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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.

user107952
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  • I think a good candidate for $(\mathbb N,+)$ might be "the free category generated by the graph with one vertex and one edge." If one has already defined what abelian monoids are, then you'd probably get the natural numbers with both operations as a monoid object in that category. – rschwieb Nov 09 '23 at 16:04
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    With anything that big "defining" anything that small, what you really do is choose a model, hopefully canonical but not uniquely well-suited by any means. – J.G. Nov 09 '23 at 16:27
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    I think this is the approach taken in the silly-but-contentful book Mathematics made difficult. – Noah Schweber Nov 09 '23 at 17:21
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    How about: use the category where the objects are $\mathbb{N}$ and the morphisms $n \to m$ are the $m \times n$ matrices over some field (such as $\mathbb{F}_2$ which is easy to define, as opposed to something like $\mathbb{Q}$ where you could argue there's a circular definition problem). Then addition is the direct sum on this abelian category, and multiplication is the tensor product (i.e. the monoidal operation in the structure as a monoidal closed category). – Daniel Schepler Nov 10 '23 at 00:29

1 Answers1

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"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.

Jonas Linssen
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  • Point 2 and point 4 are kind of the same, since the "free living endomorphism" is the delooping of the free monoid on one element. Then $(\mathrm{Map}(\mathbb{N},\mathbb{N}),\circ,\mathrm{id})$ corresponds to the functor category of endofunctors on the free living endomorphism, so you can get multiplication that way as well. – N. Virgo Nov 10 '23 at 02:50