Polished A-A and added new lines for broken enumerates.

src/cat/cat/cat-func.tex

@@ -13,7 +13,8 @@
         \item[(CAT1)] For any $A, B, A', B' \in \obj{\catc}$, $\mor{A, B}$ and $\mor{A', B'}$ are disjoint or equal, where $\mor{A, B} = \mor{A', B'}$ if and only if $A = A'$ and $B = B'$.
         \item[(CAT2)] For any $A \in \obj{\catc}$, there exists $\text{Id}_A \in \mor{A, A}$ such that $f \circ \text{Id}_A = f$ and $\text{Id}_A \circ g = g$ for all $B, C \in \obj{\catc}$, $f \in \mor{A, B}$, and $g \in \mor{C, A}$.
         \item[(CAT3)] For any $A, B, C, D \in \obj{\catc}$, $f \in \mor{A, B}$, $g \in \mor{B, C}$, and $h \in \mor{C, D}$, $(h \circ g) \circ f = h \circ (g \circ f)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The elements of $\obj{\catc}$ are the \textbf{objects} of $\catc$, and elements of $\mor{A, B}$ are the \textbf{morphisms/arrows} from $A$ to $B$.
 \end{definition}
 

src/cat/cat/tensor.tex

@@ -32,7 +32,8 @@
         A_i \ar@{->}[u]^{\iota_i} \ar@{->}[ru]_{T_i} & 
         }
         \]
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     The module $A = \bigoplus_{i \in I}A_i$ is the \textbf{direct sum} of $\seqi{A}$. 
 \end{definition}
@@ -204,7 +205,8 @@
         \[
         (x_1, \cdots, \alpha x_j, \cdots, x_n) - \alpha(x_1, \cdots, x_n)
         \]
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     (1), (2): Let $\bigotimes_{j = 1}^n E_j = M/N$ and
     \[

src/cat/cat/universal.tex

@@ -7,7 +7,8 @@
     \begin{enumerate}
         \item \textbf{universally attracting} if for every $A \in \obj{\catc}$, there exists a unique $f \in \mor{A, P}$.
         \item \textbf{universally repelling} if for every $A \in \obj{\catc}$, there exists a unique $f \in \mor{P, A}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If $P$ is universally attracting or repelling, then $P$ is a \textbf{universal object}.
 
     If $P, Q \in \obj{\catc}$ are both universally attracting/repelling, then they are isomorphic.
@@ -62,12 +63,14 @@
     \begin{enumerate}
         \item For any $i \in I$, $i \lesssim i$.
         \item For any $i, j, k \in I$ such that $i \lesssim j$ and $j \lesssim k$, $i \lesssim k$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     and one of the following holds:
     \begin{enumerate}
         \item[(3U)] For any $i, j \in I$, there exists $k \in I$ with $i, j \lesssim k$.
         \item[(3D)] For any $i, j \in I$, there exists $k \in I$ with $k \lesssim i, j$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     The directed set is \textbf{upward-directed} if it satisfies (3U), and \textbf{downward-directed} if it satisfies (3D).
 \end{definition}
 
@@ -85,7 +88,8 @@
         \item For each $i \in I$, $f^i_i = \text{Id}_{A_i}$.
         \item For each $i, j \in I$ with $i \lesssim j$, $f^i_j \in \mor{A_i, A_j}$.
         \item For each $i, j, k \in I$ with $i \lesssim j \lesssim k$, $f^j_k \circ f^i_j = f^i_k$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If $I$ is upward/downward-directed, then $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ is upward/downward-directed.
 \end{definition}
 

src/cat/gluing/index.tex

@@ -8,7 +8,8 @@
     \begin{enumerate}
         \item[(a)] $\bigcup_{i \in I}U_i = X$.
         \item[(b)] For each $i, j \in I$, either $U_i \cap U_j = \emptyset$, or $f_i|_{U_i \cap U_j} = f_j|_{U_i \cap U_j}$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then there exists a unique $f: X \to Y$ such that $f|_{U_i} = f_i$ for all $i \in I$.
 \end{lemma}
 \begin{proof}
@@ -16,7 +17,8 @@
     \begin{enumerate}
         \item By assumption (a), $\bracs{x|(x, y) \in \Gamma} = \bigcup_{i \in I}U_i = X$.
         \item For any $x \in X$, there exists $y \in Y$ with $(x, y) \in \Gamma$, and $i \in I$ such that $(x, y) \in \Gamma_i$. If $(x, y') \in \Gamma_j \subset \Gamma$, then $x \in U_i \cap U_j \ne \emptyset$. By assumption (b), $y = y'$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     Thus $\Gamma$ is the graph of a function $f: X \to Y$ with $f|_{U_i} = f_i$ for all $i \in I$.
 \end{proof}
 
@@ -27,7 +29,8 @@
         \item[(a)] $\bigcup_{V \in \fF}V = E$.
         \item[(b)] For each $V, W \in \fF$, $T_V|_{V \cap W} = T_W|_{V \cap W}$.
         \item[(c)] $\fF$ is upward-directed with respect to includion.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then there exists a unique $T \in \hom(E; F)$ such that $T|_{V} = T_V$ for all $V \in \fF$.
 \end{lemma}
 \begin{proof}

src/cat/tricks/dyadic.tex

@@ -42,7 +42,8 @@
     \begin{enumerate}
         \item[(a)] for each $n \in \natp$, $g_{n+1} + g_{n+1} \le g_n$.
         \item[(b)] For each $x, y \in G$, $x + y \ge x, y$. 
-    \end{enumerate}
+    \end\{enumerate\}
+
     
     For each $x \in \mathbb{D} \cap [0, 1)$, let $\phi(x) = \sum_{n \in M(x)}g_n$, then
     \begin{enumerate}