Replace hand-drawn commutative diagrams with TeX (#99)

* preamble: add diagram package tikz-cd

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

* content/3.2: rewrite diagrams in tikz-cd

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

* content/3.9: rewrite diagrams in tikz-cd

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

* content/3.12: rewrite diagrams in tikz-cd

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

* 3.12: fix typo, missing \ on cat

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

* 3.12: adjust id.jpg size to improve pagination

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

* 2.4: fix typo, missing \ on cat

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

* content/3.12: use \id from category.tex instead of \rm{id}

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

* content/3.14: rewrite diagrams in tikz-cd

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

* content/3.12: correct \rho -> \lambda on commutative diagram

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

* README: add myself among the contributors

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

README.md

@@ -57,6 +57,7 @@ Thanks to the following people for contributing corrections/conversions:
 * Paolo G. Giarrusso
 * Adi Shavit
 * Mico from the TeX.StackExchange community
+* Marcello Seri
 * ...and many others!
 
 License

src/content/2.4/Representable Functors.tex

@@ -145,7 +145,7 @@ We've seen that, for every choice of an object $a$ in $\cat{C}$,
 we get a functor from $\cat{C}$ to $\Set$. This kind of
 structure-preserving mapping to $\Set$ is often called a
 \newterm{representation}. We are representing objects and morphisms of
-$cat{C}$ as sets and functions in $\Set$.
+$\cat{C}$ as sets and functions in $\Set$.
 
 The functor $\cat{C}(a, -)$ itself is sometimes called representable.
 More generally, any functor $F$ that is naturally isomorphic to

src/content/3.12/Enriched Categories.tex

@@ -89,12 +89,30 @@ The associator and the unitors must satisfy coherence conditions:
 
 \begin{figure}[H]
 \centering
-\includegraphics[width=80mm]{images/assoc.jpg}
+\begin{tikzcd}[row sep=large]
+((a \otimes b) \otimes c) \otimes d
+\arrow[d, "\alpha_{(a \otimes b)cd}"]
+\arrow[rr, "\alpha_{abc} \otimes \id_d"]
+  & & (a \otimes (b \otimes c)) \otimes d
+      \arrow[d, "\alpha_{a(b \otimes c)d}"] \\
+(a \otimes b) \otimes (c \otimes d)
+\arrow[rd, "\alpha_{ab(c \otimes d)}"]
+  & & a \otimes ((b \otimes c) \otimes d)
+      \arrow[ld, "\id_a \otimes \alpha_{bcd}"] \\
+  & a \otimes (b \otimes (c \otimes d))
+\end{tikzcd}
 \end{figure}
 
 \begin{figure}[H]
 \centering
-\includegraphics[width=60mm]{images/idcoherence.jpg}
+\begin{tikzcd}[row sep=large]
+(a \otimes i) \otimes b
+\arrow[dr, "\rho_{a} \otimes \id_b"']
+\arrow[rr, "\alpha_{aib}"]
+  & & a \otimes (i \otimes b)
+      \arrow[dl, "\id_a \otimes \lambda_b"] \\
+  & a \otimes b
+\end{tikzcd}
 \end{figure}
 
 \noindent
@@ -156,7 +174,7 @@ where $i$ is the tensor unit in $\cat{V}$.
 
 \begin{figure}[H]
 \centering
-\includegraphics[width=60mm]{images/id.jpg}
+\includegraphics[width=40mm]{images/id.jpg}
 \end{figure}
 
 \noindent
@@ -165,20 +183,48 @@ $\cat{V}$:
 
 \begin{figure}[H]
 \centering
-\includegraphics[width=80mm]{images/compcoherence.jpg}
+\begin{tikzcd}[column sep=large]
+(\cat{C}(c,d) \otimes \cat{C}(b,c)) \otimes \cat{C}(a,b)
+\arrow[r, "\circ\otimes\id"]
+\arrow[dd, "\alpha"]
+  & \cat{C}(b,d) \otimes \cat{C}(a,b)
+    \arrow[dr, "\circ"] \\
+  & & \cat{C}(a,d) \\
+\cat{C}(c,d) \otimes (\cat{C}(b,c) \otimes \cat{C}(a,b))
+\arrow[r, "\id\otimes\circ"]
+  & \cat{C}(c,d) \otimes \cat{C}(a,c)
+    \arrow[ur, "\circ"]
+\end{tikzcd}
 \end{figure}
 
