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Concept
Cartesian closed category
Summary
In , a is Cartesian closed if, roughly speaking, any morphism defined on a of two 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 , whose internal language, linear type systems, are suitable for both quantum and classical computation.
Named after (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 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 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 , 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 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.
For each object Y, the counit of the exponential adjunction is a natural transformation
called the (internal) evaluation map.
Official source
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