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'Cartesian closed category'
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| Title of object: |
Cartesian closed category |
| Canonical Name: |
CartesianClosedCategory |
| Type: |
Definition |
| Created on: |
2007-01-20 00:48:49 |
| Modified on: |
2007-01-20 01:09:04 |
| Classification: |
msc:18D15 |
| Defines: |
Cartesian closed |
Revision comment (for changes between this and next version):
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Content:
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}
\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 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 $B$.
\end{enumerate}
Furthermore, we require that the corresponding right adjoints for these functors to be
\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 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 $B$.
\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 $A$.
\end{enumerate}
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.
Examples of Cartesian closed categories are the categoy of sets \textbf{Set} and the category of small categories \textbf{Cat}.
\begin{thebibliography}{8}
\bibitem{sm} S. Mac Lane, {\em Categories for the Working Mathematician}, Springer, New York (1971).
\end{thebibliography} |
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