| Version current |
Version 10 |
| \begin{definition} |
\begin{definition} |
| Let us consider two arbitrary categories $\mathcal{C}$ and $\mathcal{D}$. |
Let us consider two arbitrary categories $\mathcal{C}$ and $\mathcal{D}$. |
| A \emph{natural equivalence of categories}* is said to exist between two categories $\mathcal{C}$ and |
A \emph{natural equivalence of categories}* is said to exist between two categories $\mathcal{C}$ and |
| $\mathcal{D}$ if and only if there is a covariant functor $ E: \mathcal{C} \to \mathcal{D}$ which |
$\mathcal{D}$ if and only if there is a covariant functor $ E: \mathcal{C} \to \mathcal{D}$ which |
| is full and faithful, and that also has a full and faithful adjoint (that is either |
is full and faithful, and that also has a full and faithful adjoint (that is either |
| a left- or right- adjoint). |
a left- or right- adjoint). |
| \end{definition} |
\end{definition} |
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| * See ref. $[288]$ in the \PMlinkname{bibliography for category theory and algebraic topology}{CategoricalOntologyABibliographyOfCategoryTheory}. |
* See ref. $[288]$ in the \PMlinkname{bibliography for category theory and algebraic topology}{CategoricalOntologyABibliographyOfCategoryTheory}. |
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| \subsubsection{Examples:} |
\subsubsection{Examples:} |
| \begin{enumerate} |
\begin{enumerate} |
| \item \PMlinkname{Category of cross modules of groups, and the category of categorical groups}{NaturalEquivalenceOfC_GAndC_MCategories} |
\item \PMlinkname{Category of cross modules of groups, and the category of categorical groups}{NaturalEquivalenceOfC_GAndC_MCategories} |
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| \item Category $\bf{modB}$ of finite-dimensional right-B modules, and the category $\mathcal{C} / \Sigma T$ of ideals of morphisms of a category $\mathcal{C}$ which factor through a direct some of ideal copies $\Sigma T$. |
\item Category $\bf{modB}$ of finite-dimensional right-B modules, and the category $\mathcal{C} / \Sigma T$ of ideals of morphisms of a category $\mathcal{C}$ which factor through a direct some of ideal copies $\Sigma T$. |
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| \item The category of crossed modules of $R$--algebroids is equivalent to the category of double $R$--algebroids with thin structure (Brown and Mosa, 1986, 2008.) |
\item The category of crossed modules of $R$--algebroids is equivalent to the category of double $R$--algebroids with thin structure (Brown and Mosa, 1986, 2008.) |
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| \item The categories of crossed modules of algebroids and of double algebroids with a connection pair are equivalent. |
\item The categories of crossed modules of algebroids and of double algebroids with a connection pair are equivalent. |
| \end{enumerate} |
\end{enumerate} |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{EL1945} |
\bibitem{EL1945} |
| Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, \emph{Transactions of the American Mathematical Society} \textbf{58}: 231-294. |
Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, \emph{Transactions of the American Mathematical Society} \textbf{58}: 231-294. |
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| \end{thebibliography} |
\end{thebibliography} |
