diff --git a/src/content/1.10/natural-transformations.tex b/src/content/1.10/natural-transformations.tex index 85f5b2d0..f535dd04 100644 --- a/src/content/1.10/natural-transformations.tex +++ b/src/content/1.10/natural-transformations.tex @@ -77,7 +77,12 @@ \begin{figure}[H] \centering -\includegraphics[width=0.4\textwidth]{images/3_naturality.jpg} +\begin{tikzcd}[row sep=large,column sep=large,cells={nodes={font=\fontsize{10}{6}\selectfont}}] + & F a \arrow[dd, dashed, pos=0.7, "F f"] \arrow[dr, "\alpha_a"] & \\ + a \arrow[rr, crossing over] \arrow[ur] \arrow[dd, "f"] & & G a \arrow[dd, "G_f"] \\ + & F b \arrow[dr, "\alpha_b"] & \\ + b \arrow[rr] \arrow[ur] & & G b \\ +\end{tikzcd} \end{figure} \noindent @@ -95,7 +100,10 @@ \begin{figure}[H] \centering -\includegraphics[width=0.4\textwidth]{images/4_transport.jpg} +\begin{tikzcd}[row sep=huge,column sep=huge,cells={nodes={font=\fontsize{10}{6}\selectfont}}] + F a \arrow[r, "\alpha_a"] \arrow[d, swap, bend right=30, "F f"] & G a \arrow[d, "G f"]\\ + F b \arrow[r, dashed, "\alpha_b"] \arrow[u, swap, dashed, bend right=30, "(F f)^{-1}"] & G b +\end{tikzcd} \end{figure} \noindent @@ -406,7 +414,11 @@ \section{Functor Category} \begin{figure}[H] \centering -\includegraphics[width=0.4\textwidth]{images/5_vertical.jpg} +\begin{tikzcd}[row sep=large,column sep=large,cells={nodes={font=\fontsize{10}{6}\selectfont}}] + & & F a \arrow[dr, "\alpha a"] \arrow[dd, dashed, out=0, in=0, looseness=2.5, "(\beta\cdot\alpha) a"] & \\ + a \arrow[urr, "F"] \arrow[rrr, "G"] \arrow[drr, "H"] & & & G a \arrow[dl, "\beta a"] \\ + & & H a & \\ +\end{tikzcd} \end{figure} \noindent @@ -416,7 +428,13 @@ \section{Functor Category} \begin{figure}[H] \centering -\includegraphics[width=0.35\textwidth]{images/6_verticalnaturality.jpg} +\begin{tikzcd}[row sep=large,column sep=large,cells={nodes={font=\fontsize{10}{6}\selectfont}}] + & & & F a \arrow[dl, "\alpha_a"] \arrow[dd, "F f"] \\ + a \arrow[urrr, "F"] \arrow[rr, "G"] \arrow[dr, "H"] \arrow[dd, "f"] & & G a \arrow[dl, "\beta_a"] & \\ + & H a & & F b \arrow[dl, "\alpha_b"] \\ + b \arrow[urrr, dashed] \arrow[rr, dashed] \arrow[dr] & & G b \arrow[dl, "\beta_b"] \arrow[from=uu, crossing over, "G f"] & \\ + & H b \arrow[from=uu, crossing over, pos=0.41, "H f"] & & \\ +\end{tikzcd} \end{figure} \noindent @@ -439,7 +457,9 @@ \section{Functor Category} \begin{figure}[H] \centering -\includegraphics[width=0.3\textwidth]{images/6a_vertical.jpg} +\begin{tikzcd}[column sep=huge,cells={nodes={font=\fontsize{10}{6}\selectfont}}] + \mathcal{C} \arrow[r, bend left=65, pos=.37, "F"] \arrow[r, pos=.2, "G"] \arrow[r, bend right=65, pos=.2, "H"] \arrow[r, bend left=65, ""{name=U, below}] \arrow[r, ""{name=M, below}] \arrow[r, bend right=65, ""{name=D, below}] & \mathcal{D} \arrow[Rightarrow, from=U, to=M, "\alpha"] \arrow[Rightarrow, from=M, to=D, "\beta"] +\end{tikzcd} \end{figure} \noindent @@ -548,7 +568,9 @@ \section{2-Categories} \begin{figure}[H] \centering -\includegraphics[width=0.4\textwidth]{images/10_horizontal.jpg} +\begin{tikzcd}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + \mathcal{C} \arrow[r, bend left=50, "F"{name=U, above}] \arrow[r, bend