Exponential object

Encyclopedia

In mathematics

, specifically in category theory

, an

in set theory

. Categories with all finite products

and exponential objects are called cartesian closed categories

. An exponential object may also be called a

and let

–×

Explicitly, the definition is as follows. An object

is an exponential object if for any object

such that the following diagram commutes

:

If the exponential object

(Note: In functional programming languages, the morphism

function in some programming language

s, which evaluates quoted expressions.)

, the exponential object is the set of all functions from to . The map is just the evaluation map which sends the pair (

form of :

In the category of topological spaces

, the exponential object

. The evaluation map is the same as in the category of sets. If

However, the category of locally compact topological spaces is not cartesian closed either, since

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, specifically in category theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, an

**exponential object**is the categorical equivalent of a function spaceFunction space

In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

in set theory

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

. Categories with all finite products

Product (category theory)

In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...

and exponential objects are called cartesian closed categories

Cartesian closed category

In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in...

. An exponential object may also be called a

**power object**or**map object**(but note that the term "power object" means something different in topos theory, analogous to "power set").## Definition

Let*C*be a category with binary productsProduct (category theory)

In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...

and let

*Y*and*Z*be objects of*C*. The exponential object*Z*^{Y}can be defined as a universal morphism from the functorFunctor

In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....

–×

*Y*to*Z*. (The functor –×*Y*from*C*to*C*maps objects*X*to*X*×*Y*and morphisms φ to φ×id_{Y}).Explicitly, the definition is as follows. An object

*Z*^{Y}, together with a morphismis an exponential object if for any object

*X*and morphism*g*: (*X*×*Y*) →*Z*there is a unique morphismsuch that the following diagram commutes

Commutative diagram

In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...

:

If the exponential object

*Z*^{Y}exists for all objects*Z*in*C*, then the functor which sends*Z*to*Z*^{Y}is a right adjoint to the functor –×*Y*. In this case we have a natural bijection between the hom-sets(Note: In functional programming languages, the morphism

*eval*is often called*apply*

, and the syntax is often writtenApply

In mathematics and computer science, Apply is a function that applies functions to arguments. It is central to programming languages derived from lambda calculus, such as LISP and Scheme, and also in functional languages...

*curry*

(Currying

In mathematics and computer science, currying is the technique of transforming a function that takes multiple arguments in such a way that it can be called as a chain of functions each with a single argument...

*g*). The morphism*eval*here must not to be confused with the evalEval

In some programming languages, eval is a function which evaluates a string as though it were an expression and returns a result; in others, it executes multiple lines of code as though they had been included instead of the line including the eval...

function in some programming language

Programming language

A programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely....

s, which evaluates quoted expressions.)

## Examples

In the category of setsCategory of sets

In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...

, the exponential object is the set of all functions from to . The map is just the evaluation map which sends the pair (

*f*,*y*) to*f*(*y*). For any map the map is the curriedCurrying

In mathematics and computer science, currying is the technique of transforming a function that takes multiple arguments in such a way that it can be called as a chain of functions each with a single argument...

form of :

In the category of topological spaces

Category of topological spaces

In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous...

, the exponential object

*Z*^{Y}exists provided that*Y*is a locally compact Hausdorff space. In that case, the space*Z*^{Y}is the set of all continuous functions from*Y*to*Z*together with the compact-open topologyCompact-open topology

In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly-used topologies on function spaces, and is applied in homotopy theory and functional analysis...

. The evaluation map is the same as in the category of sets. If

*Y*is not locally compact Hausdorff, the exponential object may not exist (the space*Z*^{Y}still exists, but it may fail to be an exponential object since the evaluation function need not be continuous). For this reason the category of topological spaces fails to be cartesian closed.However, the category of locally compact topological spaces is not cartesian closed either, since

*Z*^{Y}need not be locally compact for locally compact spaces*Z*and*Y*.## External links

- Interactive Web page which generates examples of exponential objects and other categorical constructions. Written by Jocelyn Paine.