Replacing \textfb{Set} with a macro $\Set$

src/content/1.2/Types and Functions.tex

@@ -120,11 +120,11 @@ \section{What Are Types?}\label{what-are-types}
 sets tricky. There are problems with polymorphic functions that involve
 circular definitions, and with the fact that you can't have a set of all
 sets; but as I promised, I won't be a stickler for math. The great thing
-is that there is a category of sets, which is called \textbf{Set}, and
-we'll just work with it. In \textbf{Set}, objects are sets and morphisms
+is that there is a category of sets, which is called $\Set$, and
+we'll just work with it. In $\Set$, objects are sets and morphisms
 (arrows) are functions.
 
-\textbf{Set} is a very special category, because we can actually peek
+$\Set$ is a very special category, because we can actually peek
 inside its objects and get a lot of intuitions from doing that. For
 instance, we know that an empty set has no elements. We know that there
 are special one-element sets. We know that functions map elements of one
@@ -180,12 +180,12 @@ \section{What Are Types?}\label{what-are-types}
 functions, which return valid results for every possible argument.
 
 Because of the bottom, you'll see the category of Haskell types and
-functions referred to as \textbf{Hask} rather than \textbf{Set}. From
+functions referred to as \textbf{Hask} rather than $\Set$. From
 the theoretical point of view, this is the source of never-ending
 complications, so at this point I will use my butcher's knife and
 terminate this line of reasoning. From the pragmatic point of view, it's
 okay to ignore non-terminating functions and bottoms, and treat
-\textbf{Hask} as bona fide \textbf{Set}.\footnote{Nils Anders Danielsson,
+\textbf{Hask} as bona fide $\Set$.\footnote{Nils Anders Danielsson,
 John Hughes, Patrik Jansson, Jeremy Gibbons, \href{http://www.cs.ox.ac.uk/jeremy.gibbons/publications/fast+loose.pdf}{
 Fast and Loose Reasoning is Morally Correct}. This paper provides justification for ignoring bottoms in most contexts.}
 

src/content/1.3/Categories Great and Small.tex

@@ -170,7 +170,7 @@ \section{Monoid as Set}\label{monoid-as-set}
 mappend s1 s2 = (++) s1 s2
 \end{Verbatim}
 The former translates into equality of morphisms in the category
-\textbf{Hask} (or \textbf{Set}, if we ignore bottoms, which is the name
+\textbf{Hask} (or $\Set$, if we ignore bottoms, which is the name
 for never-ending calculations). Such equations are not only more
 succinct, but can often be generalized to other categories. The latter
 is called \newterm{extensional} equality, and states the fact that for any

src/content/1.5/Products and Coproducts.tex

@@ -639,7 +639,7 @@ \section{Asymmetry}\label{asymmetry}
 from the terminal end.
 
 This is not an intrinsic property of sets, it's a property of functions,
-which we use as morphisms in \textbf{Set}. Functions are, in general,
+which we use as morphisms in $\Set$. Functions are, in general,
 asymmetric. Let me explain.
 
 A function must be defined for every element of its domain set (in

src/content/1.8/Functoriality.tex

@@ -611,7 +611,7 @@ \section{Profunctors}\label{profunctors}
 
 We've seen that the function-arrow operator is contravariant in its
 first argument and covariant in the second. Is there a name for such a
-beast? It turns out that, if the target category is \textbf{Set}, such a
+beast? It turns out that, if the target category is $\Set$, such a
 beast is called a \newterm{profunctor}. Because a contravariant functor is
 equivalent to a covariant functor from the opposite category, a
 profunctor is defined as:\\

src/content/1.9/Function Types.tex

@@ -12,7 +12,7 @@
 \code{a->b} is more than that: it's a set of morphisms
 between objects \code{a} and \code{b}. A set of morphisms between
 two objects in any category is called a hom-set. It just so happens that
-in the category \textbf{Set} every hom-set is itself an object in the
+in the category $\Set$ every hom-set is itself an object in the
 same category ---because it is, after all, a \newterm{set}.
 