right=50, "F'"{name=D, below}] & \mathcal{D} \arrow[Rightarrow, from=U, to=D, "\alpha"] \arrow[r, bend left=50, "G"{name=U, above}] \arrow[r, bend right=50, "G'"{name=D, below}] & \mathcal{E} \arrow[Rightarrow, from=U, to=D, "\beta"] +\end{tikzcd} \end{figure} \noindent diff --git a/src/content/2.1/declarative-programming.tex b/src/content/2.1/declarative-programming.tex index 425fca13..b713880e 100644 --- a/src/content/2.1/declarative-programming.tex +++ b/src/content/2.1/declarative-programming.tex @@ -181,7 +181,21 @@ \begin{figure}[H] \centering -\includegraphics[width=0.35\textwidth]{images/productranking.jpg} +\begin{tikzpicture}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + \def\colspace{1} + \def\rowspace{1} + \node (P0) at (2*\rowspace, 2*\colspace) {$c'$}; + \node (P12) at (2*\rowspace, 0.8*\colspace) {$c$}; + \node (P3) at (0*\rowspace, 0*\colspace) {$a$}; + \node (P5) at (4*\rowspace, 0*\colspace) {$b$}; + + \path[commutative diagrams/.cd,every arrow, every label,line width=0.2mm] + (P0) edge node["$p'$", above] {} (P3) + (P0) edge node["$q'$"] {} (P5) + (P12) edge node[below, "$p$"] {} (P3) + (P12) edge node[below, "$q$"] {} (P5) + (P0) edge[commutative diagrams/dashrightarrow, "$m$", swap] (P12); +\end{tikzpicture} \end{figure} \noindent diff --git a/src/content/3.1/its-all-about-morphisms.tex b/src/content/3.1/its-all-about-morphisms.tex index 53bdc047..22869e07 100644 --- a/src/content/3.1/its-all-about-morphisms.tex +++ b/src/content/3.1/its-all-about-morphisms.tex @@ -60,7 +60,12 @@ \section{Natural Transformations} \begin{figure}[H] \centering -\includegraphics[width=0.35\textwidth]{images/3_naturality.jpg} +\begin{tikzcd}[row sep=large,column sep=large] + & F_a \arrow[dd, dashed, pos=0.7, "F_f"] \arrow[dr, "\alpha_a"] & \\ + a \arrow[rr, crossing over] \arrow[ur] \arrow[dd, "f"] & & G_a \arrow[dd, "G_f"] \\ + & F_b \arrow[dr, "\alpha_b"] & \\ + b \arrow[rr] \arrow[ur] & & G_b \\ +\end{tikzcd} \end{figure} \noindent @@ -78,7 +83,33 @@ \section{Natural Transformations} \begin{figure}[H] \centering -\includegraphics[width=0.35\textwidth]{images/sheets.png} +\begin{tikzpicture}[scale=1.15] + \def\between{0.85} + \def\length{1.7} + \def\arclen{1.3} + \draw (-\between/2,0) arc (0:-90:1) -- (-1-\between/2,\length) + arc (-90:0:1) -- cycle; + \draw (\between/2,0) arc (-180:-90:1) -- (1+\between/2,\length) + arc (-90:-180:1) -- cycle; + + \def\nodecoordright{\between/2+0.5} + \def\nodecoordleft{-\between/2-0.5} + \def\nodecoordhigh{\length*0.85} + \def\nodecoordlow{\length*0} + \node (Ga) at (\nodecoordright, \nodecoordhigh) {$Ga$}; + \node (Fa) at (\nodecoordleft, \nodecoordhigh) {$Fa$}; + \node (Gb) at (\nodecoordright, \nodecoordlow) {$Gb$}; + \node (Fb) at (\nodecoordleft, \nodecoordlow) {$Fb$}; + \node (a) at (4*\nodecoordleft, \nodecoordhigh) {$a$}; + \node (b) at (4*\nodecoordleft, \nodecoordlow) {$b$}; + + \path[commutative diagrams/.cd,every arrow, every label,line width=0.2mm] + (Fa) edge node {$\alpha_a$} (Ga) + (Fa) edge node[swap] {$Ff$} (Fb) + (Ga) edge node {$Gf$} (Gb) + (Fb) edge node {$\alpha_b$} (Gb) + (a) edge node[swap] {$f$} (b); +\end{tikzpicture} \end{figure} \noindent diff --git a/src/content/3.15/monads-monoids-and-categories.tex