nLab cartesian power

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Given a set SS and a cardinal number κ\kappa, the κ\kappath cartesian power S κS^\kappa of SS is the κ\kappa-fold cartesian product of SS with itself.

In particular, the S 2S^2 is the cartesian square of SS, the set of ordered pairs of elements of SS; and S 0S^{\aleph_0} is the set of infinite sequences of elements of SS.

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

If we think of κ\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

Last revised on May 26, 2022 at 19:20:44. See the history of this page for a list of all contributions to it.