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.john c. baez and mike stay, "physics, topology, logic and computation: a rosetta stone", (2009) arxiv 0903.0340 in new structures for physics, ed. bob coecke, lecture notes in physics vol. 813, springer, berlin, 2011, pp. 95-174. == definition == the category c is called cartesian closeds. mac lane, "categories for the working mathematician" 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 xÃy in c. * any two objects y and z of c have an exponential z'y 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 âÃy (i.e. the functor from c to c that maps objects x to xÃy and morphisms Ï to ÏÃidy) has a right adjoint, usually denoted ây, for all objects y in c. for locally small categories, this can be expressed by the existence of a bijection between the hom-sets :\mathrm{hom}(x\times y,z) \cong \mathrm{hom}(x,z^y) 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. == examples == examples of cartesian closed categories include: * the category set of all sets, with functions as morphisms, is cartesian closed. the product xÃy is the cartesian product of x and y, and z'y is the set of all functions from y to z. the adjointness is expressed by the following fact: the function f : xÃy â z is naturally identified with the curried function g : x â z'y defined by g(x)(y) = f(x,y) for all x in x and '
Closed_categories Lambda_calculus