 The same is not true of other categories where hom-sets are external to
@@ -26,7 +26,7 @@
 \end{wrapfigure}
 
 \noindent
-It's the self-referential nature of the category \textbf{Set} that makes
+It's the self-referential nature of the category $\Set$ that makes
 function types special. But there is a way, at least in some categories,
 to construct objects that represent hom-sets. Such objects are called
 \newterm{internal} hom-sets.
@@ -35,7 +35,7 @@ \section{Universal Construction}\label{universal-construction}
 
 Let's forget for a moment that function types are sets and try to
 construct a function type, or more generally, an internal hom-set, from
-scratch. As usual, we'll take our cues from the \textbf{Set} category,
+scratch. As usual, we'll take our cues from the $\Set$ category,
 but carefully avoid using any properties of sets, so that the
 construction will automatically work for other categories.
 
@@ -52,7 +52,7 @@ \section{Universal Construction}\label{universal-construction}
 The obvious pattern that connects these three types is called
 \newterm{function application} or \newterm{evaluation}. Given a candidate for
 a function type, let's call it \code{z} (notice that, if we are not in
-the category \textbf{Set}, this is just an object like any other
+the category $\Set$, this is just an object like any other
 object), and the argument type \code{a} (an object), the application
 maps this pair to the result type \code{b} (an object). We have three
 objects, two of them fixed (the ones representing the argument type and
@@ -77,7 +77,7 @@ \section{Universal Construction}\label{universal-construction}
 well talk about the whole \newterm{product} of the function type \code{z}
 and the argument type \code{a}. The product \code{z×a} is an object,
 and we can pick, as our application morphism, an arrow \code{g} from
-that object to \code{b}. In \textbf{Set}, \code{g} would be the
+that object to \code{b}. In $\Set$, \code{g} would be the
 function that maps every pair \code{(f,\ x)} to \code{f\ x}.
 
 So that's the pattern: a product of two objects \code{z} and
@@ -102,7 +102,7 @@ \section{Universal Construction}\label{universal-construction}
 
 Let's review the universal construction. We start with a pattern of
 objects and morphisms. That's our imprecise query, and it usually yields
-lots and lots of hits. In particular, in \textbf{Set}, pretty much
+lots and lots of hits. In particular, in $\Set$, pretty much
 everything is connected to everything. We can take any object
 \code{z}, form its product with \code{a}, and there's going to be a
 function from it to \code{b} (except when \code{b} is an empty set).
@@ -202,7 +202,7 @@ \section{Universal Construction}\label{universal-construction}
 \noindent
 Of course, there is no guarantee that such an object \code{a\ensuremath{\Rightarrow}b} exists
 for any pair of objects \code{a} and \code{b} in a given category.
-But it always does in \textbf{Set}. Moreover, in \textbf{Set}, this
+But it always does in $\Set$. Moreover, in $\Set$, this
 object is isomorphic to the hom-set \emph{Set(a, b)}.
 
 This is why, in Haskell, we interpret the function type
@@ -219,7 +219,7 @@ \section{Currying}\label{currying}
 g :: z × a -> b
 \end{Verbatim}
 Being a morphism from a product comes as close as it gets to being a
-function of two variables. In particular, in \textbf{Set}, \code{g} is
+function of two variables. In particular, in $\Set$, \code{g} is
 a function from pairs of values, one from the set \code{z} and one
 from the set \code{a}.
 