 \noindent
 Unit laws are likewise expressed in terms of unitors:
 
 \begin{figure}[H]
-\centering
-\includegraphics[width=70mm]{images/rightid.jpg}
-\end{figure}
-
-\begin{figure}[H]
-\centering
-\includegraphics[width=70mm]{images/leftid.jpg}
+  \centering
+  \begin{subfigure}
+    \centering
+    \begin{tikzcd}[row sep=large]
+      \cat{C}(a,b) \otimes i
+      \arrow[rr, "\id \otimes j_a"]
+      \arrow[dr, "\rho"]
+        & & \cat{C}(a,b) \otimes \cat{C}(a,a)
+            \arrow[dl, "\circ"] \\
+        & \cat{C}(a,b)
+    \end{tikzcd}
+  \end{subfigure}
+  \hspace{1cm}
+  \begin{subfigure}
+    \centering
+    \begin{tikzcd}[row sep=large]
+      i \otimes \cat{C}(a,b)
+      \arrow[rr, "j_b \otimes \id"]
+      \arrow[dr, "\lambda"]
+        & & \cat{C}(b,b) \otimes \cat{C}(a,b)
+            \arrow[dl, "\circ"] \\
+        & \cat{C}(a,b)
+    \end{tikzcd}
+  \end{subfigure}
 \end{figure}
 
 \section{Preorders}
@@ -265,9 +311,9 @@ us that the distance from $a$ to $a$ is always zero.
 Check!
 
 Now let's talk about composition. We start with the tensor product of
-two abutting hom-objects, $cat{C}(b, c) \otimes \cat{C}(a, b)$. We have defined
+two abutting hom-objects, $\cat{C}(b, c) \otimes \cat{C}(a, b)$. We have defined
 the tensor product as the sum of the two distances. Composition is a
-morphism in $\cat{V}$ from this product to $cat{C}(a, c)$. A morphism
+morphism in $\cat{V}$ from this product to $\cat{C}(a, c)$. A morphism
 in $\cat{V}$ is defined as the greater-or-equal relation. In other words,
 the sum of distances from $a$ to $b$ and from $b$
 to $c$ is greater than or equal to the distance from $a$
@@ -297,7 +343,16 @@ preservation of composition means that the following diagram commute:
 
 \begin{figure}[H]
 \centering
-\includegraphics[width=80mm]{images/functorcomp.jpg}
+\begin{tikzcd}[column sep=large, row sep=large]
+  \cat{C}(b,c) \otimes \cat{C}(a,b)
+  \arrow[r, "\circ"]
+  \arrow[d, "F_{bc} \otimes F_{ab}"]
+    & \cat{C}(a,c)
+      \arrow[d, "F_{ac}"] \\
+  \cat{D}(F\ b, F\ c) \otimes \cat{D}(F\ a, F\ b)
+  \arrow[r,  "\circ"]
+    & \cat{D}(F\ a, F\ c)
+\end{tikzcd}
 \end{figure}
 
 \noindent
@@ -306,7 +361,12 @@ morphisms in $\cat{V}$ that ``select'' the identity:
 
 \begin{figure}[H]
 \centering
-\includegraphics[width=70mm]{images/functorid.jpg}
+\begin{tikzcd}[row sep=large]
+    & i \arrow[dl, "j_a"'] \arrow[dr, "j_{F\ a}"] & \\
+  \cat{C}(a,a)
+  \arrow[rr, "F_{aa}"]
+    & & \cat{D}(F\ a, F\ a)
+\end{tikzcd}
 \end{figure}
 
 \section{Self Enrichment}

src/content/3.14/Lawvere Theories.tex

@@ -463,7 +463,10 @@ $\cat{L}(n, 1)$.
 
 \begin{figure}[H]
 \centering
-\includegraphics[width=60mm]{images/liftl.png}
+\begin{tikzcd}[column sep=large]
+\cat{L}(m, 1) \arrow[r] & \cat{L}(n, 1)\\
+{}^m \bullet \arrow[r, "f"'] & \bullet^n
+\end{tikzcd}
 \end{figure}
 
 \noindent
@@ -490,7 +493,16 @@ identified.
 
 \begin{figure}[H]
 \centering
-\includegraphics[width=80mm]{images/equalize1.png}
+\begin{tikzcd}
+  & a^n \times \cat{L}(m, 1)
+    \arrow[dl, "\langle f {,} \id \rangle"']
+    \arrow[dr, "\langle \id {,} f \rangle"] 
+    & \\
+a^n \times \cat{L}(m, 1)
+  & \scalebox{2.5}[1]{\sim}
+    & a^n \times \cat{L}(n, 1) \\
+    & f \Colon m \to n &
+\end{tikzcd}
 \end{figure}
 
 \noindent
@@ -566,7 +578,16 @@ by lifting $0 \to n$ in two different ways.
 
 \begin{figure}[H]
 \centering
-\includegraphics[width=60mm]{images/equalize2.png}
+\begin{tikzcd}
+  & a^n \times \cat{L}(0, 1)
+    \arrow[dl, "\langle f {,} \id \rangle"']
+    \arrow[dr, "\langle \id {,} f \rangle"] 
+    & \\
+a^0 \times \cat{L}(0, 1)
+  & \scalebox{2.5}[1]{\sim}
+    & a^n \times \cat{L}(n, 1) \\
+    & f \Colon 0 \to n &
+\end{tikzcd}
 \end{figure}
 
