Is there a category theoretic definition of the naturals as an analogue to the set theoretic definition?

Question

And similarly, is there a category theoretic construction of the real numbers?

Answer 1

Let $\mathcal{C}$ be the variety of algebras consisting of sets with a single 0-ary operation (otherwise thought of as a constant element) $i$, a single unary operation $S$, and no identities. Then $(\mathbb{N}, i := 0, S := (n \mapsto n + 1))$ is an initial object of $\mathcal{C}$ (and therefore every initial object of $\mathcal{C}$ is uniquely isomorphic to $\mathbb{N}$ with these operations). This essentially gives $\mathbb{N}$ in a way very similar to the operations in the language of Peano arithmetic.

For other characterizations, $(\mathbb{N}, 0, +)$ is also the free monoid on one generator; and $(\mathbb{N}, 0, 1, +, \cdot)$ is the initial semiring.