Exponential object
Encyclopedia
In mathematics
, specifically in category theory
, an exponential object is the categorical equivalent of a function space
in set theory
. Categories with all finite products
and exponential objects are called cartesian closed categories
. 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").
and let Y and Z be objects of C. The exponential object ZY can be defined as a universal morphism from the functor
–×Y to Z. (The functor –×Y from C to C maps objects X to X×Y and morphisms φ to φ×idY).
Explicitly, the definition is as follows. An object ZY, together with a morphism
is an exponential object if for any object X and morphism g : (X×Y) → Z there is a unique morphism
such that the following diagram commutes
:
If the exponential object ZY exists for all objects Z in C, then the functor which sends Z to ZY 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 written curry
(g). The morphism eval here must not to be confused with the eval
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 (f, y) to f(y). For any map 
the map 
is the curried
form of
:
In the category of topological spaces
, the exponential object ZY exists provided that Y is a locally compact Hausdorff space. In that case, the space ZY is the set of all continuous functions from Y to Z together with the compact-open topology
. 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 ZY 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 ZY need not be locally compact for locally compact spaces Z and Y.
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 space
Function 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 ZY can be defined as a universal morphism from the functor
Functor
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×idY).
Explicitly, the definition is as follows. An object ZY, together with a morphism
is an exponential object if for any object X and morphism g : (X×Y) → Z there is a unique morphism
such 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...
:
(Note: In functional programming languages, the morphism eval is often called apply
Apply
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...
, and the syntax
is often written curryCurrying
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 eval
Eval
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 ZY exists provided that Y is a locally compact Hausdorff space. In that case, the space ZY is the set of all continuous functions from Y to Z together with the compact-open topology
Compact-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 ZY 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 ZY 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.