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\title{Homework 3}
\author{}
\date{}

\begin{document}

\maketitle

Do the following problems in Dummit and Foote.

\begin{itemize}
  \item Section 10.3 problem 20
  \item Section 18.1 problems 1--4
\end{itemize}

Let $\mathcal{C}$ be a category, and $A$, $B$ objects.
A \textit{coproduct} of $A$ and $B$ consists of an object
$C$ together with morphisms $i_A:A\to C$ and $i_B:B\to C$
such that the following property is satisfied.

\bigbreak\noindent{\textbf{Universal property of the coproduct}}:
Given any object $M$ with morphisms $\theta_A:A\to M$ and
$\theta_B:B\to M$ there is a unique morphism $\Theta:C\to M$
such that $\theta_A=\Theta\circ i_A$ and $\theta_B=\Theta\circ i_B$.

\[\begin{tikzcd}
  A \arrow{r}{i_A}\arrow[swap]{dr}{\theta_A} & C \arrow[dashed]{d}{\Theta}
 & B \arrow[swap]{l}{i_B}\arrow{dl}{\theta_B}\\
  & M &
\end{tikzcd}
\]

\medbreak
A \textit{product} of $A$ and $B$ consists of an object $P$
together with morphisms $p_A:P\to A$ and $p_B:P\to B$
such that given any object $N$ with morphisms
$\phi_A:N\to A$ and $\phi_B:N\to B$ there is a
unique morphism $\Phi:N\to P$ such that
$\phi_A= p_A\circ\Phi$ and $\phi_B=p_B\circ\Phi$.

\[\begin{tikzcd} A & P\arrow[swap]{l}{p_A}\arrow{r}{p_B} & B\\
& N \arrow{ul}{\phi_A}\arrow[dashed]{u}{\Phi}\arrow[swap]{ur}{\phi_B}\end{tikzcd}\]

\vfill\eject
\noindent
\textbf{Problem A.} Explain briefly why if the coproduct or product exists
in a category (for fixed $A$ and $B$) it is unique up to isomorphism.

\bigbreak\noindent
\textbf{Problem B.} In the category of $R$-modules, prove that
the Cartesian product $A\times B$ is both a product and a coproduct.

\bigbreak\noindent
\textbf{Problem C.} What are the product and coproduct in the
category of sets?



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