Using the macro \Cat

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

@@ -127,11 +127,11 @@ \section{Properties of Composition}\label{properties-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 $\code{id}_A$ (\newterm{identity} on $A$). In math
+object A is called $\idarrow[A]$ (\newterm{identity} on $A$). In math
 notation, if \code{f} goes from $A$ to $B$ then
-\[f\circ{}\code{id}_A = f\]
+\[f \circ \idarrow[A] = f\]
 and
-\[\code{id}_B\circ{}f = f\]
+\[\idarrow[B] \circ f = f\]
 \end{enumerate}
 When dealing with functions, the identity arrow is implemented as the
 identity function that just returns back its argument. The

src/content/1.10/Natural Transformations.tex

@@ -539,8 +539,8 @@ \section{Functor Category}\label{functor-category}
 building so far. We started with a category, which is a collection of
 objects and morphisms. Categories themselves (or, strictly speaking
 \emph{small} categories, whose objects form sets) are themselves objects
-in a higher-level category \textbf{Cat}. Morphisms in that category are
-functors. A Hom-set in \textbf{Cat} is a set of functors. For instance
+in a higher-level category $\Cat$. Morphisms in that category are
+functors. A Hom-set in $\Cat$ is a set of functors. For instance
 Cat(C, D) is a set of functors between two categories C and D.
 
 \begin{figure}
@@ -551,15 +551,15 @@ \section{Functor Category}\label{functor-category}
 A functor category {[}C, D{]} is also a set of functors between two
 categories (plus natural transformations as morphisms). Its objects are
 the same as the members of Cat(C, D). Moreover, a functor category,
-being a category, must itself be an object of \textbf{Cat} (it so
+being a category, must itself be an object of $\Cat$ (it so
 happens that the functor category between two small categories is itself
 small). We have a relationship between a Hom-set in a category and an
 object in the same category. The situation is exactly like the
 exponential object that we've seen in the last section. Let's see how we
-can construct the latter in \textbf{Cat}.
+can construct the latter in $\Cat$.
 
 As you may remember, in order to construct an exponential, we need to
-first define a product. In \textbf{Cat}, this turns out to be relatively
+first define a product. In $\Cat$, this turns out to be relatively
 easy, because small categories are \emph{sets} of objects, and we know
 how to define cartesian products of sets. So an object in a product
 category C × D is just a pair of objects, \code{(c, d)}, one from C
@@ -570,27 +570,27 @@ \section{Functor Category}\label{functor-category}
 \code{g :: d -> d'}. These pairs of morphisms
 compose component-wise, and there is always an identity pair that is
 just a pair of identity morphisms. To make the long story short,
-\textbf{Cat} is a full-blown cartesian closed category in which there is
+$\Cat$ is a full-blown cartesian closed category in which there is
 an exponential object D\textsuperscript{C} for any pair of categories.
-And by ``object'' in \textbf{Cat} I mean a category, so
+And by ``object'' in $\Cat$ I mean a category, so
 D\textsuperscript{C} is a category, which we can identify with the
 functor category between C and D.
 
 \section{2-Categories}\label{categories}
 
-With that out of the way, let's have a closer look at \textbf{Cat}. By
-definition, any Hom-set in \textbf{Cat} is a set of functors. But, as we
+With that out of the way, let's have a closer look at $\Cat$. By
+definition, any Hom-set in $\Cat$ is a set of functors. But, as we
 have seen, functors between two objects have a richer structure than
 just a set. They form a category, with natural transformations acting as
-morphisms. Since functors are considered morphisms in \textbf{Cat},
+morphisms. Since functors are considered morphisms in $\Cat$,
 natural transformations are morphisms between morphisms.
 
 This richer structure is an example of a 2-category, a generalization of
 a category where, besides objects and morphisms (which might be called
 1-morphisms in this context), there are also 2-morphisms, which are
 morphisms between morphisms.
 
-In the case of \textbf{Cat} seen as a 2-category we have:
+In the case of $\Cat$ seen as a 2-category we have:
 
 \begin{itemize}
 \tightlist
@@ -620,7 +620,7 @@ \section{2-Categories}\label{categories}
 With two kinds of composition in a 2-category, the question arises: How
 do they interact with each other?
 
-Let's pick two functors, or 1-morphisms, in \textbf{Cat}:
+Let's pick two functors, or 1-morphisms, in $\Cat$:
 
 \begin{Verbatim}[commandchars=\\\{\}]
 F :: C -> D
@@ -719,7 +719,7 @@ \section{2-Categories}\label{categories}
 transformations, and it's part of the functor category. But what about
 the horizontal composition? What category does that live in?
 
-The way to figure this out is to look at \textbf{Cat} sideways. Look at
+The way to figure this out is to look at $\Cat$ sideways. Look at
 natural transformations not as arrows between functors but as arrows
 between categories. A natural transformation sits between two
 categories, the ones that are connected by the functors it transforms.
@@ -731,7 +731,7 @@ \section{2-Categories}\label{categories}
 \end{figure}
 
 \noindent
-Let's focus on two objects of \textbf{Cat} --- categories C and D. There
+Let's focus on two objects of $\Cat$ --- categories C and D. There
 is a set of natural transformations that go between functors that
 connect C to D. These natural transformations are our new arrows from C
 to D. By the same token, there are natural transformations going between
@@ -753,7 +753,7 @@ \section{2-Categories}\label{categories}
 the evident diagrams needed to prove this fact.
 
