It occurred to me earlier today that one can form a category by taking the objects to be the positive natural numbers and postulating a morphisms from $a$ to $b$ iff $a$ divides $b$. (That is, there is a map from $4$ to $8$, but not from $4$ to $7$.)
In this setting, the initial object is $1$, a product is the $\gcd$ and a coproduct is the $\operatorname{lcm}$.
You could describe prime numbers as the objects to which there are exactly 2 morphisms. Does this have a name?
My knowledge of category theory (and number theory) is very slim, so I was wondering if there was anything interesting that could be said about this connection? That is, can one explain (or at least motivate) deeper ideas in elementary number theory in terms categorical language? Can one go the other way and find other categories that share crucial properties with this one (existence of prime-like objects, maybe) and motivate questions there?
(I did a little googling, but didn't find anything that seemed relevant.)
Thank you.
Best Answer
Every preorder $(P,\leq)$ can be regarded as a category. The object set is $P$, and a morphism $x \to y$ exists (and is unique) iff $x \leq y$. These are precisely those categories in which every diagram commutes. But still, it is interesting to apply category theory to preorders. A limit is just an infimum, a colimit is a supremum. In particular, initial (terminal) objects are least (largest) elements. Monotonic maps are just functors. Galois connections are just adjunctions between preorders.
We can define ideals and prime ideals in $P$. If $p \in P$ is an element, one says that $p$ is a meet prime iff the generated ideal ${\downarrow}p$ is a prime ideal, i.e. iff $p$ is not the largest element and $\inf(x,y) \leq p$ implies $x \leq p$ or $y \leq p$.
If $P = \mathbb{N}^+$ and $\leq$ is the relation of divisibility reversed(!), then prime elements are precisely the usual prime numbers. You can omit this reversion by looking at join prime elements.
More generally, if $P$ is the preorder of ideals of a ring $R$, ordered by inclusion, then prime elements are precisely the prime ideals of $R$ in the sense of ring theory.
One can generalize many notions of number theory / ring theory to preorders resp. lattices (see Wikipedia for a start; I have also found many papers by a quick google research). In order to answer your interesting question "That is, can one explain (or at least motivate) deeper ideas in elementary number theory in terms categorical language?" I would like to see an explicit "deeper idea in elementary number theory" first - then we may try to explain it in category-theoretic terms (although I doubt that we will gain anything from that).