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\lhead{{\sf Summer Bridge Course (Algebra)}: Friday July 17}
\chead{}\rhead{\thepage}
\lfoot{Due Monday, July 20 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
\pagestyle{fancy}


\begin{document}

%\(\;\) \vspace{-4mm}


\noindent \textbf{Topics}: Basic category theory. \\

\noindent \textbf{Do}: Answer the following questions. \\

%%--------------------------------------------------------------------- 

%% HW 9, Problem 1
\begin{exercise}
  If \(f\colon A\to B\) is a morphism in a category \(\mathcal C\), then \(g\colon B\to A\) is its \emph{inverse} if \(g\circ f=\Id_A\) and \(f\circ g=\Id_B\). Prove that if \(f\) has an inverse, then it is unique. \\
\end{exercise}

%\begin{solution}
%\end{solution}

%%--------------------------------------------------------------------- 

%% HW 9, Problem 2
\begin{exercise}
  Let \(X\) be an object in a category \(\mathcal{C}\). A \emph{universal pair} \((U,\upsilon)\) for \(Y\) with respect to  a property consists of an object \(U\) and morphisms \(\upsilon\colon U\to Y\), such that every other morphism \(f\colon X\to Y\) with the same property factors through \(\upsilon\) uniquely. That is, there exists a unique morphism \(g\colon X\to U\) such that \(f=\upsilon\circ g\), i.e., it makes the following diagram commute:
  %%
  %% Commutative diagrams showing universal pairs
  \[
  \begin{tikzpicture}[scale=1.6]
    \node (X) at (-1,1) {\(X\)};
    \node (U) at (0,0) {\(U\)};
    \node (Y) at (1,1) {\(Y\)};
    \draw[->] (U) -- (Y) node[midway,below right]{\footnotesize \(\upsilon\)};
    \draw[->] (X) -- (Y) node[midway,above]{\footnotesize \(f\)};
    \draw[->,dashed] (X) -- (U) node[midway,below left]{\footnotesize \(g\)};
  \end{tikzpicture}
  \]
  Prove that if \(Y\) has a universal pair \((U,\upsilon)\) with respect to  
  some property, then \(U\) is unique up to isomorphism. \\
\end{exercise}

%\begin{solution}
%\end{solution}

%%--------------------------------------------------------------------- 

%% HW 9, Problem 3
\begin{exercise}
  Let \(A\) and \(B\) be objects in the category \(\mathbf{Set}\).
  
  \begin{enumerate}[label=(\roman*)]
  \item Prove that the Cartesian product \(A\times B\), together with the
    projection maps
    \[
    \pi_A\colon A\times B\longto A,\qquad \pi_B\colon A\times B\longto B,
    \]
    is a product of \(A\) and \(B\) in \(\mathbf{Set}\).
    
  \item Prove that the disjoint union \(A\sqcup B\), together with the
    inclusion maps
    \[
    \iota_A\colon A\longto A\sqcup B,\qquad \iota_B\colon B\longto A\sqcup B,
    \]
    is a coproduct of \(A\) and \(B\) in \(\mathbf{Set}\).
    
  \item Explain why the ordinary union \(A\cup B\) is not always a coproduct
    of \(A\) and \(B\) in \(\mathbf{Set}\).
  \end{enumerate}
\end{exercise}

%\begin{solution}
%\end{solution}

%%--------------------------------------------------------------------- 

\end{document}