Cartesian closed category

In category theory, a category C is called cartesian closed if it satisfies the following three properties:

• it has a terminal object
• any two objects have a product
• for every object X in C, the functor −×X from C to C has a right adjoint

The right adjoint of −×X is usually denoted by HOM(X,−). The adjointness means that the set of morphisms in C from Y×X to Z is naturally identified with the set of morphism from Y to HOM(X,Z), for any three objects X, Y and Z in C.

The term "cartesian closed" is used because one thinks of Y×X as akin to the cartesian product of two sets.

Examples

Examples of cartesian closed categories include:

• The category Set of all sets, with functions as morphisms, is cartesian closed. The product Y×X is the cartesian product of Y and X, and HOM(X,Z) is the set of all functions from X to Z. The adjointness is expressed by the following fact: the function f : Y×X → Z is naturally identified with the function g : Y → HOM(X,Z) defined by g(y)(x) = f(y,x) 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 C is a small category, then the category of all functors from C to Set (with natural transformations as morphisms) is a cartesian closed category.

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 don't have right adjoints.

• The category of abelian groups is not cartesian closed, for the same reason.

• The category of topological spaces is not cartesian closed.

Applications

In cartesian closed categories, a "function of two variables" can always be represented as a "function of one variable". In other contexts, this is known as currying; it has lead to the realization that lambda calculus can be formulated in any cartesian closed category.

In algebraic topology, cartesian closed categories are particularly easy to work with, and it is regrettable that 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.