The natural numbers can be thought of as being the category of finite sets modulo isomorphisms, since:

  • $|A \times B| = |A| \times |B|$
  • $|A + B| = |A| + |B|$
  • $|A^B| = |A|^{|B|}$
  • $|\text{Aut}(X)| = |X|!$

What category can we form that has real numbers as objects and categorical products, sums, exponentials, etc. are the regular operations on real numbers?


Unfortunately, this isn't possible, just from the conditions about the product and coproduct. This is a result of numbers having inverses under $+$ and $\times$, which doesn't happen in $\mathbb{N}$

First note that the description of products ensures that $1$ is an terminal object, since for all $x$ the product of $1$ and $x$ is $x$. This is because for any morphism $f:x \rightarrow 1$, by the naturality of the projection morphisms, we have that

$$x \times 1 \xrightarrow{f \times Id} 1 \times 1 \xrightarrow{\pi_2} 1$$

is equal to

$$x \times 1 \xrightarrow{\pi_2} 1 \xrightarrow{Id} 1$$

In other words, by chasing around the diagram and using the fact that $\pi_2: 1 \times 1 \rightarrow 1$ is invertible, the only possible morphism from $x$ to $1$ is $x \cong x \times 1 \xrightarrow{\pi_2} 1$.

Similarly, $0$ must be initial.

Now, for any $x$, we have that $x + (-x) = 0$. For any other $y$, we have that $Hom(x,y) \times Hom((-x),y) = Hom(0,y)$ which is a singleton and so $x$ (and $(-x)$) is initial as well and therefore isomorphic to $0$.

Even if you leave out the negative reals a similar problem occurs with $x \times \frac{1}{x} = 1$ proving that every object (but $0$) is terminal.

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