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representability of $G$. - - - - \begin{claim} If $G$ is representable, then $F$ is representable. \end{claim} -We start noting that $H$ being an equivalence implies readily that $K$ is an equivalence as $H$ and $K$ are symmetric in the condition of being a category equivalence. The rest of the proof here follows the exact same line of thought as above with the only difference that we compose the hom-functor with $H$ instead of $K$. Hence, we forego providing every detail.\\ -Now, suppose $G$ is representable, we can find $D \in \mathcal{D}$ s.t. $$ +We start noting that $H$ being an equivalence implies readily that $K$ is an equivalence as $H$ and $K$ are symmetric in the condition of being a category equivalence. The rest of the proof here follows the exact same line of thought as above with the only difference that we compose the hom-functor with $H$ instead of $K$. \\ +Now, supposing, $G$ is representable, we can find $D \in \mathcal{D}$ s.t. $$ \beta: G \cong \text{Hom}_{\mathcal{D}}(D, -). $$ Then, $$ F \cong G \circ H \cong \text{Hom}_{\mathcal{D}}(D, -) \circ H $$ -and we can find – details are skipped – a $C \in \mathcal{C}$ s.t. +and we can find a $C \in \mathcal{C}$ such that $$ F \cong \text{Hom}_{\mathcal{C}}(C, -) $$ @@ -1745,7 +1741,7 @@ where $C = K(D) \in \mathcal{C}$. This completes the proof. \\ We can summarize: -\begin{cor} Representability of functors is \textbf{not} preserved under category equivalence. +\begin{cor} Representability of functors is preserved under category equivalence. \end{cor} diff --git a/Exercises/Exercises.toc b/Exercises/Exercises.toc index 09c3fa1..64a9902 100644 --- a/Exercises/Exercises.toc +++ b/Exercises/Exercises.toc @@ -73,7 +73,7 @@ \contentsline {subsection}{\tocsubsection {}{12.2}{}}{35}{subsection.12.2}% \contentsline {subsection}{\tocsubsection {}{12.3}{}}{35}{subsection.12.3}% \contentsline {subsection}{\tocsubsection {}{12.4}{}}{36}{subsection.12.4}% -\contentsline {subsection}{\tocsubsection {}{12.5}{}}{37}{subsection.12.5}% +\contentsline {subsection}{\tocsubsection {}{12.5}{}}{36}{subsection.12.5}% \contentsline {subsection}{\tocsubsection {}{12.6}{}}{37}{subsection.12.6}% \contentsline {subsection}{\tocsubsection {}{12.7}{}}{37}{subsection.12.7}% \contentsline {subsection}{\tocsubsection {}{12.8}{}}{37}{subsection.12.8}% -- GitLab