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# Cartesian closed category

A category $\mathcal{C}$ with finite products is said to be *Cartesian closed* if each of the following functors has a right adjoint

1. $\textbf{0}:\mathcal{C}\to\textbf{1}$, where 1 is the trivial category with one object $0$, and $\textbf{0}(A)=0$

2. the diagonal functor $\delta:\mathcal{C}\to\mathcal{C}\times\mathcal{C}$, where $\delta(A)=(A,A)$, and

3. 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 $B$.

Furthermore, we require that the corresponding right adjoints for these functors to be

1. 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 $\mathcal{C}$.

2. the product (bifunctor) $(-\times-):\mathcal{C}\times\mathcal{C}\to\mathcal{C}$ given by $(-\times-)(A,B)\mapsto A\times B$, the product of $A$ and $B$.

3. 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 $A$.

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 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 Cat (terminal object: any trivial category; product object: any finite product of categores; exponential object: any functor category), and every elementary topos.

# References

- 1 S. Mac Lane, Categories for the Working Mathematician, Springer, New York (1971).

## Mathematics Subject Classification

18D15*no label found*

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