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:

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What is a universal property?

The following three constructions have something in common:

  • 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.

Do you already see the similarity? Let me rephrase them to make it more obvious:

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