@@ -230,7 +230,7 @@ \section{Currying}\label{currying}
 \begin{Verbatim}[commandchars=\\\{\}]
 h :: z -> (a⇒b)
 \end{Verbatim}
-In \textbf{Set}, this just means that \code{h} is a function that
+In $\Set$, this just means that \code{h} is a function that
 takes one variable of type \code{z} and returns a function from
 \code{a} to \code{b}. That makes \code{h} a higher order function.
 Therefore the universal construction establishes a one-to-one
@@ -408,7 +408,7 @@ \section{Cartesian Closed
 Although I will continue using the category of sets as a model for types
 and functions, it's worth mentioning that there is a larger family of
 categories that can be used for that purpose. These categories are
-called \newterm{cartesian closed}, and \textbf{Set} is just one example of
+called \newterm{cartesian closed}, and $\Set$ is just one example of
 such a category.
 
 A cartesian closed category must contain:
@@ -442,7 +442,7 @@ \section{Cartesian Closed
 (b + c) × a = b × a + c × a
 \end{Verbatim}
 is called a \newterm{bicartesian closed} category. We'll see in the next
-section that bicartesian closed categories, of which \textbf{Set} is a
+section that bicartesian closed categories, of which $\Set$ is a
 prime example, have some interesting properties.
 
 \section{Exponentials and Algebraic Data
@@ -468,8 +468,8 @@ \subsection{Zeroth Power}\label{zeroth-power}
 set of morphisms going from the initial object to an arbitrary object
 \code{a}. By the definition of the initial object, there is exactly
 one such morphism, so the hom-set \emph{C(0, a)} is a singleton set. A
-singleton set is the terminal object in \textbf{Set}, so this identity
-trivially works in \textbf{Set}. What we are saying is that it works in
+singleton set is the terminal object in $\Set$, so this identity
+trivially works in $\Set$. What we are saying is that it works in
 any bicartesian closed category.
 
 In Haskell, we replace 0 with \code{Void}; 1 with the unit type
@@ -492,7 +492,7 @@ \subsection{Powers of One}\label{powers-of-one}
 \begin{Verbatim}[commandchars=\\\{\}]
 1\textsuperscript{a} = 1
 \end{Verbatim}
-This identity, when interpreted in \textbf{Set}, restates the definition
+This identity, when interpreted in $\Set$, restates the definition
 of the terminal object: There is a unique morphism from any object to
 the terminal object. In general, the internal hom-object from \code{a}
 to the terminal object is isomorphic to the terminal object itself.
@@ -510,7 +510,7 @@ \subsection{First Power}\label{first-power}
 This is a restatement of the observation that morphisms from the
 terminal object can be used to pick ``elements'' of the object
 \code{a}. The set of such morphisms is isomorphic to the object
-itself. In \textbf{Set}, and in Haskell, the isomorphism is between
+itself. In $\Set$, and in Haskell, the isomorphism is between
 elements of the set \code{a} and functions that pick those elements,
 \code{()->a}.
 

src/content/2.2/Limits and Colimits.tex

@@ -210,7 +210,7 @@ \section{Limit as a Natural Isomorphism}\label{limit-as-a-natural-isomorphism}
 But what are the functors that are related by this transformation?
 
 One functor is the mapping of \code{c} to the set
-\code{C(c, Lim D)}. It's a functor from \mathtext{C} to \textbf{Set} ---
+\code{C(c, Lim D)}. It's a functor from \mathtext{C} to $\Set$ ---
 it maps objects to sets. In fact it's a contravariant functor. Here's
 how we define its action on morphisms: Let's take a morphism \code{f}
 from \code{c'} to \code{c}:
@@ -311,13 +311,13 @@ \section{Limit as a Natural Isomorphism}\label{limit-as-a-natural-isomorphism}
 c -> Nat(\ensuremath{\Delta_c}, D)
 \end{Verbatim}
 What I have just done is to show you that we have two (contravariant)
-functors from \mathtext{C} to \textbf{Set}. I haven't made any assumptions
+functors from \mathtext{C} to $\Set$. I haven't made any assumptions
 --- these functors always exist.
 