 There is one more piece of notation that might come in handy in the
-future. In this new sideways interpretation of \textbf{Cat} there are
+future. In this new sideways interpretation of $\Cat$ there are
 two ways of getting from object to object: using a functor or using a
 natural transformation. We can, however, re-interpret the functor arrow
 as a special kind of natural transformation: the identity natural

src/content/3.12/Enriched Categories.tex

@@ -431,18 +431,18 @@ \section{Self Enrichment}\label{self-enrichment}
 
 \section{Relation to 2-Categories}\label{relation-to-2-categories}
 
-I talked about 2-categories in the context of \textbf{Cat}, the category
+I talked about 2-categories in the context of $\Cat$, the category
 of (small) categories. The morphisms between categories are functors,
 but there is an additional structure: natural transformations between
 functors. In a 2-category, the objects are often called zero-cells;
 morphisms, 1-cells; and morphisms between morphisms, 2-cells. In
-\textbf{Cat} the 0-cells are categories, 1-cells are functors, and
+$\Cat$ the 0-cells are categories, 1-cells are functors, and
 2-cells are natural transformations.
 
 But notice that functors between two categories form a category too; so,
-in \textbf{Cat}, we really have a \newterm{hom-category} rather than a
+in $\Cat$, we really have a \newterm{hom-category} rather than a
 hom-set. It turns out that, just like $\Set$ can be treated as a
-category enriched over $\Set$, \textbf{Cat} can be treated as a
-category enriched over \textbf{Cat}. Even more generally, just like
+category enriched over $\Set$, $\Cat$ can be treated as a
+category enriched over $\Cat$. Even more generally, just like
 every category can be treated as enriched over $\Set$, every
-2-category can be considered enriched over \textbf{Cat}.
+2-category can be considered enriched over $\Cat$.

src/content/3.15/Monads, Monoids, and Categories.tex

@@ -51,7 +51,7 @@ \section{Bicategories}\label{bicategories}
 \end{figure}
 
 \noindent
-The category of categories \textbf{Cat} is an immediate example. We have
+The category of categories $\Cat$ is an immediate example. We have
 categories as 0-cells, functors as 1-cells, and natural transformations
 as 2-cells. The laws of a 2-category tell us that 1-cells between any
 two 0-cells form a category (in other words, \code{C(a, b)} is a
@@ -73,7 +73,7 @@ \section{Bicategories}\label{bicategories}
 the corresponding category laws.
 
 Let's see what this means in our canonical example of a 2-category
-\textbf{Cat}. The hom-category \code{Cat(a, a)} is the category of
+$\Cat$. The hom-category \code{Cat(a, a)} is the category of
 endofunctors on \code{a}. Endofunctor composition plays the role of a
 tensor product in it. The identity functor is the unit with respect to
 this product. We've seen before that endofunctors form a monoidal
@@ -202,7 +202,7 @@ \section{Monads}\label{monads}
 By now you should be pretty familiar with the definition of a monad as a
 monoid in the category of endofunctors. Let's revisit this definition
 with the new understanding that the category of endofunctors is just one
-small hom-category of endo-1-cells in the bicategory \textbf{Cat}. We
+small hom-category of endo-1-cells in the bicategory $\Cat$. We
 know it's a monoidal category: the tensor product comes from the
 composition of endofunctors. A monoid is defined as an object in a
 monoidal category --- here it will be an endofunctor \code{T} ---
@@ -251,7 +251,7 @@ \section{Monads}\label{monads}
 \noindent
 That's a much more general definition of a monad using only 0-cells,
 1-cells, and 2-cells. It reduces to the usual monad when applied to the
-bicategory \textbf{Cat}. But let's see what happens in other
+bicategory $\Cat$. But let's see what happens in other
 bicategories.
 
 Let's construct a monad in \textbf{Span}. We pick a 0-cell, which is a

src/content/3.2/Adjunctions.tex

@@ -67,9 +67,9 @@ \section{Adjunction and Unit/Counit
 can indeed use the equality of elements of a set to equality-compare
 objects.
 
-But, remember, \textbf{Cat} is really a 2-category. Hom-sets in a
+But, remember, $\Cat$ is really a 2-category. Hom-sets in a
 2-category have additional structure --- there are 2-morphisms acting
-between 1-morphisms. In \textbf{Cat}, 1-morphisms are functors, and
+between 1-morphisms. In $\Cat$, 1-morphisms are functors, and
 2-morphisms are natural transformations. So it's more natural (can't
 avoid this pun!) to consider natural isomorphisms as substitutes for
 equality when talking about functors.

src/content/3.6/Monads Categorically.tex

@@ -451,7 +451,7 @@ \section{Monads as Monoids}\label{monads-as-monoids}
 any two functors can be composed --- the target category of one has to
 be the source category of the other. That's just the usual rule of
 composition of morphisms --- and, as we know, functors are indeed
-morphisms in the category \textbf{Cat}. But just like endomorphisms
+morphisms in the category $\Cat$. But just like endomorphisms
 (morphisms that loop back to the same object) are always composable, so
 are endofunctors. For any given category \emph{C}, endofunctors from
 \emph{C} to \emph{C} form the functor category \code{{[}C, C{]}}. Its