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\title{Math 728\\Fall 2004\\Homework 5}
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\centerline{\it Due Wednesday October 6, 2004.}

\centerline{}

A category, $\mathfrak{C}$, is a collection of objects together with the following.
\begin{itemize}
\item[a)] A collection of disjoint sets, one for each pair of objects $A,B\in\mathfrak{C}$, denoted $\text{hom}(A,B)$. An element $f\in\text{hom}(A,B)$ is called a morphism from $A$ to $B$ and is sometimes written $f:A\longrightarrow B$.
\item[b)] For each triple $(A,B,C)$ of objects in $\mathfrak{C}$, we have a function
\[
\text{hom}(B,C)\times\text{hom}(A,B)\longrightarrow\text{hom}(A,C)
\]

\noindent (if $f:A\longrightarrow B$ and $g:B\longrightarrow C$ we write $(g,f)\mapsto g\circ f$). This is called the composite and is subject to the following two axioms:
\begin{itemize}
\item[i)] $h\circ(g\circ f)=(h\circ g)\circ f$.
\item[ii)] For any object $A$, there is a morphism $1_A:A\longrightarrow A$ such that for all $f:B\longrightarrow A$ and $g:A\longrightarrow B$, $g\circ 1_A=g$ and $1_A\circ f=f$.
\end{itemize}
\end{itemize}

A functor from the category $\mathfrak{C}$ to the category $\mathfrak{D}$ is a pair of functions (both denoted by $F$) such that $F(C)$ is an object of $\mathfrak{D}$ for all objects $C\in\mathfrak{C}$. Also if $f:A\longrightarrow B$ is a morphism, then $F(f):F(A)\longrightarrow F(B)$ is a morphism with the following conditions.
\begin{itemize}
\item[a)] $F(1_A)=1_{F(A)}$ for all objects $A$ in $\mathfrak{C}$.
\item[b)] $F(g\circ f)=F(g)\circ F(f)$ (in this case the functor is called covariant). OR
\item[b$^{\prime}$)] $F(g\circ f)=F(f)\circ F(g)$ (in this case the functor is called contravariant).
\end{itemize}


\centerline{}

\noindent 1. Show that the following form categories.
\begin{itemize}
\item[a)] (5 pt) Commutative rings with identity (with ring homomorphisms).
\item[b)] (5 pt) Abelian groups (with group homomorphisms).
\item[c)] (5 pt) $R-$modules (with $R-$module homomorphisms).
\end{itemize}

\centerline{}

\noindent 2. (5 pt) Let $\mathfrak{C}$ be the category of commutative rings with identity and let $U(R)$ denote the units of $R$. Show that the assignment $R\mapsto U(R)$ defines a functor from the catogory of commutative rings with identity to the category of abelian groups (how does this work on morphisms?).

\centerline{}

\noindent 3. Consider a fixed $R-$module $D$. We have seen that for all $R-$modules $A$, $\text{Hom}_R(D,A)$ is again an $R-$module.
\begin{itemize}
\item[a)] (5 pt) Show that the assignment $A\mapsto\text{Hom}_R(D,A)$ defines a covariant functor from the category of $R-$modules to itself (how does it work on morphisms?).
\item[b)] (5 pt) Show that the assignment $A\mapsto\text{Hom}_R(A,D)$ defines a contravariant functor from the category of $R-$modules to itself (how does it work on morphisms?).
\end{itemize}



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