b/src/content/3.15/monads-monoids-and-categories.tex index b87a16f1..91dc02fd 100644 --- a/src/content/3.15/monads-monoids-and-categories.tex +++ b/src/content/3.15/monads-monoids-and-categories.tex @@ -48,7 +48,9 @@ \section{Bicategories} \begin{figure}[H] \centering -\includegraphics[width=0.35\textwidth]{images/twocat.png} +\begin{tikzcd}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + a \arrow[r, bend left=50, "f"{name=U, above}] & b \arrow[l, bend left=50, "g"{name=D}] \arrow[Rightarrow, from=U, to=D, "\alpha"] +\end{tikzcd} \caption{$0$-cells $a, b$; $1$-cells $f, g$; and a $2$-cell $\alpha$.} \end{figure} @@ -123,7 +125,17 @@ \section{Bicategories} \begin{figure}[H] \centering -\includegraphics[width=0.35\textwidth]{images/span.png} +\begin{tikzpicture}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + \def\colspace{1.2} + \def\rowspace{1.5} + \node (x) at (2*\rowspace, 2*\colspace) {$x$}; + \node (a) at (1*\rowspace, 1*\colspace) {$a$}; + \node (b) at (3*\rowspace, 1*\colspace) {$b$}; + + \path[commutative diagrams/.cd,every arrow, every label,line width=0.2mm] + (x) edge node[swap] {$f$} (a) + (x) edge node {$g$} (b); +\end{tikzpicture} \end{figure} \noindent @@ -138,7 +150,21 @@ \section{Bicategories} \begin{figure}[H] \centering -\includegraphics[width=0.5\textwidth]{images/compspan.png} +\begin{tikzpicture}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + \def\colspace{1.2} + \def\rowspace{1.5} + \node (P1) at (1*\rowspace, 1*\colspace) {$x$}; + \node (P2) at (3*\rowspace, 1*\colspace) {$y$}; + \node (P3) at (0*\rowspace, 0*\colspace) {$a$}; + \node (P4) at (2*\rowspace, 0*\colspace) {$b$}; + \node (P5) at (4*\rowspace, 0*\colspace) {$c$}; + + \path[commutative diagrams/.cd,every arrow, every label,line width=0.2mm] + (P1) edge node[swap] {$f$} (P3) + (P1) edge node {$g$} (P4) + (P2) edge node[swap] {$f'$} (P4) + (P2) edge node {$g'$} (P5); +\end{tikzpicture} \end{figure} \noindent @@ -156,7 +182,25 @@ \section{Bicategories} \begin{figure}[H] \centering -\includegraphics[width=0.5\textwidth]{images/pullspan.png} +\begin{tikzpicture}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + \def\colspace{1.2} + \def\rowspace{1.5} + \node (z) at (2*\rowspace, 2*\colspace) {$z$}; + \node (x) at (1*\rowspace, 1*\colspace) {$x$}; + \node (y) at (3*\rowspace, 1*\colspace) {$y$}; + \node (a) at (0*\rowspace, 0*\colspace) {$a$}; + \node (b) at (2*\rowspace, 0*\colspace) {$b$}; + \node (c) at (4*\rowspace, 0*\colspace) {$c$}; + \node (P0b) at (2*\rowspace, 1.7*\colspace) {\rotatebox[origin=c]{-45}{\Large$\lrcorner$}}; + + \path[commutative diagrams/.cd,every arrow, every label,line width=0.2mm] + (z) edge node[swap] {$h$} (x) + (z) edge node {$h'$} (y) + (x) edge node[swap] {$f$} (a) + (x) edge node[swap] {$g$} (b) + (y) edge node {$f'$} (b) + (y) edge node {$g'$} (c); +\end{tikzpicture} \end{figure} \noindent @@ -170,7 +214,21 @@ \section{Bicategories} \begin{figure}[H] \centering -\includegraphics[width=0.4\textwidth]{images/morphspan.png} +\begin{tikzpicture}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + \def\colspace{1.2} + \def\rowspace{1.2} + \node (P0) at (2*\rowspace, 2*\colspace) {$x$}; + \node (P12) at (2*\rowspace, 0.8*\colspace) {$y$}; + \node (P3) at (0*\rowspace, 0*\colspace) {$a$}; + \node (P5) at (4*\rowspace, 0*\colspace) {$b$}; + + \path[commutative diagrams/.cd,every arrow, every label,line width=0.2mm] + (P0) edge node {} (P3) + (P0) edge node {} (P5) + (P12) edge node {} (P3) + (P12) edge node[swap] {} (P5) + (P0) edge[commutative diagrams/Rightarrow, "$h$"] (P12); +\end{tikzpicture} \caption{A $2$-cell in $\cat{Span}$.