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src/cat/cat-func.tex

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+\chapter{Categories and Functors}
+\label{chap:categories}
+
+
+
+\begin{definition}[Category]
+\label{definition:category}
+    A \textbf{category} $\catc$ is a collection of objects $\obj{\catc}$, such that for any $A, B, C \in \obj{\catc}$, there exists sets $\mor{A, B}$, $\mor{B, C}$, and a composition law
+    \[
+    \mor{A, B} \times \mor{B, C} \to \mor{A, C}
+    \]
+    that satisfies the following axioms:
+    \begin{enumerate}
+        \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}
+    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}
+
+\begin{definition}[Isomorphism]
+\label{definition:isomorphism}
+    Let $\catc$ be a category, $A, B \in \obj{\catc}$, and $f \in \mor{A, B}$, then $f$ is an \textbf{isomorphism} if there exists $g \in \mor{B, A}$ such that $g \circ f = \text{Id}_A$ and $f \circ g = \text{Id}_B$.
+
+    For any $A, B \in \obj{\catc}$, $A$ and $B$ are \textbf{isomorphic} if there exists an isomorphism $f \in \mor{A, B}$.
+\end{definition}
+
+
+\begin{definition}[Functor]
+\label{definition:functor}
+    Let $\mathfrak{A}$ and $\mathfrak{B}$ be categories. A \textbf{covariant functor} is a rule that assigns each $A \in \obj{\mathfrak{A}}$ to some $\lambda(A) \in \obj{\mathfrak{B}}$, and each $f \in \mor{A, B}$ to some $\lambda(f) \in \mor{\lambda(A), \lambda(B)}$, such that
+    \begin{enumerate}
+        \item[(FN1)] For any $A \in \obj{\mathfrak{A}}$, $\lambda(\text{Id}_A) = \text{Id}_{\lambda(A)}$.
+        \item[(FN2)] For any $A, B, C \in \obj{\mathfrak{A}}$, $f \in \mor{A, B}$, and $g \in \mor{B, C}$, $\lambda(g \circ f) = \lambda(g) \circ \lambda(f)$.
+    \end{enumerate}
+
+    A \textbf{contravariant} functor is a rule that assigns each $A \in \obj{\mathfrak{A}}$ to some $\lambda(A) \in \obj{\mathfrak{B}}$, and each $f \in \mor{A, B}$ to some $\lambda(f) \in \mor{\lambda(B), \lambda(A)}$, that satisfies (FN1) and
+    \begin{enumerate}
+        \item[(FN2')] For any $A, B, C \in \obj{\mathfrak{A}}$, $f \in \mor{A, B}$, and $g \in \mor{B, C}$, $\lambda(g \circ f) = \lambda(f) \circ \lambda(g)$.
+    \end{enumerate}
+\end{definition}

src/cat/index.tex

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+\part{Categories}
+\label{part:part-categories}
+
+\textit{I do not know much about categories, however some concepts from it are useful in phrasing certain properties.}
+
+\input{./src/cat/cat-func.tex}
+\input{./src/cat/universal.tex}

src/cat/universal.tex

@@ -0,0 +1,137 @@
+\chapter{Universal Construction}
+\label{chap:universal}
+
+\begin{definition}[Universal Object]
+\label{definition:universal}
+    Let $\catc$ be a category and $P \in \obj{\catc}$, then $P$ is...
+    \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}
+    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.
+\end{definition}
+\begin{proof}
+    By assumption, there exists morphisms $f \in \mor{P, Q}$ and $g \in \mor{Q, P}$. Since $f \circ g \in \mor{Q, Q}$ and $g \circ f \in \mor{P, P}$ are unique, $f \circ g = \text{Id}_Q$ and $g \circ f = \text{Id}_P$. Thus $f$ is an isomorphism.
+\end{proof}
+
+\begin{definition}[Product]
+\label{definition:product}
+    Let $\catc$ be a category and $\seqi{A} \subset \obj{\catc}$. A \textbf{product} of $\seqi{A}$ is a pair $(P, \seqi{\pi})$ where
+    \begin{enumerate}
+        \item $P \in \obj{\catc}$.
+        \item For each $i \in I$, $\pi_i \in \mor{P, A_i}$.
+        \item[\textbf{(U)}] For any pair $(C, \seqi{f})$ satisfying (1) and (2), there exists a unique $f \in \mor{C, P}$ such that the following diagram commutes
+
+        \[
+        \xymatrix{
+        C \ar@{->}[rd]^{f_i} \ar@{->}[d]_{f} &  \\
+        P \ar@{->}[r]_{\pi_i} & A_i
+        }
+        \]
+
+        for all $i \in I$.
+    \end{enumerate}
+\end{definition}
+
+\begin{definition}[Coproduct]
+    \label{definition:coproduct}
+    Let $\catc$ be a category and $\seqi{A} \subset \obj{\catc}$. A \textbf{product} of $\seqi{A}$ is a pair $(P, \seqi{\iota})$ where
+    \begin{enumerate}
+        \item $P \in \obj{\catc}$.
+        \item For each $i \in I$, $\iota_i \in \mor{A_i P}$.
+        \item[\textbf{(U)}] For any pair $(C, \seqi{f})$ satisfying (1) and (2), there exists a unique $f \in \mor{P, C}$ such that the following diagram commutes
+
+        \[
+        \xymatrix{
+         & C \\
+        A_i \ar@{->}[r]_{\iota_i} \ar@{->}[ru]^{f_i} & P \ar@{->}[u]_{f}
+        }
+        \]
+
+        for all $i \in I$.
+    \end{enumerate}
+\end{definition}
+
+\begin{definition}[Directed Set]
+\label{definition:directed-set}
+    Let $I$ be a set and $\lesssim$ be a relation on $I$, then $(I, \lesssim)$ is a \textbf{directed set} if
+    \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}
+    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}
+    The directed set is \textbf{upward-directed} if it satisfies (3U), and \textbf{downward-directed} if it satisfies (3D).
+\end{definition}
+
+\begin{definition}[Directed System]
+\label{definition:directed-system}
+    Let $\catc$ be a category and $(I, \lesssim)$ be a directed set. A \textbf{directed system} is a pair $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ such that:
+    \begin{enumerate}
+        \item $\seqi{A} \subset \obj{\catc}$.
+        \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}
+    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}
+
+\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:
+    \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:
+
+        \[
+        \xymatrix{
+        A_i \ar@{->}[rd]_{f^i_A} \ar@{->}[r]^{f^i_j} & A_j \ar@{->}[d]^{f^j_A} \\
+         & A
+        }
+        \]
+
+        \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
+
+        \[
+        \xymatrix{
+        A_j \ar@{->}[d]_{f^j_A} \ar@{->}[rd]^{g^i_B} &  \\
+        A \ar@{->}[r]_{g} & B
+        }
+        \]
+
+        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:
+\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:
+
+    \[
+    \xymatrix{
+    A_i \ar@{->}[r]^{f^i_j} & A_j \\
+    A \ar@{->}[u]^{f^A_i} \ar@{->}[ru]_{f^A_j} &
+    }
+    \]
+
+    \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
+
+    \[
+    \xymatrix{
+     & A_i \\
+    B \ar@{->}[r]_{g} \ar@{->}[ru]^{g^B_i} & A \ar@{->}[u]_{f^A_i}
+    }
+    \]
+
+    for all $i \in I$.
+\end{enumerate}
+\end{definition}