Cartesian Closed Category - Examples

Examples

Examples of cartesian closed categories include:

• The category Set of all sets, with functions as morphisms, is cartesian closed. The product X×Y is the cartesian product of X and Y, and ZY is the set of all functions from Y to Z. The adjointness is expressed by the following fact: the function f : X×Y → Z is naturally identified with the curried function g : X → ZY defined by g(x)(y) = f(x,y) for all x in X and y in Y.
• The category of finite sets, with functions as morphisms, is cartesian closed for the same reason.
• If G is a group, then the category of all G-sets is cartesian closed. If Y and Z are two G-sets, then ZY is the set of all functions from Y to Z with G action defined by (g.F)(y) = g.(F(g-1.y)) for all g in G, F:Y → Z and y in Y.
• The category of finite G-sets is also cartesian closed.
• The category Cat of all small categories (with functors as morphisms) is cartesian closed; the exponential CD is given by the functor category consisting of all functors from D to C, with natural transformations as morphisms.
• If C is a small category, then the functor category SetC consisting of all covariant functors from C into the category of sets, with natural transformations as morphisms, is cartesian closed. If F and G are two functors from C to Set, then the exponential FG is the functor whose value on the object X of C is given by the set of all natural transformations from (X,−) × G to F.
  • The earlier example of G-sets can be seen as a special case of functor categories: every group can be considered as a one-object category, and G-sets are nothing but functors from this category to Set
  • The category of all directed graphs is cartesian closed; this is a functor category as explained under functor category.
• In algebraic topology, cartesian closed categories are particularly easy to work with. Neither the category of topological spaces with continuous maps nor the category of smooth manifolds with smooth maps is cartesian closed. Substitute categories have therefore been considered: the category of compactly generated Hausdorff spaces is cartesian closed, as is the category of Frölicher spaces.
• In order theory, complete partial orders (cpos) have a natural topology, the Scott topology, whose continuous maps do form a cartesian closed category (that is, the objects are the cpos, and the morphisms are the Scott continuous maps). Both currying and apply are continuous functions in the Scott topology, and currying, together with apply, provide the adjoint.
• A Heyting algebra is a Cartesian closed (bounded) lattice. An important example arises from topological spaces. If X is a topological space, then the open sets in X form the objects of a category O(X) for which there is a unique morphism from U to V if U is a subset of V and no morphism otherwise. This poset is a cartesian closed category: the "product" of U and V is the intersection of U and V and the exponential UV is the interior of U∪(X\V).

The following categories are not cartesian closed:

• The category of all vector spaces over some fixed field is not cartesian closed; neither is the category of all finite-dimensional vector spaces. While they have products (called direct sums), the product functors do not have right adjoints. (They are, however, symmetric monoidal closed categories: the set of linear transformations between two vector spaces forms another vector space, so they are closed, and if one replaces the product by the tensor product, a similar isomorphism exists between the Hom spaces.)
• The category of abelian groups is not cartesian closed, for the same reason.