In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories 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.[1][2]
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:
![](http://upload.wikimedia.org/wikipedia/commons/thumb/3/31/ExponentialObject-01.svg/194px-ExponentialObject-01.svg.png)
This assignment of a unique to each
establishes an isomorphism (bijection) of hom-sets,
If exists for all objects
in
, then the functor
defined on objects by
and on arrows by
, is a right adjoint to the product functor
. For this reason, the morphisms
and
are sometimes called exponential adjoints of one another.[3]
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 from the product functor
to the object
. This universal morphism consists of an object
and a morphism
.
Examples
In the category of sets, an exponential object is the set of all functions
.[4] The map
is just the evaluation map, which sends the pair
to
. For any map
the map
is the curried form of
:
A Heyting algebra is just a bounded lattice that has all exponential objects. Heyting implication,
, is an alternative notation for
. The above adjunction results translate to implication (
) being right adjoint 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 that case, the space
is the set of all continuous functions from
to
together with the compact-open topology. The evaluation map is the same as in the category of sets; it is continuous with the above topology.[5] 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 spanned by the compactly generated Hausdorff spaces.
In functional programming languages, 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 languages, which evaluates quoted expressions.
See also
Notes
References
- Adámek, Jiří; Horst Herrlich; George Strecker (2006) [1990]. Abstract and Concrete Categories (The Joy of Cats). John Wiley & Sons.
- Awodey, Steve (2010). "Chapter 6: Exponentials". Category theory. Oxford New York: Oxford University Press. ISBN 978-0199237180.
- Mac Lane, Saunders (1998). "Chapter 4: Adjoints". Categories for the working mathematician. New York: Springer. ISBN 978-0387984032.
External links
- Interactive Web page which generates examples of exponential objects and other categorical constructions. Written by Jocelyn Paine.