Definition
The category C is called Cartesian closed if and only if it satisfies the following three properties:
- It has a terminal object.
- Any two objects X and Y of C have a product X×Y in C.
- Any two objects Y and Z of C have an exponential ZY in C.
The first two conditions can be combined to the single requirement that any finite (possibly empty) family of objects of C admit a product in C, because of the natural associativity of the categorical product and because the empty product in a category is the terminal object of that category.
The third condition is equivalent to the requirement that the functor –×Y (i.e. the functor from C to C that maps objects X to X×Y and morphisms φ to φ×idY) has a right adjoint, usually denoted –Y, for all objects Y in C. For locally small categories, this can be expressed by the existence of a bijection between the hom-sets
which is natural in both X and Z.
If a category is such that all its slice categories are cartesian closed, then it is called locally cartesian closed.
Read more about this topic: Cartesian Closed Category
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