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 ${\text{Hom}}_C(X,Y)$ in an arbitrary cartesian closed category $C$ for the first set of reasons listed above.