 Incidentally, the first of these functors plays an important role in
 category theory, and we'll see it again when we talk about Yoneda's
 lemma. There is a name for contravariant functors from any category
-\mathtext{C} to \textbf{Set}: they are called ``presheaves''. This one is
+\mathtext{C} to $\Set$: they are called ``presheaves''. This one is
 called a \newterm{representable presheaf}. The second functor is also a
 presheaf.
 
@@ -337,7 +337,7 @@ \section{Limit as a Natural Isomorphism}\label{limit-as-a-natural-isomorphism}
 I'm not going to go through the proof of this statement. The procedure
 is pretty straightforward if not tedious. When dealing with natural
 transformations, you usually focus on components, which are morphisms.
-In this case, since the target of both functors is \textbf{Set}, the
+In this case, since the target of both functors is $\Set$, the
 components of the natural isomorphism will be functions. These are
 higher order functions, because they go from the hom-set to the set of
 natural transformations. Again, you can analyze a function by
@@ -409,7 +409,7 @@ \section{Examples of Limits}\label{examples-of-limits}
 f . p = g . p
 \end{Verbatim}
 The way to think about it is that, if we restrict our attention to
-\textbf{Set}, the image of the function \code{p} selects a subset of
+$\Set$, the image of the function \code{p} selects a subset of
 \code{a}. When restricted to this subset, the functions \code{f} and
 \code{g} are equal.
 
@@ -680,7 +680,7 @@ \section{Continuity}\label{continuity}
 \end{Verbatim}
 When its second argument is fixed, the hom-set functor (which becomes
 the representable presheaf) maps colimits in \mathtext{C} to limits in
-\textbf{Set}; and when its first argument is fixed, it maps limits to
+$\Set$; and when its first argument is fixed, it maps limits to
 limits.
 
 In Haskell, a hom-functor is the mapping of any two types to a function

src/content/2.3/Free Monoids.tex

@@ -155,7 +155,7 @@ \section{Free Monoid Universal Construction}\label{free-monoid-universal-constru
 \textbf{Mon} to sets, but we can also map morphisms of \textbf{Mon}
 (homomorphisms) to functions. Again, this seems sort of trivial, but it
 will become useful soon. This mapping of objects and morphisms from
-\textbf{Mon} to \textbf{Set} is in fact a functor. Since this functor
+\textbf{Mon} to $\Set$ is in fact a functor. Since this functor
 ``forgets'' the monoidal structure --- once we are inside a plain set,
 we no longer distinguish the unit element or care about multiplication
 --- it's called a \newterm{forgetful functor}. Forgetful functors come up
@@ -183,17 +183,17 @@ \section{Free Monoid Universal Construction}\label{free-monoid-universal-constru
 those blobs. Here's how it works:
 
 We start with a set of generators, \code{x}. That's a set in
-\textbf{Set}.
+$\Set$.
 
 The pattern we are going to match consists of a monoid \code{m} --- an
-object of \textbf{Mon} --- and a function \code{p} in \textbf{Set}:
+object of \textbf{Mon} --- and a function \code{p} in $\Set$:
 
 \begin{Verbatim}[commandchars=\\\{\}]
 p :: x -> U m
 \end{Verbatim}
 where \code{U} is our forgetful functor from \textbf{Mon} to
-\textbf{Set}. This is a weird heterogeneous pattern --- half in
-\textbf{Mon} and half in \textbf{Set}.
+$\Set$. This is a weird heterogeneous pattern --- half in
+\textbf{Mon} and half in $\Set$.
 
 The idea is that the function \code{p} will identify the set of
 generators inside the X-ray image of \code{m}. It doesn't matter that

src/content/2.4/Representable Functors.tex

@@ -3,7 +3,7 @@
 mathematics --- at least it used to be. Category theory tries to step
 away from set theory, to some extent. For instance, it's a known fact
 that the set of all sets doesn't exist, but the category of all sets,
-\textbf{Set}, does. So that's good. On the other hand, we assume that
+$\Set$, does. So that's good. On the other hand, we assume that
 morphisms between any two objects in a category form a set. We even
 called it a hom-set. To be fair, there is a branch of category theory
 where morphisms don't form sets. Instead they are objects in another
@@ -46,14 +46,14 @@
 \section{The Hom Functor}\label{the-hom-functor}
 
 Every category comes equipped with a canonical family of mappings to
-\textbf{Set}. Those mappings are in fact functors, so they preserve the
+$\Set$. Those mappings are in fact functors, so they preserve the
 structure of the category. Let's build one such mapping.
 
