| Version 5 |
Version 4 |
| A category $\mathcal{C}$ with finite products is said to be \emph{Cartesian closed} if each of the following functors has a right adjoint |
A category $\mathcal{C}$ with finite products is said to be \emph{Cartesian closed} if each of the following functors has a right adjoint |
| \begin{enumerate} |
\begin{enumerate} |
| \item $\textbf{0}:\mathcal{C}\to \textbf{1}$, where $\textbf{1}$ is the trivial category with one object $0$, and $\textbf{0}(A)=0$ |
\item $\textbf{0}:\mathcal{C}\to \textbf{1}$, where $\textbf{1}$ is the trivial category with one object $0$, and $\textbf{0}(A)=0$ |
| \item the diagonal functor $\delta: \mathcal{C}\to \mathcal{C}\times\mathcal{C}$, where $\delta(A)=(A,A)$, and |
\item the diagonal functor $\delta: \mathcal{C}\to \mathcal{C}\times\mathcal{C}$, where $\delta(A)=(A,A)$, and |
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\item for any object $B$, the functor $(-\times B):\mathcal{C}\to\mathcal{C}$, where $(-\times B)(A)=A\times B$, the product of $A$ and $B$.
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\item for any object $B$, the functor $(\cdot \times B):\mathcal{C}\to\mathcal{C}$, where $(\cdot \times B)(A)=A\times B$, the product of $A$ and $B$.
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| \end{enumerate} |
\end{enumerate} |
| Furthermore, we require that the corresponding right adjoints for these functors to be |
Furthermore, we require that the corresponding right adjoints for these functors to be |
| \begin{enumerate} |
\begin{enumerate} |
| \item any functor $\textbf{1}\to\mathcal{C}$, where $0$ is mapped to an object $T$ in $\mathcal{C}$. $T$ is necessarily a terminal object of $\mathcal{C}$. |
\item any functor $\textbf{1}\to\mathcal{C}$, where $0$ is mapped to an object $T$ in $\mathcal{C}$. $T$ is necessarily a terminal object of $\mathcal{C}$. |
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\item the product (\PMlinkname{bifunctor}{Bifunctor}) $(-\times -): \mathcal{C} \times \mathcal{C}\to \mathcal{C}$ given by $(-\times -)(A,B)\mapsto A\times B$, the product of $A$ and $B$.
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\item the product (bifunctor) $(\cdot\times\cdot):\mathcal{C}\times \mathcal{C}\to \mathcal{C}$ given by $(\cdot\times\cdot)(A,B)\mapsto A\times B$, the product of $A$ and $B$.
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\item for any object $B$, the exponential functor $(-^B):\mathcal{C}\to\mathcal{C}$ given by $(-^B)(A)=A^B$, the exponential object from $B$ to $A$.
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\item for any object $B$, the exponential functor $(\cdot^B):\mathcal{C}\to\mathcal{C}$ given by $(\cdot^B)(A)=A^B$, the exponential object from $B$ to $A$.
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| \end{enumerate} |
\end{enumerate} |
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| In other words, a Cartesian closed category $C$ is a category with finite products, has a terminal objects, and has exponentials. It can be shown that a Cartesian closed category is the same as a finitely complete category having exponentials. |
In other words, a Cartesian closed category $C$ is a category with finite products, has a terminal objects, and has exponentials. It can be shown that a Cartesian closed category is the same as a finitely complete category having exponentials. |
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| Examples of Cartesian closed categories are the category of sets \textbf{Set} ( terminal object: any singleton; product: any Cartesian product of a finite number of sets; exponential object: the set of functions from one set to another) the category of small categories \textbf{Cat} (terminal object: any trivial category; product object: any finite product of categores; exponential object: any functor category), and every elementary topos. |
Examples of Cartesian closed categories are the category of sets \textbf{Set} ( terminal object: any singleton; product: any Cartesian product of a finite number of sets; exponential object: the set of functions from one set to another) the category of small categories \textbf{Cat} (terminal object: any trivial category; product object: any finite product of categores; exponential object: any functor category), and every elementary topos. |
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| \begin{thebibliography}{8} |
\begin{thebibliography}{8} |
| \bibitem{sm} S. Mac Lane, {\em Categories for the Working Mathematician}, Springer, New York (1971). |
\bibitem{sm} S. Mac Lane, {\em Categories for the Working Mathematician}, Springer, New York (1971). |
| \end{thebibliography} |
\end{thebibliography} |
