Cartesian closed categoryCartesianClosedCategory2007-01-20 00:48:492008-10-20 19:07:56DefinitionCartesian closed\usepackage{amssymb,amscd}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
\usepackage{xypic}
\usepackage{pst-plot}
\usepackage{psfrag}
% define commands here
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 $(-\times B):\mathcal{C}\to\mathcal{C}$, where $(-\times B)(A)=A\times B$, 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 (\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 $B$.
\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 $A$.
\end{enumerate}
In other words, a Cartesian closed category $\mathcal{C}$ 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 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.
\begin{thebibliography}{8}
\bibitem{sm} S. Mac Lane, {\em Categories for the Working Mathematician}, Springer, New York (1971).
\end{thebibliography}