How to define addition and multiplication of natural numbers using category theory?

Question

Is there a way to define addition and multiplication of natural numbers using the language of category theory? Like, one could say that "Addition is the unique functor that satisfies..." and "Multiplication is the unique functor that satisfies...". I would be very interested in such a definition.

Answer 1

"Using category theory" is a bit too general of a question. So there are multiple answers.

1. There is a notion of a natural numbers object in any suitable category $\mathcal{C}$. One can use the universal property to define addition and multiplication and verify the Peano-axioms internal to $\mathcal{C}$.

2. The natural numbers with addition are the free monoid on one element. Using plain category theory one can show that the forgetful functor $\mathsf{Mon} \rightarrow \mathsf{Set}$ admits a left adjoint, giving you access to $(\mathbb{N},+)$. Multiplication comes from this universal property by using that $(\operatorname{Map}(\Bbb N, \Bbb N),\circ,\operatorname{id})$ is a monoid.

3. The natural numbers are the set of isomorphism classes of the category of finite sets. Addition and multiplication come from the symmetric monoidal structures given by disjoint union and cartesian product.

4. The natural numbers with addition can be regarded as the free living endomorphism, ie. the category generated by the graph with one vertex and one loop, as suggested by @rschwieb. If you also want multiplication, I think you should regard the natural numbers as the free commutative-monoid-enriched category on one object and one loop.