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Cartesian closed category (Definition)

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 $ \textbf{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.

Bibliography

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



"Cartesian closed category" is owned by CWoo. [ full author list (2) ]
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See Also: complete category, index of categories

Also defines:  Cartesian closed
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Cross-references: elementary topos, functor category, Cat, category of small categories, functions, number, Cartesian product, singleton, category of sets, exponentials, finitely complete category, has exponentials, exponential object, exponential functor, terminal object, diagonal functor, object, trivial category, right adjoint, functors, products, finite, category
There are 7 references to this entry.

This is version 6 of Cartesian closed category, born on 2007-01-20, modified 2008-10-20.
Object id is 8802, canonical name is CartesianClosedCategory.
Accessed 1323 times total.

Classification:
AMS MSC: 18D15 (Category theory; homological algebra :: Categories with structure :: Closed categories )
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