} \end{figure} @@ -207,7 +265,17 @@ \section{Monads} \begin{figure}[H] \centering -\includegraphics[width=0.3\textwidth]{images/monad.png} +\begin{tikzpicture}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + \def\colspace{1.8} + \def\rowspace{1.8} + \node (P0) at (1*\rowspace, 1*\colspace) {$T\circ T$}; + \node (P1) at (1*\rowspace, 0*\colspace) {$T$}; + \node (P2) at (0*\rowspace, 0*\colspace) {$I$}; + + \path[commutative diagrams/.cd,every arrow, every label,line width=0.2mm] + (P0) edge node {$\mu$} (P1) + (P2) edge node {$\eta$} (P1); +\end{tikzpicture} \end{figure} \noindent @@ -228,7 +296,9 @@ \section{Monads} \begin{figure}[H] \centering -\includegraphics[width=0.3\textwidth]{images/bimonad.png} +\begin{tikzcd}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + a \arrow[loop above, "id"{name=id}] \arrow[loop, "T"{name=T}, looseness=18, out=130,in=50] \arrow[loop, "T\circ T"{name=T2}, looseness=25, out=140,in=40] \arrow[Rightarrow, from=id, to=T, "\eta"] \arrow[Rightarrow, from=T2, to=T, "\mu"'] +\end{tikzcd} \end{figure} \noindent @@ -249,7 +319,17 @@ \section{Monads} \begin{figure}[H] \centering -\includegraphics[width=0.3\textwidth]{images/spanmonad.png} +\begin{tikzpicture}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + \def\colspace{1.2} + \def\rowspace{1.5} + \node (P0) at (2*\rowspace, 2*\colspace) {$Arr$}; + \node (P1) at (1*\rowspace, 1*\colspace) {$Ob$}; + \node (P2) at (3*\rowspace, 1*\colspace) {$Ob$}; + + \path[commutative diagrams/.cd,every arrow, every label,line width=0.2mm] + (P0) edge node[swap] {$dom$} (P1) + (P0) edge node {$cod$} (P2); +\end{tikzpicture} \end{figure} \noindent @@ -272,7 +352,21 @@ \section{Monads} \begin{figure}[H] \centering -\includegraphics[width=0.4\textwidth]{images/spanunit.png} +\begin{tikzpicture}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + \def\colspace{1.2} + \def\rowspace{1.2} + \node (P0) at (2*\rowspace, 2*\colspace) {$Ob$}; + \node (P12) at (2*\rowspace, 0.8*\colspace) {$Arr$}; + \node (P3) at (0*\rowspace, 0*\colspace) {$Ob$}; + \node (P5) at (4*\rowspace, 0*\colspace) {$Ob$}; + + \path[commutative diagrams/.cd,every arrow, every label,line width=0.2mm] + (P0) edge node[swap] {$id$} (P3) + (P0) edge node {$id$} (P5) + (P12) edge node {$dom$} (P3) + (P12) edge node[swap] {$cod$} (P5) + (P0) edge[commutative diagrams/Rightarrow, "$\eta$"] (P12); +\end{tikzpicture} \end{figure} \noindent @@ -293,7 +387,27 @@ \section{Monads} \begin{figure}[H] \centering -\includegraphics[width=0.5\textwidth]{images/spanmul.png} +\begin{tikzpicture}[cells={nodes={font=\bfseries}}] + \def\colspace{1.2} + \def\rowspace{1.5} + \node (P0) at (2*\rowspace, 2*\colspace) {$Arr\otimes Arr$}; + \node (P1) at (1*\rowspace, 1*\colspace) {$Arr$}; + \node (P2) at (3*\rowspace, 1*\colspace) {$Arr$}; + \node (P3) at (0*\rowspace, 