Table of contents |
2 Examples 3 Applications 4 Equational theory |
The category C is called cartesian closed iff it satisfies the following three properties:
The term "cartesian closed" is used because one thinks of Y×X as akin to the cartesian product of two sets.
Examples of cartesian closed categories include:
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.
Certain cartesian closed categories, the topoi, have been proposed as a general setting for mathematics.Definition
The right adjoint of −×Y, applied to the object Z, is written as HOM(Y,Z), Y=>Z or ZY; we will use the exponential notation in the sequel. The adjointness condition means that the set of morphisms in C from X×Y to Z is naturally identified with the set of morphisms from X to ZY, for any three objects X, Y and Z in C.Examples
The following categories are not cartesian closed:Applications