Exponential notation for function spaces

If \(X\) and \(Y\) are sets, the set of functions from \(X\) to \(Y\) (often written \(X \to Y\)) is sometimes also written \(Y^X\). This latter notation, which we’ll call exponential notation, is related to the notation for finite powers of sets (e.g., \(Y^3\) for the set of triples of elements of \(Y\)) as well as the notation of exponentiation for numbers.

Without further ado, here are some reasons this is good notation.

• A function \(f : X \to Y\) can be thought of as an “$X$ wide” tuple of elements of \(Y\). That is, a tuple of elements of \(Y\) where the positions in the tuple are given by elements of \(X\), generalizing the notation \(Y^n\) which denotes the set of \(n\) wide tuples of elements of \(Y\). Note that if \(|X| = n\), then \(Y^X \cong Y^n\).

• This notion of exponentiation together with cartesian product as multiplication and disjoint union as addition satisfy the same relations as exponentiation, multiplication, and addition of natural numbers. Namely,

• \(Z^{X \times Y} \cong (Z^X)^Y\) (this isomorphism is called currying)

• \(Z^{X + Y} \cong Z^X \times Z^Y\)

• \(Z^1 \cong Z\) (where \(1\) is a one element set, since there is one function into \(Z\) for every element of \(Z\))

• \(Z^0 \cong 1\) (where \(0\) is the empty set, since there is one function from the empty set to any set)

More generally, \(Y^X\) is good notation for the exponential object representing \(\text{Hom}_{\mathcal{C}}(X, Y)\) in an arbitrary cartesian closed category \(\mathcal{C}\) for the first set of reasons listed above.