Minor edits

Signed-off-by: zeramorphic <[email protected]>

iii/cat/01_definitions_and_examples.tex

@@ -178,7 +178,7 @@ \subsection{Natural transformations}
         \end{tikzcd}\]
         In particular, the existence of \( \alpha_x \) proves that \( fx \leq gx \).
         Thus a natural transformation \( f \to g \) exists if and only if \( fx \leq gx \) pointwise for all \( x \in P \).
-        Note that every commutative square in a poset commutes.
+        Note that every square of morphisms in a poset commutes.
         \item Let \( u, v : G \rightrightarrows H \) be group homomorphisms.
         For \( g \in G \), the naturality square is
         \[\begin{tikzcd}
@@ -260,7 +260,7 @@ \subsection{Equivalence of categories}
         and
         \[ G : \mathbf{Part} \to \mathbf{Set}_\star;\quad G(A) = A \cup \qty{A};\quad G(A \xrightarrow f B \text{ partial})(x) = \begin{cases}
             f(x) & \text{if } f \text{ is defined at } x \\
-            V & \text{otherwise}
+            B & \text{otherwise}
         \end{cases} \]
         Note that \( FG = 1_{\mathbf{Part}} \), but \( GF \) is not equal to \( 1_{\mathbf{Set}_\star} \).
         It is not possible for these two categories to be isomorphic, because there is an isomorphism class of \( \mathbf{Part} \) that has only one member, namely \( \qty{\varnothing} \), but this cannot occur in \( \mathbf{Set}_\star \).
@@ -359,7 +359,7 @@ \subsection{Equivalence of categories}
 \end{proof}
 We call a subcategory full if its inclusion functor is full.
 \begin{definition}
-    A category of \emph{skeletal} if every isomorphism class has a single member.
+    A category is called \emph{skeletal} if every isomorphism class has a single member.
     A \emph{skeleton} of \( \mathcal C \) is a full subcategory \( \mathcal C' \) containing exactly one object for each isomorphism class.
 \end{definition}
 Note that an equivalence of skeletal categories is bijective on objects, and hence is an isomorphism of categories.

iii/cat/02_yoneda_lemma.tex

@@ -209,7 +209,7 @@ \subsection{Representable functors}
     Hence, \( e \) is a monomorphism.
     Monomorphisms that occur in this way are called \emph{regular}.
     \item Dually, there is also a notion of coequaliser, giving rise to an epimorphism.
-    We again epimorphisms \emph{regular} if they arise in this way.
+    We again call epimorphisms \emph{regular} if they arise in this way.
 \end{enumerate}
 In \( \mathbf{Set} \), the categorical product is the Cartesian product, and the categorical coproduct is the disjoint union.
 The equaliser of \( f, g : A \rightrightarrows B \) is the set
@@ -361,5 +361,5 @@ \subsection{Projectivity}
     % any coproduct of projectives is projective
     Then \( P \) is pointwise projective, since the \( \mathcal C(A, -) \) are.
     There is a natural transformation \( \alpha : P \to F \) where the \( (A, x) \)-indexed term is \( \Psi(x) : \mathcal C(A, -) \to F \).
-    This is pointwise epic, since any \( x \in FA \) is in the image of \( \Psi(x) \). 
+    This is pointwise epic, since any \( x \in FA \) is in the image of \( \Psi(x) \).
 \end{proof}

iii/cat/04_limits.tex

@@ -37,7 +37,7 @@ \subsection{Cones over diagrams}
 \end{example}
 \begin{definition}
     Let \( D \) be a diagram of shape \( J \) in \( \mathcal C \).
-    A \emph{cone over \( D \)} consists of an object \( C \in \ob \mathcal C \) called the \emph{apex} of the cone, together with morphisms \( \lambda_j : A \to D(j) \) called the \emph{legs} of the cone, such that all triangles of the following form commute.
+    A \emph{cone over \( D \)} consists of an object \( A \in \ob \mathcal C \) called the \emph{apex} of the cone, together with morphisms \( \lambda_j : A \to D(j) \) called the \emph{legs} of the cone, such that all triangles of the following form commute.
     % https://q.uiver.app/#q=WzAsMyxbMSwwLCJBIl0sWzAsMSwiRChqKSJdLFsyLDEsIkQoaicpIl0sWzAsMSwiXFxsYW1iZGFfaiIsMl0sWzEsMiwiRChcXGFscGhhKSIsMl0sWzAsMiwiXFxsYW1iZGFfe2onfSJdXQ==
 \[\begin{tikzcd}
 	& A \\
@@ -400,6 +400,7 @@ \subsection{General adjoint functor theorem}
     for some \( i \in I \) and \( g : B_i \to B \).
     This set \( I \) is called a solution-set at \( A \).
 \end{theorem}
+The solution-set condition can be equivalently phrased as the assertion that the categories \( (A \downarrow G) \) all have \emph{weakly initial} sets of objects: every object of \( (A \downarrow G) \) admits a morphism from a member of the solution set.
 \begin{proof}
     If \( F \dashv G \), then \( G \) preserves all limits that exist in its domain, so in particular it preserves small limits, and \( \qty{\eta_A : A \to GFA} \) is a solution-set at \( A \) for any \( A \).
     Now suppose \( A \in \ob \mathcal C \).
@@ -441,7 +442,7 @@ \subsection{General adjoint functor theorem}
     \end{enumerate}
 \end{example}
 
-\subsection{Well-poweredness}
+\subsection{Special adjoint functor theorem}
 \begin{definition}
     Let \( A \in \ob \mathcal C \).
     A \emph{subobject} of \( A \) is a monomorphism with codomain \( A \); dually, a \emph{quotient} of \( A \) is an epimorphism with domain \( A \).
@@ -485,8 +486,6 @@ \subsection{Well-poweredness}
 \end{tikzcd}\]
     So \( \ell \) and \( m \) are both factorisations of \( (h\ell, k\ell) \) through the pullback, so \( \ell = m \).
 \end{proof}
-
-\subsection{Special adjoint functor theorem}
 \begin{theorem}
     Let \( \mathcal C, \mathcal D \) be locally small, and suppose that \( \mathcal D \) is complete, well-powered, and has a coseparating set.
     Then a functor \( G : \mathcal D \to \mathcal C \) preserves all small limits if and only if it has a left adjoint.