# What is an exponential object?

One interpretation for a power of natural numbers, such as $10^6$, is that it is counting a certain collection of functions: in this case, the number of functions from a six-element set to a ten-element set. (One way to think about it is that $10^6$ is the number of six-digit combinations from 000000 to 999999, and each combination corresponds to a function from $\{1,\ldots,6\}$ to $\{0,\dots,9\}$: the first digit is the image of $1$, the second digit is the image of $2$, etc.)

In general, the number $m^n$ is the number of functions from an $n$-element set to a $m$-element set. This is somewhat analogous to standard interpretations of other arithmetical operations:

• Kernels: If $f:G\to H$ and $g:H\to K$ are two group homomorphisms, then the composite $g\circ f$ is the trivial homomorphism $G\to K$ if and only if the image of $f$ is contained in the kernel of $g$.
• Polynomial rings: If $A$ is any $\mathbb{R}$-algebra, then an $\mathbb{R}$-algebra homomorphism $\mathbb{R}[x]\to A$ is entirely determined by where it sends $x$.
• Topological products: The product topology on a product of topological spaces $\prod_{n=1}^\infty X_n$ is a little counterintuitive—at first glance, the box topology seems more natural—but the product topology has the property that a function $(f_1,f_2,\dots):A\to\prod_{n=1}^\infty X_n$ is continuous if and only if each component map $f_n:A\to X_n$ is continuous.