From b48f5859f1f951a271f66ceed8d477b51ed4674d Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Thu, 29 Jan 2026 12:06:44 -0500
Subject: [PATCH] Fixed some typos.

---
 src/cat/cat/universal.tex | 63 ++++++++++++++++++++++++++++++++++++---
 1 file changed, 59 insertions(+), 4 deletions(-)

diff --git a/src/cat/cat/universal.tex b/src/cat/cat/universal.tex
index cb6f1e6..865c4be 100644
--- a/src/cat/cat/universal.tex
+++ b/src/cat/cat/universal.tex
@@ -101,7 +101,7 @@
         }
         \]
 
-        \item[(U)] For any pair $(B, \bracsn{g^i_A}_{i \in I})$ satisfying (1) and (2), there exists a unique $g \in \mor{A, B}$ such that the following diagram commutes
+        \item[(U)] For any pair $(B, \bracsn{g^i_B}_{i \in I})$ satisfying (1) and (2), there exists a unique $g \in \mor{A, B}$ such that the following diagram commutes
 
         \[
         \xymatrix{
@@ -128,12 +128,12 @@ Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim
     }
     \]
 
-    \item[(U)] For any pair $(B, \bracsn{g^A_i}_{i \in I})$, there exists a unique $g \in \mor{B, A}$ such that the following diagram commutes
+    \item[(U)] For any pair $(B, \bracsn{g^B_i}_{i \in I})$, there exists a unique $g \in \mor{B, A}$ such that the following diagram commutes
 
     \[
     \xymatrix{
      & A_i \\
-    B \ar@{->}[r]_{g} \ar@{->}[ru]^{g^B_i} & A \ar@{->}[u]_{f^A_i}
+    B \ar@{->}[r]_{S} \ar@{->}[ru]^{g^B_i} & A \ar@{->}[u]_{f^A_i}
     }
     \]
 
@@ -141,6 +141,61 @@ Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim
 \end{enumerate}
 \end{definition}
 
+\begin{proposition}
+\label{proposition:direct-limit}
+    Let $R$ be a ring and $(\seqi{A}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of $R$-modules, then there exists $(A, \bracsn{T^i_A}_{i \in I})$ such that:
+    \begin{enumerate}
+        \item For each $i \in I$, $T^i_A \in \hom({A_i, A})$.
+        \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
+
+        \[
+        \xymatrix{
+        A_i \ar@{->}[rd]_{T^i_A} \ar@{->}[r]^{T^i_j} & A_j \ar@{->}[d]^{T^j_A} \\
+         & A
+        }
+        \]
+
+        \item[(U)] For any pair $(B, \bracsn{S^i_B}_{i \in I})$ satisfying (1) and (2), there exists a unique $S \in \hom({A, B})$ such that the following diagram commutes
+
+        \[
+        \xymatrix{
+        A_i \ar@{->}[d]_{T^i_A} \ar@{->}[rd]^{S^i_B} &  \\
+        A \ar@{->}[r]_{g} & B
+        }
+        \]
+
+        for all $i \in I$.
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    Let $M = \bigoplus_{i \in I}A_i$. For any $i, j \in I$ with $i \lesssim j$ and $x \in A_i$, let $x_{i, j} \in M$ such that for any $k \in I$,
+    \[
+    \pi_k(x_{i, j}) = \begin{cases}
+        x &k = i \\
+        T^i_j x &k = j \\
+        0 &k \ne i, j
+    \end{cases}
+    \]
+    Let $N \subset M$ be the submodule generated by $\bracs{x_{i, j}|i, j \in I, i \lesssim j, x \in A_i}$, $A = M/N$, and $\pi: M \to M/N$ be the canonical map.
+
+    (1): For each $i \in I$, let
+    \[
+    T^i_M: A_i \to M \quad \pi_k T^i_M x = \begin{cases}
+        x &k = i \\
+        0 &k \ne i
+    \end{cases}
+    \]
+    and $T^i_A = \pi \circ T^i_M$.
+
+    (2): Let $i, j \in I$ with $i \lesssim j$, then for any $x \in A_i$, $T^i_Mx - T^j_M T^i_j x \in N$. Hence $T^i_Ax = T^j_A T^i_jx$.
+
+    (U): Let
+    \[
+    S_0: M \to B \quad x \mapsto \sum_{i \in I}S^i_B \pi_i x
+    \]
+    then $S_0$ is the unique linear map such that $S_0 \circ T^i_M = S^i_B$ for all $i \in I$. For any $i, j \in I$ with $i \lesssim J$, $S^i_B x = S^j_B T^i_j x$, so $\ker S_0 \supset N$. By the first isomorphism theorem, there exists a unique $S \in \hom(A; B)$ such that $S_0 = S \circ \pi$.
+\end{proof}
+
 \begin{proposition}
 \label{proposition:module-inverse-limit}
     Let $R$ be a ring and $(\seqi{A}, \bracs{T^i_j|i, j \in I, i \lesssim j)}$ be a downward-directed system of $R$-modules, then there exists $(A, \bracsn{T^A_i}_{i \in I})$ such that:
@@ -170,7 +225,7 @@ Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim
     \[
     A = \bracs{x \in \prod_{i \in I}A_i \bigg | \pi_j(x) = T^i_j\pi_i(x) \forall i, j \in I, i \lesssim j}
     \]
-    For each $i \in I$, let $T^A_i = \pi_i$, then $(A, (A, \bracsn{T^A_i}_{i \in I})$ satisfies (1) and (2) by definition of $A$.
+    For each $i \in I$, let $T^A_i = \pi_i$, then $(A, \bracsn{T^A_i}_{i \in I})$ satisfies (1) and (2) by definition of $A$.
 
     (U): Let $(B, \bracsn{S^B_i}_{i \in I})$ satisfying (1) and (2). Let
     \[