Cartesian closed category

A (mathematical ) category is called cartesian closed if - roughly expressed - the morphism again correspond to objects of the category.

Definition

If in a category with finite products, for an object of Produktfunktor

Has a rechtsadjungierten functor, it means exponentially. The adjoint functor is then often

Written.

Are all objects exponentially, ie the cartesian closed category.

Examples

• The category Set of sets ( and pictures ) is completed in Cartesian coordinates. The required rechtsadjungierte functor is given by the Adjungiertheit supplying natural equivalence in that is mapped to with.
• For any small category, the functor category is Cartesian closed. Products are formed object instance. For the exponentiation.
• The category Ab of abelian groups is not completed in Cartesian coordinates. While wearing the morphism by pointwise addition in turn the structure of an abelian group, but all abelian groups are not exponential.
• The category Top of topological spaces and continuous maps is not cartesian closed, but the category of compactly generated topological spaces separated ( and continuous maps ) is ( a topological space is compact generated if the corresponding topology is final with respect to the family of all inclusions compact subsets, in particular all pseudo- metric and all locally compact spaces are compact generated ). The exponential objects in Top are characterized in generalization of local compactness as so-called quasi- locally compact spaces.
• An association can be considered as a category. The Association shall determine in this case the morphisms, intersection and union are products and coproducts. Is the so resulting category cartesian closed, the association is a Heyting algebra.

Applications

In Cartesian closed categories often following construction will be used. For an object X we consider the set of all morphisms from X into a special room Frequently Q. Q is very simple choice: Set in considering Q = { 0,1}, in BanSp1 ( Banach spaces with continuous linear maps ) selects often than Q, the real numbers and in CBanAlg ( commutative Banach algebras with unit complex and standard reducing Algebrenhomomorphismen ) take the complex numbers. The resulting function space X * is often called the dual space. The functor which assigns to each object X * X and any morphism f: X → Y the morphism f *: Y * → X * by virtue of f * ( l): = lof assigns is called dual functor, adjoint functor or exponential functor, each of said name has a different meaning.

This construction makes it possible questions to an object X in questions to the object X * transform, which are then sometimes easier to answer. Particularly user- reflexive objects for which (X *) * = X is true.

• Category theory