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Cartesian closed category
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both quantum and classical computation.
Named after René Descartes (1596–1650), French philosopher, mathematician, and scientist, whose formulation of analytic geometry gave rise to the concept of Cartesian product, which was later generalized to the notion of categorical product.
The category C is called Cartesian closed iff it satisfies the following three properties:
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 X, Y, and Z.
Take care to note that a Cartesian closed category need not have finite limits; only finite products are guaranteed.
If a category has the property that all its slice categories are Cartesian closed, then it is called locally cartesian closed. Note that if C is locally Cartesian closed, it need not actually be Cartesian closed; that happens if and only if C has a terminal object.
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Cartesian closed category
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both quantum and classical computation.
Named after René Descartes (1596–1650), French philosopher, mathematician, and scientist, whose formulation of analytic geometry gave rise to the concept of Cartesian product, which was later generalized to the notion of categorical product.
The category C is called Cartesian closed iff it satisfies the following three properties:
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 X, Y, and Z.
Take care to note that a Cartesian closed category need not have finite limits; only finite products are guaranteed.
If a category has the property that all its slice categories are Cartesian closed, then it is called locally cartesian closed. Note that if C is locally Cartesian closed, it need not actually be Cartesian closed; that happens if and only if C has a terminal object.