nLab cartesian power

Given a set $S$ and a cardinal number $\kappa$, the $\kappa$th cartesian power $S^\kappa$ of $S$ is the $\kappa$-fold cartesian product of $S$ with itself.

In particular, the $S^2$ is the cartesian square of $S$, the set of ordered pairs of elements of $S$; and $S^{\aleph_0}$ is the set of infinite sequences of elements of $S$.

The concept generalises from Set to any category $C$ with all products; $S$ becomes an object of $C$, but $\kappa$ remains a cardinal number (still essentially an object of $Set$).

If we think of $\kappa$ as a full-fledged set in its own right (rather than just its cardinal number), then we are talking about a function set, and the generalisation is to cartesian closed categories.

See also

• n-ary operation
• multicategory
• multimorphism