text: One can use sets as exponents for other operations on sets, typically for direct sums of abelian groups, vector spaces, or modules. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example,  denotes the vector space of the infinite sequences of real numbers, and  the vector space of those sequences that have a finite number of nonzero elements. The latter has a basis consisting of the sequences with exactly one nonzero element that equals , while the Hamel bases of the former cannot be explicitly described (because their existence involves Zorn's lemma).
  In this context,  can represents the set  So,  denotes the power set of , that is the set of the functions from  to  which can be identified with the set of the subsets of , by mapping each function to the inverse image of .
  This fits in with the exponentiation of cardinal numbers, in the sense that , where  is the cardinality of .
  In category theory
  In the category of sets, the morphisms between sets  and  are the functions from  to . It results that the set of the functions from  to  that is denoted  in the preceding section can also be denoted  The isomorphism  can be rewritten
  This means the functor "exponentiation to the power " is a right adjoint to the functor "direct product with ".