 Let's fix one object \code{a} in \emph{C} and pick another object
 \code{x} also in \emph{C}. The hom-set \code{C(a, x)} is a set, an
-object in \textbf{Set}. When we vary \code{x}, keeping \code{a}
-fixed, \code{C(a, x)} will also vary in \textbf{Set}. Thus we have a
-mapping from \code{x} to \textbf{Set}.
+object in $\Set$. When we vary \code{x}, keeping \code{a}
+fixed, \code{C(a, x)} will also vary in $\Set$. Thus we have a
+mapping from \code{x} to $\Set$.
 
 \begin{figure}[H]
 \centering
@@ -163,16 +163,16 @@ \section{The Hom Functor}\label{the-hom-functor}
 \section{Representable Functors}\label{representable-functors}
 
 We've seen that, for every choice of an object \code{a} in \emph{C},
-we get a functor from \emph{C} to \textbf{Set}. This kind of
-structure-preserving mapping to \textbf{Set} is often called a
+we get a functor from \emph{C} to $\Set$. This kind of
+structure-preserving mapping to $\Set$ is often called a
 \newterm{representation}. We are representing objects and morphisms of
-\emph{C} as sets and functions in \textbf{Set}.
+\emph{C} as sets and functions in $\Set$.
 
 The functor \code{C(a, -)} itself is sometimes called representable.
 More generally, any functor \code{F} that is naturally isomorphic to
 the hom-functor, for some choice of \code{a}, is called
 \newterm{representable}. Such functor must necessarily be
-\textbf{Set}-valued, since \code{C(a, -)} is.
+$\Set$-valued, since \code{C(a, -)} is.
 
 I said before that we often think of isomorphic sets as identical. More
 generally, we think of isomorphic \newterm{objects} in a category as
@@ -203,7 +203,7 @@ \section{Representable Functors}\label{representable-functors}
 natural transformation.
 
 Let's look at the component of α at some object \code{x}. It's a
-function in \textbf{Set}:
+function in $\Set$:
 
 \begin{Verbatim}[commandchars=\\\{\}]
 α\textsubscript{x} :: C(a, x) -> F x
@@ -248,7 +248,7 @@ \section{Representable Functors}\label{representable-functors}
 α ◦ β = id = β ◦ α
 \end{Verbatim}
 We will see later that a natural transformation from \code{C(a, -)}
-to any \textbf{Set}-valued functor always exists (Yoneda's lemma) but it
+to any $\Set$-valued functor always exists (Yoneda's lemma) but it
 is not necessarily invertible.
 
 Let me give you an example in Haskell with the list functor and
@@ -328,7 +328,7 @@ \section{Representable Functors}\label{representable-functors}
 Representability for contravariant functors is similarly defined, except
 that we keep the second argument of \code{C(-, a)} fixed. Or,
 equivalently, we may consider functors from \emph{C}\textsuperscript{op}
-to \textbf{Set}, because \code{C\textsuperscript{op}(a, -)} is the same as
+to $\Set$, because \code{C\textsuperscript{op}(a, -)} is the same as
 \code{C(-, a)}.
 
 There is an interesting twist to representability. Remember that
@@ -370,7 +370,7 @@ \section{Challenges}\label{challenges}
 \tightlist
 \item
   Show that the hom-functors map identity morphisms in \emph{C} to
-  corresponding identity functions in \textbf{Set}.
+  corresponding identity functions in $\Set$.
 \item
   Show that \code{Maybe} is not representable.
 \item