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Add solution to I.5.5 #10

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9 changes: 9 additions & 0 deletions Solution_to_Algebra_Chapter_0.tex
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Expand Up @@ -1153,6 +1153,15 @@ \subsection{\textsection5. Universal properties}
What are the final objects in the category considered in \S5.3? [\S5.3]
\end{problem}
\begin{solution}
The final objects are morphisms to any singletons, i.e.\ $f:A\to\{\bullet\}$.

For any set $Z$, there exists exactly one function $\sigma:Z\to\{\bullet\}$ which maps all elements in $Z$ to $\bullet$.
Thus, for any object $(\varphi,Z)$ of the category considered in \S5.3, there exists a unique morphism determined by $\sigma$:
\[\begin{tikzcd}[column sep=tiny]
Z \arrow[rr,"\sigma"] && \{\bullet\} \\
&A \arrow[ul,"\varphi"] \arrow[ur,swap,"f"]&
\end{tikzcd}.\]
Hence, $f$ is the final object.
\end{solution}

% Problem 5.6
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