Exponential object

﻿
Exponential object

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.

Definition

Let "C" be a category with binary products and let "Y" and "Z" be objects of "C". The exponential object "Z""Y" can be defined as a universal morphism from the functor &ndash;&times;"Y" to "Z". (The functor &ndash;&times;"Y" from "C" to "C" maps objects "X" to "X"&times;"Y" and morphisms &phi; to &phi;&times;id"Y").

Explicitly, the definition is as follows. An object "Z""Y", together with a morphism

:$mathrm\left\{eval\right\}colon \left(Z^Y imes Y\right) ightarrow Z,$

is an exponential object if for any object "X" and morphism "g" : ("X"&times;"Y") &rarr; "Z" there is a unique morphism

:$lambda gcolon X o Z^Y,$

such that the following diagram commutes:

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 &ndash;&times;"Y". In this case we have a natural bijection between the hom-sets:$mathrm\left\{Hom\right\}\left(X imes Y,Z\right) cong mathrm\left\{Hom\right\}\left(X,Z^Y\right).$

(Note: In functional programming languages, the morphism "eval" is often called "apply", and the syntax $lambda g$ is often written "curry"("g"). The morphism "eval" here must not to be confused with the eval function in some programming languages, which evaluates quoted expressions.)

Examples

In the category of sets, the exponential object $Z^Y$ is the set of all functions from $Y$ to $Z$. The map $mathrm\left\{eval\right\}colon \left(Z^Y imes Y\right) o Z$ is just the evaluation map which sends the pair ("f", "y") to "f"("y"). For any map $gcolon \left(X imes Y\right) ightarrow Z$ the map $lambda gcolon X o Z^Y$ is the curried form of $g$::$lambda g\left(x\right)\left(y\right) = g\left(x,y\right).,$

In the category of topological spaces, 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 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 "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.

References

*cite book|last=Adámek|first=Jiří|coauthors=Horst Herrlich, George Strecker|title=Abstract and Concrete Categories (The Joy of Cats)|url=http://katmat.math.uni-bremen.de/acc/|origyear=1990|year=2006

Wikimedia Foundation. 2010.

Look at other dictionaries:

• exponential object — noun Given objects Y and Z, the exponential object is uniquely defined by the following universal property: for any object X with arrow , there can always be constructed an arrow which induces an arrow , nbsp; which is unique in satisfying where …   Wiktionary

• Exponential — may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay *Exponential growth *Exponential map, in differential… …   Wikipedia

• Exponential decay — A quantity undergoing exponential decay. Larger decay constants make the quantity vanish much more rapidly. This plot shows decay for decay constants of 25, 5, 1, 1/5, and 1/25 for x from 0 to 5. A quantity is said to be subject to exponential… …   Wikipedia

• Exponential function — The natural exponential function y = ex In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex is its own derivative …   Wikipedia

• Exponential calculus — Calculus Cal cu*lus, n.; pl. {Calculi}. [L, calculus. See {Calculate}, and {Calcule}.] 1. (Med.) Any solid concretion, formed in any part of the body, but most frequent in the organs that act as reservoirs, and in the passages connected with… …   The Collaborative International Dictionary of English

• List of exponential topics — This is a list of exponential topics, by Wikipedia page. See also list of logarithm topics. *Accelerating change *Artin Hasse exponential *Bacterial growth *Baker Campbell Hausdorff formula *Cell growth *Barometric formula *Basic infection number …   Wikipedia

• Mayfair Exponential Game System — The Mayfair Exponential Game System or MEGS is a rules system developed for role playing games. It is noteworthy for its use of an exponential system for measuring nearly everything in the game. This system makes it possible to have both… …   Wikipedia

• Ordered exponential — The ordered exponential (also called the path ordered exponential) is a mathematical object, defined in non commutative algebras, which is equivalent to the exponential function of the integral in the commutative algebras. Therefore it is a… …   Wikipedia

• Exponentiation — Exponent redirects here. For other uses, see Exponent (disambiguation). Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent (or power) n. When n is a positive integer, exponentiation… …   Wikipedia

• List of mathematics articles (E) — NOTOC E E₇ E (mathematical constant) E function E₈ lattice E₈ manifold E∞ operad E7½ E8 investigation tool Earley parser Early stopping Earnshaw s theorem Earth mover s distance East Journal on Approximations Eastern Arabic numerals Easton s… …   Wikipedia