Cartesian closed category

A ( mathematical ) category is called Cartesian closed if - roughly speaking - the sets of morphisms again correspond to objects of the category.

definition

If in a category with finite products for an object the product functor ${\displaystyle X}$

${\displaystyle Y\mapsto Y\times X}$

has a right adjoint functor, it is called exponential . The adjoint functor then becomes frequent ${\displaystyle X}$

${\displaystyle Z\mapsto Z^{X}}$ or ${\displaystyle Z\mapsto \mathbf {Hom} (X,Z)}$

written.

If all objects are exponential, the category is called Cartesian closed .

Examples

• The category set of sets (and figures) is closed in Cartesian. The required right adjoint functor is given by, the natural equivalence providing the adjointness is given by mapping to with . ${\displaystyle Z\mapsto \operatorname {Mor} (X,Z)}$ ${\displaystyle f\colon Y\times X\to Z}$ ${\displaystyle g\colon Y\to \operatorname {Mor} (X,Z)}$ ${\displaystyle g(y)\colon x\mapsto f(x,y)}$
• For every small category that is functor completed Cartesian. Products are formed by object: . The following applies to the exponentiation . ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mathbf {Set} ^{{\mathcal {C}}^{\text{op}}}}$ ${\displaystyle (F\times G)(A)\cong F(A)\times G(A)}$ ${\displaystyle G^{F}(A)\cong \mathbf {Set} ^{{\mathcal {C}}^{\text{op}}}({\mathcal {C}}(-,A)\times F,G)}$
• The category from the Abelian groups is not complete Cartesian. Although the sets of morphisms have the structure of an Abelian group due to pointwise addition, not all Abelian groups are exponential.
• The category Top of topological spaces and continuous mappings is not Cartesian closed, but the category of compactly generated separated topological spaces (and continuous mappings) is (a topological space is compactly generated if the corresponding topology is final with respect to the family of inclusions of all compact subsets, in particular all pseudometric and all locally compact spaces are generated compactly). The exponential objects in Top are characterized as a generalization of local compactness as so-called quasi- local compact spaces.
• An association can be viewed as a category. The association rules determine the morphisms, average and association are products and co-products. If the resulting category is Cartesian closed, the union is a Heyting algebra .

Applications

The following construction is often used in Cartesian closed categories. For an object one considers the set of all morphisms from in a particular space . The choice is often very simple: in Set one considers , in BanSp ₁ ( Banach spaces with continuous linear mappings) one often chooses the real numbers and in CBanAlg (commutative complex Banach algebras with unit and norm-reducing algebra homomorphisms) one takes the complex numbers. The function space created in this way is often called a dual space . The functor that any object that assigns and each morphism the morphism virtue assigns dual functor is adjoint functor or exponential functor called, each of these names has a different meaning. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle Q}$ ${\displaystyle Q}$ ${\displaystyle Q=\left\{0,1\right\}}$ ${\displaystyle Q}$ ${\displaystyle X^{*}}$ ${\displaystyle X}$ ${\displaystyle X^{*}}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle f^{*}\colon Y^{*}\to X^{*}}$ ${\displaystyle f^{*}(l):=l\circ f}$

This construction makes it possible to transform questions about an object into questions about the object , which are then sometimes easier to answer. The reflective objects for which applies are particularly comfortable . ${\displaystyle X}$ ${\displaystyle X^{*}}$ ${\displaystyle (X^{*})^{*}=X}$