Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$ Suppose $n$ is a non-negative 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:

$$\xymatrix@C=0.5cm{& B \ar@{.>}[d]^{h} \ar@/_4ex/[dddl]_{p_1} \ar@/^4ex/[dddr]^{p_n} & \\ & P \ar[ddl]_{\pi_1}="1" \ar[ddr]^{\pi_n}="2" & \\ & & \\ A & \cdots & A \ar@{}"1";"2"|-{\cdots}}$$

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 $A$