\triangleright}$}; + \end{scope} + \begin{scope}[shift={(11,0)}] + \begin{scope} + \clip (-1,0) rectangle (2,2); + \fill[fill=leftcolor] (-2,0) rectangle (0,2); + \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); + \node[above=2pt] at (0, 2) {$X$}; + \node[below=2pt] at (0, 0) {$Y$}; + \node[below=2pt] at (1.5, 0) {$R$}; + + \draw[fill=black] (0, 1) circle (0.07); + \draw[fill=black] (0.54, 1.46) circle (0.07) node[above=0pt] {$\eta$}; + \node[left=7pt, above=-2pt] at (0, 1) {$f$}; + \end{scope} + + \begin{scope}[shift={(16,0)}] + \begin{scope} + \fill[fill=leftcolor] (-1,0) rectangle (0,2); + \fill[fill=green!20] (0,0) rectangle (2,2); + + \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=0.5pt] (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) {$R$}; + + \draw[fill=black] (0, 0.7) circle (0.07) node[left=0pt] {$f$}; + \draw[fill=black] (0, 1.4) circle (0.07) node[left=0pt] {$\eta_X$}; + \end{scope} + + \end{tikzpicture} + \end{center} + As the string diagram shows, we find $\phi_{X,Y}$ conincides with the wire bending map + \[ + \triangleright: \mathrm{Hom}_{\mathsf{D}}\left(L(X), Y\right)\xlongrightarrow{\sim} \mathrm{Hom}_{\mathsf{C}}(X, R(Y)) + \] + which is natural in both $X$ and $Y$. Hence $\phi$ is a natural isomorphism. It is straightforward to check that $T$ is a bijection. +} \section{Monoidal Category} \dfn{Monoidal Category}{