In
mathematics, specifically in
category theory, an exponential object or map object is the
categorical generalization of a
function space in
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
.
Categories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
* Categories (Peirce)
* ...
with all
finite products and exponential objects are called
cartesian closed categories. Categories (such as
subcategories of
Top) without adjoined products may still have an exponential law.
Definition
Let
be a category, let
and
be
objects of
, and let
have all
binary products with
. An object
together with a
morphism is an ''exponential object'' if for any object
and morphism
there is a unique morphism
(called the ''transpose'' of
) such that the following diagram
commutes:
![ExponentialObject-01](https://upload.wikimedia.org/wikipedia/commons/3/31/ExponentialObject-01.svg)
This assignment of a unique
to each
establishes an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
(
bijection) of
hom-set
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s,
If
exists for all objects
in
, then the
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
defined on objects by
and on arrows by
, is a
right adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
to the product functor
. For this reason, the morphisms
and
are sometimes called ''exponential adjoints'' of one another.
Equational definition
Alternatively, the exponential object may be defined through equations:
* Existence of
is guaranteed by existence of the operation
.
* Commutativity of the diagrams above is guaranteed by the equality
.
* Uniqueness of
is guaranteed by the equality
.
Universal property
The exponential
is given by a
universal morphism
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently ...
from the product functor
to the object
. This universal morphism consists of an object
and a morphism
.
Examples
In the
category 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 the total functions from ''A'' to ''B'', and the composition o ...
, an exponential object
is the set of all
functions .
The map
is just the
evaluation map
In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' t ...
, which sends the pair
to
. For any map
the map
is the
curried form of
:
:
A
Heyting algebra In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of '' ...
is just a bounded
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
that has all exponential objects. Heyting implication,
, is an alternative notation for
. The above adjunction results translate to implication (
) being
right adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
to
meet (
). This adjunction can be written as
, or more fully as:
In the
category of topological spaces, the exponential object
exists provided that
is a
locally compact Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
. In that case, the space
is the set of all
continuous functions
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
from
to
together with the
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 ...
. The evaluation map is the same as in the category of sets; it is continuous with the above topology.
[ Joseph J. Rotman, ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag ''(See Chapter 11 for proof.)''] If
is not locally compact Hausdorff, the exponential object may not exist (the space
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
need not be locally compact for locally compact spaces
and
. A cartesian closed category of spaces is, for example, given by the
full subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitive ...
spanned by the
compactly generated In mathematics, compactly generated can refer to:
* Compactly generated group, a topological group which is algebraically generated by one of its compact subsets
*Compactly generated space
In topology, a compactly generated space is a topological s ...
Hausdorff spaces.
In
functional programming language
In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that ...
s, the morphism
is often
called , and the syntax
is often
written . The morphism
here must not to be confused with the
eval
function in some
programming language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language.
The description of a programming ...
s, which evaluates quoted expressions.
See also
*
Closed monoidal category
In mathematics, especially in category theory, a closed monoidal category (or a ''monoidal closed category'') is a category that is both a monoidal category and a closed category in such a way that the structures are compatible.
A classic exa ...
Notes
References
*
*
*
External links
Interactive Web page which generates examples of exponential objects and other categorical constructions. Written b
Jocelyn Paine
{{Category theory
Objects (category theory)