MATH ALL THE THINGS

src/category.tex

@@ -0,0 +1,12 @@
+\newcommand{\cat}{%
+\mathbf%
+}
+\newcommand{\idarrow}[1][]{%
+\mathbf{id}_{#1}%
+}
+\newcommand{\Set}{\cat{Set}}
+\newcommand{\Rel}{\cat{Rel}}
+\newcommand{\Cat}{\cat{Cat}}
+\newcommand{\id}{\mathbf{id}}
+
+\newcommand{\stimes}{{\times}}

src/content/1.1/Category - The Essence of Composition.tex

@@ -5,47 +5,49 @@
 for variety, I will occasionally draw objects as piggies and arrows as
 fireworks.) But the essence of a category is \newterm{composition}. Or, if you
 prefer, the essence of composition is a category. Arrows compose, so
-if you have an arrow from object A to object B, and another arrow from
-object B to object C, then there must be an arrow --- their composition
---- that goes from A to C.
+if you have an arrow from object $A$ to object $B$, and another arrow from
+object $B$ to object $C$, then there must be an arrow --- their composition
+--- that goes from $A$ to $C$.
 
 \begin{figure}
 \centering
 \includegraphics[width=\textwidth]{images/img_1330.jpg}
-\caption{In a category, if there is an arrow going from A to B and an arrow going from B to C then there must also be a direct arrow from A to C that is their composition. This diagram is not a full category because it’s missing identity morphisms (see later).}
+\caption{In a category, if there is an arrow going from $A$ to $B$ and an arrow going from $B$ to $C$
+then there must also be a direct arrow from $A$ to $C$ that is their composition. This diagram is not a full
+category because it’s missing identity morphisms (see later).}
 \end{figure}
 
 \section{Arrows as Functions}\label{arrows-as-functions}
 
 Is this already too much abstract nonsense? Do not despair. Let's talk
 concretes. Think of arrows, which are also called \newterm{morphisms}, as
-functions. You have a function \code{f} that takes an argument of type A and
-returns a B. You have another function \code{g} that takes a B and returns a C.
-You can compose them by passing the result of \code{f} to \code{g}. You have just
-defined a new function that takes an A and returns a C.
+functions. You have a function \code{f} that takes an argument of type $A$ and
+returns a $B$. You have another function \code{g} that takes a $B$ and returns a $C$.
+You can compose them by passing the result of $f$ to $g$. You have just
+defined a new function that takes an $A$ and returns a $C$.
 
 In math, such composition is denoted by a small circle between
 functions: \ensuremath{g \circ f}. Notice the right to left order of composition. For some
 people this is confusing. You may be familiar with the pipe notation in
 Unix, as in:
 
-\begin{Verbatim}[commandchars=\\\{\}]
+\begin{Verbatim}
 lsof | grep Chrome
 \end{Verbatim}
 or the chevron \code{>>} in F\#, which both
 go from left to right. But in mathematics and in Haskell functions
-compose right to left. It helps if you read \code{g◦f} as ``g \emph{after} f.''
+compose right to left. It helps if you read \(g \circ f\) as ``g \emph{after} f.''
 
 Let's make this even more explicit by writing some C code. We have one
 function \code{f} that takes an argument of type \code{A} and
 returns a value of type \code{B}:
 
-\begin{Verbatim}[commandchars=\\\{\}]
+\begin{Verbatim}
 B f(A a);
 \end{Verbatim}
 and another:
 
-\begin{Verbatim}[commandchars=\\\{\}]
+\begin{Verbatim}
 C g(B b);
 \end{Verbatim}
 Their composition is:
@@ -64,25 +66,25 @@ \section{Arrows as Functions}\label{arrows-as-functions}
 there isn't one. So let's try some Haskell for a change. Here's the
 declaration of a function from A to B:
 
-\begin{Verbatim}[commandchars=\\\{\}]
+\begin{Verbatim}
 f :: A -> B
 \end{Verbatim}
 Similarly:
 
-\begin{Verbatim}[commandchars=\\\{\}]
+\begin{Verbatim}
 g :: B -> C
 \end{Verbatim}
 Their composition is:
 
