Editing Exponentiation (section)

==Powers of sets {{Anchor|Exponentiation over sets}}==
The [[Cartesian product]] of two [[set (mathematics)|sets]] {{mvar|S}} and {{mvar|T}} is the set of the [[ordered pair]]s <math>(x,y)</math> such that <math>x\in S</math> and <math>y\in T.</math> This operation is not properly [[commutative]] nor [[associative]], but has these properties [[up to]] [[canonical map|canonical]] [[isomorphism]]s, that allow identifying, for example, <math>(x,(y,z)),</math> <math>((x,y),z),</math> and <math>(x,y,z).</math>

This allows defining the {{mvar|n}}th power <math>S^n</math> of a set {{mvar|S}} as the set of all {{mvar|n}}-[[tuple]]s <math>(x_1, \ldots, x_n)</math> of elements of {{mvar|S}}.

When {{mvar|S}} is endowed with some structure, it is frequent that <math>S^n</math> is naturally endowed with a similar structure. In this case, the term "[[direct product]]" is generally used instead of "Cartesian product", and exponentiation denotes product structure. For example <math>\R^n</math> (where <math>\R</math> denotes the real numbers) denotes the Cartesian product of {{mvar|n}} copies of <math>\R,</math> as well as their direct product as [[vector space]], [[topological space]]s, [[ring (mathematics)|rings]], etc.

===Sets as exponents===
{{See also|Function (mathematics)#Set exponentiation}}
A {{mvar|n}}-tuple <math>(x_1, \ldots, x_n)</math> of elements of {{mvar|S}} can be considered as a [[function (mathematics)|function]] from <math>\{1,\ldots, n\}.</math> This generalizes to the following notation.

Given two sets {{mvar|S}} and {{mvar|T}}, the set of all functions from {{mvar|T}} to {{mvar|S}} is denoted <math>S^T</math>. This exponential notation is justified by the following canonical isomorphisms (for the first one, see [[Currying]]):
: <math>(S^T)^U\cong S^{T\times U},</math>
: <math>S^{T\sqcup U}\cong S^T\times S^U,</math>
where <math>\times</math> denotes the Cartesian product, and <math>\sqcup</math> the [[disjoint union]].

One can use sets as exponents for other operations on sets, typically for [[direct sum]]s of [[abelian group]]s, [[vector space]]s, or [[module (mathematics)|modules]]. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example, <math>\R^\N</math> denotes the vector space of the [[infinite sequence]]s of real numbers, and <math>\R^{(\N)}</math> the vector space of those sequences that have a finite number of nonzero elements. The latter has a [[basis (linear algebra)|basis]] consisting of the sequences with exactly one nonzero element that equals {{math|1}}, while the [[Hamel basis|Hamel base]]s of the former cannot be explicitly described (because their existence involves [[Zorn's lemma]]).

In this context, {{math|2}} can represents the set <math>\{0,1\}.</math> So, <math>2^S</math> denotes the [[power set]] of {{mvar|S}}, that is the set of the functions from {{mvar|S}} to <math>\{0,1\},</math> which can be identified with the set of the [[subset]]s of {{mvar|S}}, by mapping each function to the [[inverse image]] of {{math|1}}.

This fits in with the [[Cardinal exponentiation|exponentiation of cardinal numbers]], in the sense that {{math|1={{abs|''S''<sup>''T''</sup>}} = {{abs|''S''}}<sup>{{abs|''T''}}</sup>}}, where {{math|{{abs|''X''}}}} is the cardinality of {{math|''X''}}.

===In category theory===
{{Main|Cartesian closed category}}
In the [[category of sets]], the [[morphism]]s between sets {{mvar|X}} and {{mvar|Y}} are the functions from {{mvar|X}} to {{mvar|Y}}. It results that the set of the functions from {{mvar|X}} to {{mvar|Y}} that is denoted <math>Y^X</math> in the preceding section can also be denoted <math>\hom(X,Y).</math> The isomorphism <math>(S^T)^U\cong S^{T\times U}</math> can be rewritten
:<math>\hom(U,S^T)\cong \hom(T\times U,S).</math>
This means the functor "exponentiation to the power {{mvar|T{{space|thin}}}}" is a [[right adjoint]] to the functor "direct product with {{mvar|T{{space|thin}}}}".

This generalizes to the definition of [[exponential (category theory)|exponentiation in a category]] in which finite [[direct product]]s exist: in such a category, the functor <math>X\to X^T</math> is, if it exists, a right adjoint to the functor <math>Y\to T\times Y.</math> A category is called a ''Cartesian closed category'', if direct products exist, and the functor <math>Y\to X\times Y</math> has a right adjoint for every {{mvar|T}}.