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.