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Suppose, we do universal algebra in a "non-evil" fashion, such that every algebra carries around an equivalence relation $\cong$ that poses as equality and such that size issues don't matter (we may have proper classes as underlying "sets").

If we consider the category of finite sets with binary products $\times$, binary coproducts $+$, the empty set $0 := \emptyset$ and some one-element set $1$ together with isomorphy (is that a word?) of objects $\cong$, we get the rig (or semiring if you will) of natural numbers.

Are there "nice" and "interesting" categories equipped with finite products and coproducts that in the same fashion yield the ring of integers or the field of rationals?

I'm afraid, that this question may have a really easy answer, however...

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  • $\begingroup$ "isomorphism" is the word $\endgroup$ – Berci Aug 24 '15 at 19:55
  • $\begingroup$ @Berci I thought "Isomorphism" refers to a morphism and not to an equivalence relation between objects. $\endgroup$ – Stefan Perko Aug 24 '15 at 20:02
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I'm afraid, that this question may have a really easy answer, however...

Unfortunately it does. Assume $A×B ≅ 1$ (the terminal object). Then for every $C$ we have $$1 ≅ \mathrm{Hom}(C, 1) ≅ \mathrm{Hom}(C, A × B) ≅ \mathrm{Hom}(C, A) × \mathrm{Hom}(C, B),$$ so $\mathrm{Hom}(C, A)$ and $\mathrm{Hom}(C, B)$ are both one-element sets, and $A$ and $B$ both terminal (you can easily show this directly). Dually, $A + B = 0$ implies $A$, $B$ initial. You will find some ideas on categories that represent $ℤ$ and $ℚ$ by googling categorification of integers/rationals. They will necessarily be more complicated than this however.

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