Added projective limits.

src/cat/cat/universal.tex

@@ -89,7 +89,7 @@
 
 \begin{definition}[Direct Limit]
 \label{definition:direct-limit}
-    Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system, then a \textbf{direct limit} of $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ is a pair $(A, \bracsn{f^i_A}_{i \in I})$ such that:
+    Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system, then a \textbf{direct limit} of $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$, denoted $\lim_{\longrightarrow}A_i$, is a pair $(A, \bracsn{f^i_A}_{i \in I})$ such that:
     \begin{enumerate}
         \item For each $i \in I$, $f^i_A \in \mor{A_i, A}$.
         \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
@@ -101,7 +101,7 @@
         }
         \]
 
-        \item[(U)] For any pair $(B, \bracsn{g^i_A}_{i \in I})$ satsifying (1) and (2), there exists a unique $g \in \mor{A, B}$ such that the following diagram commutes
+        \item[(U)] For any pair $(B, \bracsn{g^i_A}_{i \in I})$ satisfying (1) and (2), there exists a unique $g \in \mor{A, B}$ such that the following diagram commutes
 
         \[
         \xymatrix{
@@ -112,12 +112,11 @@
 
         for all $i \in I$.
     \end{enumerate}
-
 \end{definition}
 
 \begin{definition}[Inverse Limit]
 \label{definition:inverse-limit}
-Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system, then an \textbf{inverse limit} of $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ is a pair $(A, \bracsn{f^A_i}_{i \in I})$ such that:
+Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ be an downward-directed system, then an \textbf{inverse limit} of $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$, denoted $\lim_{\longleftarrow}A_i$, is a pair $(A, \bracsn{f^A_i}_{i \in I})$ such that:
 \begin{enumerate}
     \item For each $i \in I$, $f^A_i \in \mor{A, A_i}$.
     \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
@@ -141,3 +140,45 @@ Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim
     for all $i \in I$.
 \end{enumerate}
 \end{definition}
+
+\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:
+    \begin{enumerate}
+        \item For each $i \in I$, $T^A_i \in \hom(A; A_i)$.
+        \item For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
+        \[
+        \xymatrix{
+        A_i \ar@{->}[r]^{T^i_j} & A_j \\
+        A \ar@{->}[u]^{T^A_i} \ar@{->}[ru]_{T^A_j} &
+        }
+        \]
+        \item[(U)] For any pair $(B, \bracsn{S^B_i}_{i \in I})$ satisfying (1) and (2), there exists a unique $S \in \hom(B; A)$ such that the following diagram commutes
+
+        \[
+        \xymatrix{
+         & A_i \\
+        B \ar@{->}[r]_{S} \ar@{->}[ru]^{S^B_i} & A \ar@{->}[u]_{T^A_i}
+        }
+        \]
+
+        for all $i \in I$.
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    Let
+    \[
+    A = \bracs{x \in \prod_{i \in I}A_i \bigg | \pi_j(x) = T^i_j\pi_i(x) \forall i, j \in I, i \lesssim j}
+    \]
+    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 $A$.
+
+    (U): Let $(B, \bracsn{S^B_i}_{i \in I})$ satisfying (1) and (2). Let
+    \[
+    S: B \to \prod_{i \in I}A_i \quad \pi_i(Sx) = S^B_i
+    \]
+    then for any $x \in B$ and $i, j \in I$ with $i \lesssim j$,
+    \[
+    \pi_j (Sx) = S^B_jx = T^i_j S^B_ix = T^i_j \pi_i(S x)
+    \]
+    so $S \in \hom(B; A)$, and the diagram commutes. Since any map $f: B \to A$ is uniquely determined by its composition with the projections, $S$ is unique.
+\end{proof}