-\begin{Verbatim}[commandchars=\\\{\}]
+\begin{Verbatim}
 g . f
 \end{Verbatim}
 Once you see how simple things are in Haskell, the inability to express
 straightforward functional concepts in C++ is a little embarrassing. In
 fact, Haskell will let you use Unicode characters so you can write
 composition as:
 
-\begin{Verbatim}[commandchars=\\\{\}]
+\begin{Verbatim}
 g ◦ f
 \end{Verbatim}
 You can even use Unicode double colons and arrows:
@@ -102,17 +104,14 @@ \section{Properties of Composition}\label{properties-of-composition}
 
 \begin{enumerate}
 \item
-Composition is associative. If you have three morphisms, f, g, and h,
+Composition is associative. If you have three morphisms, $f$, $g$, and $h$,
 that can be composed (that is, their objects match end-to-end), you
 don't need parentheses to compose them. In math notation this is
 expressed as:
-
-\begin{Verbatim}[commandchars=\\\{\}]
-h◦(g◦f) = (h◦g)◦f = h◦g◦f
-\end{Verbatim}
+\[h\circ{}(g\circ{}f) = (h\circ{}g)\circ{}f = h\circ{}g\circ{}f\]
 In (pseudo) Haskell:
 
-\begin{Verbatim}[commandchars=\\\{\}]
+\begin{Verbatim}
 f :: A -> B
 g :: B -> C
 h :: C -> D
@@ -124,18 +123,15 @@ \section{Properties of Composition}\label{properties-of-composition}
 be not as obvious in other categories.
 
 \item
-For every object A there is an arrow which is a unit of composition.
+For every object $A$ there is an arrow which is a unit of composition.
 This arrow loops from the object to itself. Being a unit of composition
-means that, when composed with any arrow that either starts at A or ends
-at A, respectively, it gives back the same arrow. The unit arrow for
-object A is called \code{id\textsubscript{A}} (\newterm{identity} on A). In math
-notation, if \code{f} goes from A to B then
-
-\code{f◦id\textsubscript{A} = f}
-
+means that, when composed with any arrow that either starts at $A$ or ends
+at $A$, respectively, it gives back the same arrow. The unit arrow for
+object A is called $\idarrow[A]$ (\newterm{identity} on $A$). In math
+notation, if \code{f} goes from $A$ to $B$ then
+\[f \circ \idarrow[A] = f\]
 and
-
-\code{id\textsubscript{B}◦f = f}
+\[\idarrow[B] \circ f = f\]
 \end{enumerate}
 When dealing with functions, the identity arrow is implemented as the
 identity function that just returns back its argument. The
@@ -152,7 +148,7 @@ \section{Properties of Composition}\label{properties-of-composition}
 In Haskell, the identity function is part of the standard library
 (called Prelude). Here's its declaration and definition:
 
-\begin{Verbatim}[commandchars=\\\{\}]
+\begin{Verbatim}
 id :: a -> a
 id x = x
 \end{Verbatim}
@@ -180,7 +176,7 @@ \section{Properties of Composition}\label{properties-of-composition}
 
 The identity conditions can be written (again, in pseudo-Haskell) as:
 
-\begin{Verbatim}[commandchars=\\\{\}]
+\begin{Verbatim}
 f . id == f
 id . f == f
 \end{Verbatim}
@@ -229,7 +225,7 @@ \section{Composition is the Essence of
 time. One of the most cited papers in psychology,
 \href{http://en.wikipedia.org/wiki/The_Magical_Number_Seven,_Plus_or_Minus_Two}{The
 Magical Number Seven, Plus or Minus Two}, postulated that we can only
-keep \ensuremath{7 \pm 2} ``chunks'' of information in our minds. The details of our
+keep $7 \pm 2$ ``chunks'' of information in our minds. The details of our
 understanding of the human short-term memory might be changing, but we
 know for sure that it's limited. The bottom line is that we are unable
 to deal with the soup of objects or the spaghetti of code. We need