0*\colspace) {$Ob$}; + \node (P4) at (2*\rowspace, 0*\colspace) {$Ob$}; + \node (P5) at (4*\rowspace, 0*\colspace) {$Ob$}; + \node (P0b) at (2*\rowspace, 1.7*\colspace) {\rotatebox[origin=c]{-45}{$\lrcorner$}}; + + + \path[commutative diagrams/.cd,every arrow, every label,line width=0.2mm] + (P0) edge node {} (P1) + (P0) edge node {} (P2) + (P1) edge node[swap] {$dom$} (P3) + (P1) edge node[swap] {$cod$} (P4) + (P2) edge node[swap] {$dom$} (P4) + (P2) edge node[swap] {$cod$} (P5); +\end{tikzpicture} + \end{figure} \noindent diff --git a/src/content/3.6/monads-categorically.tex b/src/content/3.6/monads-categorically.tex index 3012ca6e..b3ce0d4a 100644 --- a/src/content/3.6/monads-categorically.tex +++ b/src/content/3.6/monads-categorically.tex @@ -115,7 +115,9 @@ \begin{figure}[H] \centering -\includegraphics[width=0.4\textwidth]{images/assoc1.png} +\begin{tikzcd}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + C \arrow[r, bend left=50, "T^2"{name=U, above}] \arrow[r, bend right=50, "T"{name=D, below}] & C \arrow[Rightarrow, from=U, to=D, "\mu"] \arrow[r, bend left=50, "T"{name=U, above}] \arrow[r, bend right=50, "T"{name=D, below}] & C \arrow[Rightarrow, from=U, to=D, "I_T"] +\end{tikzcd} \end{figure} \noindent @@ -132,7 +134,17 @@ \begin{figure}[H] \centering -\includegraphics[width=0.3\textwidth]{images/assoc2.png} +\begin{tikzpicture}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + \def\colspace{1.8} + \def\rowspace{2} + \node (P0) at (0*\rowspace, 1*\colspace) {$T^3$}; + \node (P1) at (1*\rowspace, 0*\colspace) {$T$}; + \node (P2) at (0*\rowspace, 0*\colspace) {$T^2$}; + + \path[commutative diagrams/.cd,every arrow, every label,line width=0.2mm] + (P0) edge node {$T\circ\mu$} (P2) + (P2) edge node {$\mu$} (P1); +\end{tikzpicture} \end{figure} \noindent @@ -143,7 +155,20 @@ \begin{figure}[H] \centering -\includegraphics[width=0.3\textwidth]{images/assoc.png} +\begin{tikzpicture}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + \def\colspace{1.8} + \def\rowspace{2} + \node (P0) at (0*\rowspace, 1*\colspace) {$T^3$}; + \node (P1) at (1*\rowspace, 0*\colspace) {$T$}; + \node (P2) at (0*\rowspace, 0*\colspace) {$T^2$}; + \node (P4) at (1*\rowspace, 1*\colspace) {$T^2$}; + + \path[commutative diagrams/.cd,every arrow, every label,line width=0.2mm] + (P0) edge node {$T\circ\mu$} (P2) + (P2) edge node {$\mu$} (P1) + (P0) edge node {$T\circ\mu$} (P4) + (P4) edge node {$\mu$} (P1); +\end{tikzpicture} \end{figure} \noindent @@ -156,7 +181,21 @@ \begin{figure}[H] \centering -\includegraphics[width=0.4\textwidth]{images/unitlawcomp-1.png} +\begin{tikzpicture}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + \def\colspace{1.5} + \def\rowspace{2} + \node (P0) at (0*\rowspace, 1*\colspace) {$I\circ T$}; + \node (P1) at (1*\rowspace, 1*\colspace) {$T^2$}; + \node (P2) at (2*\rowspace, 1*\colspace) {$T\circ I$}; + \node (P3) at (1*\rowspace, 0*\colspace) {$T$}; + + \path[commutative diagrams/.cd,every arrow, every label,line width=0.2mm] + (P0) edge node {$\eta\,\circ\, T$} (P1) + (P1) edge node {$T\,\circ\,\eta$} (P2) + (P0) edge node {} (P3) + (P1) edge node {$\eta$} (P3) + (P2) edge node {} (P3); +\end{tikzpicture} \end{figure} \noindent @@ -276,7 +315,11 @@ \section{Monoidal Categories} \begin{figure}[H] \centering -\includegraphics[width=0.5\textwidth]{images/assocmon.png} +\begin{tikzcd}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + (a\times a) \times a \arrow[rr, "\alpha"] & & a\times (a\times a) \\ + a\times a \arrow[from=1-1, "\mu\times id"] & & a\times a \arrow[from=1-3, "id\times\mu"] \\ + & a \arrow[from=2-1, "\mu"] \arrow[from=2-3, "\mu"'] & \\ +\end{tikzcd} \end{figure} \noindent @@ -351,12 +394,19 @@ \section{Monoid in a Monoidal Category} \begin{figure}[H] \centering -\includegraphics[width=0.5\textwidth]{images/assoctensor.jpg} +\begin{tikzcd}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + (m\otimes m) \otimes m \arrow[rr, "\alpha"] & & m\otimes (m\otimes m) \\ + m\otimes m \arrow[from=1-1, "\mu\otimes id"] & & m\otimes m \arrow[from=1-3, "id\otimes\mu"] \\ + & m \arrow[from=2-1, "\mu"] \arrow[from=2-3, "\mu"'] & \\ +\end{tikzcd} \end{figure} \begin{figure}[H] \centering -\includegraphics[width=0.5\textwidth]{images/unitmon.jpg} +\begin{tikzcd}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + i\otimes m \arrow[r, "\eta\otimes id"] \arrow[dr, "\lambda"] & m\otimes m \arrow[d, "\mu"] & m\otimes i \arrow[l, "id\otimes\eta"] \arrow[dl, "\rho"] \\ + & m & \\ +\end{tikzcd} \end{figure} \noindent @@ -396,7 +446,9 @@ \section{Monads as Monoids} \begin{figure}[H] \centering -\includegraphics[width=0.4\textwidth]{images/horizcomp.png} +\begin{tikzcd}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + C \arrow[r, bend left=50, "F"{name=U, above}] \arrow[r, bend right=50, "F'"{name=D, below}] & C \arrow[Rightarrow, from=U, to=D] \arrow[r, bend left=50, "G"{name=U, above}] \arrow[r, bend right=50, "G'"{name=D, below}] & C \arrow[Rightarrow, from=U, to=D] +\end{tikzcd} \end{figure} \noindent @@ -418,12 +470,42 @@ \section{Monads as Monoids} \begin{figure}[H] \centering -\includegraphics[width=0.4\textwidth]{images/assoc.png} +\begin{figure}[H] +\centering +\begin{tikzpicture}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + \def\colspace{1.8} + \def\rowspace{2} + \node (P0) at (0*\rowspace, 1*\colspace) {$T^3$}; + \node (P1) at (1*\rowspace, 0*\colspace) {$T$}; + \node (P2) at (0*\rowspace, 0*\colspace) {$T^2$}; + \node (P4) at (1*\rowspace, 1*\colspace) {$T^2$}; + + \path[commutative diagrams/.cd,every arrow, every label,line width=0.2mm] + (P0) edge node {$T\circ\mu$} (P2) + (P2) edge node {$\mu$} (P1) + (P0) edge node {$T\circ\mu$} (P4) + (P4) edge node {$\mu$} (P1); +\end{tikzpicture} +\end{figure} \end{figure} \begin{figure}[H] \centering -\includegraphics[width=0.5\textwidth]{images/unitlawcomp.png} +\begin{tikzpicture}[cells={nodes={font=\fontsize{10}{6}\selectfont}}] + \def\colspace{1.8} + \def\rowspace{2} + \node (P0) at (0*\rowspace, 1*\colspace) {$T\circ I$}; + \node (P1) at (1*\rowspace, 1*\colspace) {$T^2$}; + \node (P2) at (2*\rowspace, 1*\colspace) {$I\circ T$}; + \node (P3) at (1*\rowspace, 0*\colspace) {$T$}; + + \path[commutative diagrams/.cd,every arrow, every label,line width=0.2mm] + (P0) edge node {$\eta\,\circ\, T$} (P1) + (P1) edge node {$T\,\circ\,\eta$} (P2) + (P0) edge node {} (P3) + (P1) edge node {$\eta$} (P3) + (P2) edge node {} (P3); +\end{tikzpicture} \end{figure} \noindent diff --git a/src/content/3.8/f-algebras.tex b/src/content/3.8/f-algebras.tex index 7effb5ee..8f356a80 100644 --- a/src/content/3.8/f-algebras.tex +++ b/src/content/3.8/f-algebras.tex @@ -209,7 +209,10 @@ \section{Category of F-Algebras} \begin{figure}[H] \centering -\includegraphics[width=0.3\textwidth]{images/alg.png} +\begin{tikzcd}[column sep=large,row sep=large,cells={nodes={font=\fontsize{10}{6}\selectfont}}] + F a \arrow[r, "F m"] \arrow[d, "f"] & F b \arrow[d, "g"] \\ + a \arrow[r, "m"] & b \\ +\end{tikzcd} \end{figure} \noindent @@ -228,7 +231,10 @@ \section{Category of F-Algebras} \begin{figure}[H] \centering -\includegraphics[width=0.3\textwidth]{images/alg2.png} +\begin{tikzcd}[column sep=large,row sep=large,cells={nodes={font=\fontsize{10}{6}\selectfont}}] + F i \arrow[r, "F m"] \arrow[d, "j"] & Fa \arrow[d, "f"] \\ + i \arrow[r, "m"] & a \\ +\end{tikzcd} \end{figure} \noindent @@ -243,7 +249,10 @@ \section{Category of F-Algebras} \begin{figure}[H] \centering -\includegraphics[width=0.3\textwidth]{images/alg3a.png} +\begin{tikzcd}[column sep=large,row sep=large,cells={nodes={font=\fontsize{10}{6}\selectfont}}] + F i \arrow[r, "F m"] \arrow[d, "j"] & F(F i) \arrow[d, "F j"] \\ + i \arrow[r, "m"] & Fi \\ +\end{tikzcd} \end{figure} \noindent @@ -252,7 +261,10 @@ \section{Category of F-Algebras} \begin{figure}[H] \centering -\includegraphics[width=0.3\textwidth]{images/alg3.png} +\begin{tikzcd}[column sep=large,row sep=large,cells={nodes={font=\fontsize{10}{6}\selectfont}}] + F(F i) \arrow[r, "F j"] \arrow[d, "F j"] & F i \arrow[d, "j"] \\ + F i \arrow[r, "j"] & i \\ +\end{tikzcd} \end{figure} \noindent @@ -262,7 +274,10 @@ \section{Category of F-Algebras} \begin{figure}[H] \centering -\includegraphics[width=0.6\textwidth]{images/alg4.png} +\begin{tikzcd}[column sep=large,row sep=large,cells={nodes={font=\fontsize{10}{6}\selectfont}}] + F i \arrow[r, "F m"] \arrow[d, "j"] & F(F i) \arrow[d, "F j"] \arrow[r, "F j"] & F i \arrow[d, "j"] \\ + i \arrow[r, "m"] & F i \arrow[r, "j"] & i\\ +\end{tikzcd} \end{figure} \noindent @@ -319,7 +334,10 @@ \section{Catamorphisms} \begin{figure}[H] \centering -\includegraphics[width=0.4\textwidth]{images/alg5.png} +\begin{tikzcd}[column sep=large,row sep=large,cells={nodes={font=\fontsize{10}{6}\selectfont}}] + f(Fix\,f) \arrow[r, "f\,map\,m"] \arrow[d, "Fix"] & fa \arrow[d, "alg"] \\ + Fix\,f \arrow[r, "m"] & a +\end{tikzcd} \end{figure} \noindent @@ -333,7 +351,10 @@ \section{Catamorphisms} \begin{figure}[H] \centering -\includegraphics[width=0.4\textwidth]{images/alg6.png} +\begin{tikzcd}[column sep=large,row sep=large,cells={nodes={font=\fontsize{10}{6}\selectfont}}] + f(Fix\,f) \arrow[r, "f\,map\,m"] \arrow[from=d, "unFix"] & fa \arrow[d, "alg"] \\ + Fix\,f \arrow[r, "m"] & a +\end{tikzcd} \end{figure} \noindent @@ -432,7 +453,10 @@ \section{Coalgebras} \begin{figure}[H] \centering -\includegraphics[width=0.4\textwidth]{images/alg7.png} +\begin{tikzcd}[column sep=large,row sep=large,cells={nodes={font=\fontsize{10}{6}\selectfont}}] + F t & F a \arrow[l, "F m"{above}] \\ + t \arrow[u, "u"{right}] & a \arrow[u, "f"{right}] \arrow[l, "m"{above}] +\end{tikzcd} \end{figure} \noindent