You are here
Homepower of an object
Primary tabs
power of an object
Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. Suppose $n$ is a nonnegative integer. The $n$th power of $A$ is defined as the direct product of $A$ with itself $n$ times. In other words, the $n$th power of $A$ is an object $P$ in $\mathcal{C}$, together with $n$ parallel morphisms $\pi_{1},\ldots,\pi_{n}\in\hom(P,A)$, such that if there are $n$ parallel morphisms $p_{1},\ldots,p_{n}\in\hom(B,A)$, then there is a unique morphism $h:B\to P$ such that $\pi_{i}\circ h=p_{i}$, where $i=1,\ldots,n$. The commutative diagram below illustrates the situation:
$\xymatrix@C=0.5cm{&B\ar@{.>}[d]^{{h}}\ar@/_{4}ex/[dddl]_{{p_{1}}}\ar@/^{4}ex/[% dddr]^{{p_{n}}}&\\ &P\ar[ddl]_{{\pi_{1}}}="1"\ar[ddr]^{{\pi_{n}}}="2"&\\ &&\\ A&\cdots&A\ar@{}"1";"2"{\cdots}}$ 
The $n$th power of $A$ is denoted by $A^{n}$.
Below are some properties of the power of an object in a category:

Each of the projection morphisms $\pi_{i}$ is a split epimorphism.

$A^{1}\cong A$.

$A^{0}$ is a terminal object in $\mathcal{C}$.

$A^{{m+n}}\cong A^{n}\times A^{m}$, if the product exists.
For example, in the category of sets, the $n$th power of a set $A$ is the set of $n$tuples where each entry is an element of $A$.
Remark. The copower of an object is defined dually. All of the properties above can be dualized. For example, the $0$th copower of an object is an initial object. The $n$th copower of an object $A$ in Set is the disjoint union of $n$copies of $A$.
Mathematics Subject Classification
18A30 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections