Fix minor typos (#304)

* typo (date -> data)

* Replace double colon with LaTeX \Colon

* typo (\eta_aa -> \eta_a)

* Replace \emph{} with \cat{}

* Add missing colon

* typo (× -> \times)

* typo (C -> $\cat{C}$)

* fixup! 38f8fe8e41038b2fab7fa068a78cff4fd2676e7438f8fe8e41038b2fab7fa068a78cff4fd2676e74

src/content/1.9/function-types.tex

@@ -320,7 +320,7 @@ \section{Exponentials}
 specified by a pair of values: one corresponding to \code{False}, and
 one corresponding to \code{True}. The set of all possible functions
 from \code{Bool} to, say, \code{Int} is the set of all pairs of
-\code{Int}s. This is the same as the product \code{Int} × \code{Int} or,
+\code{Int}s. This is the same as the product \code{Int} \times \code{Int} or,
 being a little creative with notation, \code{Int}\textsuperscript{2}.
 
 For another example, let's look at the C++ type \code{char}, which
@@ -330,7 +330,7 @@ \section{Exponentials}
 Functions like \code{isupper} or \code{isspace} are implemented
 using tables, which are equivalent to tuples of 256 Boolean values. A
 tuple is a product type, so we are dealing with products of 256
-Booleans: \code{bool × bool × bool × ... × bool}. We know from
+Booleans: \code{bool \times bool \times bool \times ... \times bool}. We know from
 arithmetics that an iterated product defines a power. If you
 ``multiply'' \code{bool} by itself 256 (or \code{char}) times, you
 get \code{bool} to the power of \code{char}, or \code{bool}\textsuperscript{\code{char}}.

src/content/3.10/ends-and-coends.tex

@@ -15,7 +15,7 @@
 \[\cat{C}(-, =) \Colon \cat{C}^{op}\times{}\cat{C} \to \Set\]
 In general, any functor like this may be interpreted as establishing a
 relation between objects in a category. A relation may also involve two
-different categories \emph{C} and \emph{D}. A functor, which describes
+different categories $\cat{C}$ and $\cat{D}$. A functor, which describes
 such a relation, has the following signature and is called a profunctor:
 \[p \Colon \cat{D}^{op}\times{}\cat{C} \to \Set\]
 Mathematicians say that it's a profunctor from $\cat{C}$ to $\cat{D}$

src/content/3.14/lawvere-theories.tex

@@ -112,8 +112,8 @@ \section{Lawvere Theories}
 identified through isomorphisms.
 
 Using the category $\cat{F}$ we can formally define a \newterm{Lawvere
-theory} as a category $\cat{L}$ equipped with a special functor
-\[I_{\cat{L}} :: \Fop \to \cat{L}\]
+theory} as a category $\cat{L}$ equipped with a special functor:
+\[I_{\cat{L}} \Colon \Fop \to \cat{L}\]
 This functor must be a bijection on objects and it must preserve finite
 products (products in $\Fop$ are the same as
 coproducts in $\cat{F}$):
@@ -496,7 +496,7 @@ \section{Monads as Coends}
 \begin{tikzcd}
   & a^n \times \cat{L}(m, 1)
     \arrow[dl, "\langle f {,} \id \rangle"']
-    \arrow[dr, "\langle \id {,} f \rangle"] 
+    \arrow[dr, "\langle \id {,} f \rangle"]
     & \\
 a^m \times \cat{L}(m, 1)
   & \scalebox{2.5}[1]{\sim}
@@ -518,7 +518,7 @@ \section{Monads as Coends}
 \[a^n \times \cat{L}(n, 1)\]
 when we lift $\langle \id, f \rangle$. This doesn't
 mean, however, that all elements of $a^n \times \cat{L}(n, 1)$ can be
-identified with $a × \cat{L}(1, 1)$. That's because not all elements
+identified with $a \times \cat{L}(1, 1)$. That's because not all elements
 of $\cat{L}(n, 1)$ can be reached from $\cat{L}(1, 1)$. Remember
 that we can only lift morphisms from $\cat{F}$. A non-trivial $n$-ary
 operation in $\cat{L}$ cannot be constructed by lifting a morphism
@@ -579,7 +579,7 @@ \section{Lawvere Theory of Side Effects}
 \begin{tikzcd}
   & a^n \times \cat{L}(0, 1)
     \arrow[dl, "\langle f {,} \id \rangle"']
-    \arrow[dr, "\langle \id {,} f \rangle"] 
+    \arrow[dr, "\langle \id {,} f \rangle"]
     & \\
 a^0 \times \cat{L}(0, 1)
   & \scalebox{2.5}[1]{\sim}

