# Category on integers with the usual product and coproduct?

I've just been introduced to category theory. I understand the basic definitions, and I'm trying to get some intuition on how categories tick. I'm wondering: is there a category $C$ such that:

• Its objects are either the integers, the positive integers, or the nonnegative integers (I'm fine with any one of these)
• Its category-theoretic product is the same as the usual product
• Its coproduct is the same as the usual addition?

Intuitively, I think the answer is yes, but I can't come up with a construction.

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You may be interested in the concept of categorification ncatlab.org/nlab/show/vertical+categorification. In my opinion this is one of the most fundamental principals of modern mathematics. You ask for a categorification of the semiring or rig (ncatlab.org/nlab/show/rig) of natural numbers. The answer is the 2-rig of finite sets (ncatlab.org/nlab/show/2-rig). – Martin Brandenburg Aug 26 '13 at 12:44

I suppose technically we want the skeleton of $\sf Set$, but having extra isomorphic copies of objects is not such a big deal. – anon Aug 26 '13 at 1:07
@Lopsy: or, as anon says, we can take the skeleton of $\text{FinSet}$, which can be taken to have objects the non-negative integers. But as a category theorist I freely identify equivalent categories in general. – Qiaochu Yuan Aug 26 '13 at 4:01