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.