src/content/3.7/comonads.tex

@@ -377,7 +377,7 @@ \section{The Store Comonad}
 The \code{Store} comonad plays an important role as the theoretical
 basis for the \code{Lens} library. Conceptually, the
 \code{Store s a} comonad encapsulates the idea of ``focusing'' (like
-a lens) on a particular substructure of the date type \code{a} using
+a lens) on a particular substructure of the data type \code{a} using
 the type \code{s} as an index. In particular, a function of the type:
 
 \src{snippet38}

src/content/3.9/algebras-for-monads.tex

@@ -29,7 +29,7 @@
 object --- the carrier $a$ --- together with the morphism:
 \[alg \Colon m\ a \to a\]
 The first thing to notice is that the algebra goes in the opposite
-direction to $\eta_aa$. The intuition is that $\eta_a$ creates a
+direction to $\eta_a$. The intuition is that $\eta_a$ creates a
 trivial expression from a value of type $a$. The first coherence
 condition that makes the algebra compatible with the monad ensures that
 evaluating this expression using the algebra whose carrier is $a$
@@ -52,7 +52,7 @@
   \begin{subfigure}
     \centering
     \begin{tikzcd}[column sep=large, row sep=large]
-      a \arrow[rd, equal] \arrow[r, "\eta_a"] 
+      a \arrow[rd, equal] \arrow[r, "\eta_a"]
           & Ta \arrow[d, "alg"] \\
           & a
     \end{tikzcd}
@@ -196,7 +196,7 @@ \section{T-algebras}
   \begin{subfigure}
     \centering
     \begin{tikzcd}[column sep=large, row sep=large]
-      Ta \arrow[rd, equal] \arrow[r, "T \eta_a"] 
+      Ta \arrow[rd, equal] \arrow[r, "T \eta_a"]
           & T(Ta) \arrow[d, "\mu_a"] \\
           & Ta
     \end{tikzcd}
@@ -205,7 +205,7 @@ \section{T-algebras}
   \begin{subfigure}
     \centering
     \begin{tikzcd}[column sep=large, row sep=large]
-      a \arrow[rd, equal] \arrow[r, "\eta_a"] 
+      a \arrow[rd, equal] \arrow[r, "\eta_a"]
           & Ta \arrow[d, "f"] \\
           & a
     \end{tikzcd}
@@ -220,7 +220,7 @@ \section{T-algebras}
 \[F^T \dashv U^T\]
 Every adjunction gives rise to a monad. The round trip
 \[U^T \circ F^T\]
-is the endofunctor in C that gives rise to the corresponding monad.
+is the endofunctor in $\cat{C}$ that gives rise to the corresponding monad.
 Let's see what its action on an object $a$ is. The free algebra
 created by $F^T$ is $(T\ a, \mu_a)$. The forgetful functor
 $U^T$ drops the evaluator. So, indeed, we have:
@@ -292,7 +292,7 @@ \section{The Kleisli Category}
 \src{snippet06}
 We can also define a functor $G$ from $\cat{C}_T$
 back to $\cat{C}$. It takes an object $a$ from the Kleisli
-category and maps it to an object $T\ a$ in $\cat{C}$. Its action 
+category and maps it to an object $T\ a$ in $\cat{C}$. Its action
 on a morphism $f_{\cat{K}}$ corresponding to a Kleisli arrow:
 \[f \Colon a \to T\ b\]
 is a morphism in $\cat{C}$:
@@ -329,7 +329,7 @@ \section{Coalgebras for Comonads}
   \begin{subfigure}
     \centering
     \begin{tikzcd}[column sep=large, row sep=large]
-      a \arrow[rd, equal] 
+      a \arrow[rd, equal]
           & Wa \arrow[l, "\epsilon_a"'] \\
           & a \arrow[u, "coa"']
     \end{tikzcd}