Expand category theory and module tensor product sections

Added detailed definitions and universal properties for slice, coslice, and pointed object categories in category_theory.tex, including connected categories and limits. Enhanced module.tex with explicit definitions for bimodules, base change functor, and clarified tensor product constructions. Improved localization and tensor product explanations, including natural isomorphisms and universal properties.

Author: hooyuser

associative_algebra.tex

@@ -366,7 +366,8 @@ \section{Algebra over Field}
 Let $K$ be a field and $A$ be a $K$-algebra. Then $a\in A$ is algebraic over $K$ if and only if $a$ is integral over $K$.
 
 
-\section{Commutative Algebra}
+\chapter{Commutative Unital Algebra}
+\section{Basic Properties}
 \begin{definition}{Commutative Algebra}{}
     Let $R$ be a commutative ring. A \textbf{commutative $R$-algebra} is an $R$-algebra where the multiplication is commutative. Or equivalently, a commutative $R$-algebra is a commutative ring $A$ together with a ring homomorphism $R\to A$. 
 \end{definition}
@@ -375,7 +376,7 @@ \section{Commutative Algebra}
     There is a category isomorphism $R\text{-}\mathsf{CAlg}\cong \left(R/\mathsf{CRing}\right)$.
 \end{remark}
 
-\subsection{Polynomial Algebra}
+\section{Polynomial Algebra}
 \begin{definition}{Polynomial Ring}{}
     Let $R$ be a commutative ring. The \textbf{polynomial ring} in $n$ variables over $R$ is the ring $R[x_1,\cdots,x_n]$ defined as the set of all formal sums $$\sum_{\alpha\in\mathbb{N}^n}a_\alpha x^\alpha$$ where $a_\alpha\in R$ satisfies $a_\alpha=0$ for all but finitely many $\alpha\in\mathbb{N}^n$ and $x^\alpha:=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ for $\alpha=(\alpha_1,\cdots,\alpha_n)\in\mathbb{N}^n$. The addition and multiplication are defined as follows: $$\sum_{\alpha\in\mathbb{N}^n}a_\alpha x^\alpha+\sum_{\alpha\in\mathbb{N}^n}b_\alpha x^\alpha=\sum_{\alpha\in\mathbb{N}^n}(a_\alpha+b_\alpha)x^\alpha$$ and $$\left(\sum_{\alpha\in\mathbb{N}^n}a_\alpha x^\alpha\right)\left(\sum_{\beta\in\mathbb{N}^n}b_\beta x^\beta\right)=\sum_{\gamma\in\mathbb{N}^n}\left(\sum_{\alpha+\beta=\gamma}a_\alpha b_\beta\right)x^\gamma.$$
 \end{definition}
@@ -526,7 +527,9 @@ \subsection{Polynomial Algebra}
     If $\deg f\le 0$
 \end{example}
 
-\subsection{Construction}
+\section{Construction}
+
+\subsection{Free Object}
 
 \begin{definition}{Free Commutative Algebra}{free_commutative_algebra}
     Let $X$ be a set and $R$ be a commutative ring. The \textbf{free commutative $R$-algebra} on $X$, denoted by $\mathrm{Free}_{R\text{-}\mathsf{CAlg}}(X)$, together with a map $\iota:X\to \mathrm{Free}_{R\text{-}\mathsf{CAlg}}(X)$, is defined by the following universal property: for any commutative $R$-algebra $A$ and any map $f:X\to A$, there exists a unique homomorphism $\widetilde{f}:\mathrm{Free}_{R\text{-}\mathsf{CAlg}}(X)\to A$ such that the following diagram commutes
@@ -558,6 +561,8 @@ \subsection{Construction}
     \]
 \end{prf}
 
+\subsection{Coproduct}
+