 \noindent

src/content/3.2/Adjunctions.tex

@@ -156,11 +156,20 @@ diagrams commute:
 
   \begin{subfigure}
     \centering
-    \includegraphics[width=40mm]{images/triangles.png}
+    \begin{tikzcd}[column sep=large, row sep=large]
+    	L \arrow[rd, equal] \arrow[r, "L \circ \eta"] 
+          & L \circ R \circ L \arrow[d, "\epsilon \circ L"] \\
+    	    & L
+    \end{tikzcd}
   \end{subfigure}%
+  \hspace{1cm}
   \begin{subfigure}
     \centering
-    \includegraphics[width=44.5mm]{images/triangles-2.png}
+    \begin{tikzcd}[column sep=large, row sep=large]
+      R \arrow[rd, equal] \arrow[r, "\eta \circ R"] 
+          & R \circ L \circ R \arrow[d, "R \circ \epsilon"] \\
+          & R
+    \end{tikzcd}
   \end{subfigure}
 \end{figure}
 

src/content/3.9/Algebras for Monads.tex

@@ -44,13 +44,25 @@ diagrams describe the two conditions (I replaced $m$ with
 $T$ in anticipation of what follows):
 
 \begin{figure}[H]
-\centering
-\includegraphics[width=1.81250in]{images/talg1.png}
-\end{figure}
-
-\begin{figure}[H]
-\centering
-\includegraphics[width=2.40625in]{images/talg2.png}
+  \centering
+  \begin{subfigure}
+    \centering
+    \begin{tikzcd}[column sep=large, row sep=large]
+      a \arrow[rd, equal] \arrow[r, "\eta_a"] 
+          & Ta \arrow[d, "alg"] \\
+          & a
+    \end{tikzcd}
+  \end{subfigure}
+  \hspace{1cm}
+  \begin{subfigure}
+    \centering
+    \begin{tikzcd}[column sep=large, row sep=large]
+      T(Ta) \arrow[r, "T\ alg"] \arrow[d, "\mu_a"]
+          & Ta \arrow[d, "alg"] \\
+      Ta \arrow[r, "alg"]
+          & a
+    \end{tikzcd}
+  \end{subfigure}
 \end{figure}
 
 \noindent
@@ -168,7 +180,12 @@ that it's a homomorphism of T-algebras:
 
 \begin{figure}[H]
 \centering
-\includegraphics[width=2.57292in]{images/talg31.png}
+\begin{tikzcd}[column sep=large, row sep=large]
+  T(Ta) \arrow[r, "T f"] \arrow[d, "\mu_a"]
+      & Ta \arrow[d, "f"] \\
+  Ta \arrow[r, "f"]
+      & a
+\end{tikzcd}
 \end{figure}
 
 \noindent
@@ -180,8 +197,24 @@ To complete the adjunction we also need to show that the unit and the
 counit satisfy triangular identities. These are:
 
 \begin{figure}[H]
-\centering
-\includegraphics[width=\textwidth]{images/talg4.png}
+  \centering
+  \begin{subfigure}
+    \centering
+    \begin{tikzcd}[column sep=large, row sep=large]
+      Ta \arrow[rd, equal] \arrow[r, "T \eta_a"] 
+          & T(Ta) \arrow[d, "\mu_a"] \\
+          & Ta
+    \end{tikzcd}
+  \end{subfigure}%
+  \hspace{1cm}
+  \begin{subfigure}
+    \centering
+    \begin{tikzcd}[column sep=large, row sep=large]
+      a \arrow[rd, equal] \arrow[r, "\eta_a"] 
+          & Ta \arrow[d, "f"] \\
+          & a
+    \end{tikzcd}
+  \end{subfigure}
 \end{figure}
 
 \noindent
@@ -301,8 +334,25 @@ $W$. We can define a category of coalgebras that are compatible
 with a comonad. They make the following diagrams commute:
 
 \begin{figure}[H]
-\centering
-\includegraphics[width=\textwidth]{images/talg5.png}
+  \centering
+  \begin{subfigure}
+    \centering
+    \begin{tikzcd}[column sep=large, row sep=large]
+      a \arrow[rd, equal] 
+          & Wa \arrow[l, "\epsilon_a"'] \\
+          & a \arrow[u, "coa"']
+    \end{tikzcd}
+  \end{subfigure}%
+  \hspace{1cm}
+  \begin{subfigure}
+    \centering
+    \begin{tikzcd}[column sep=large, row sep=large]
+      W(Wa)
+          & Wa \arrow[l, "W\ coa"'] \\
+      Wa \arrow[u, "\delta_a"]
+          & a \arrow[u, "coa"] \arrow[l, "coa"']
+    \end{tikzcd}
+  \end{subfigure}
 \end{figure}
 
 \noindent

src/preamble.tex

@@ -53,6 +53,9 @@
 \DeclareMathDelimiter{\lbrace}{\mathopen}{parenthesis}{"7B}{largesymbols}{"08}
 \DeclareMathDelimiter{\rbrace}{\mathclose}{parenthesis}{"7D}{largesymbols}{"09}
 
+% Use tikz-cd for commutative diagrams
+\usepackage{tikz-cd}
+
 \usepackage{fancyvrb}
 \fvset{fontsize=\small}