In their 1998 paper, Baez and Dolan tell a fable about the origin of the natural numbers:
Long ago, when
shepherds wanted to see if two herds of sheep were isomorphic, they would
look for an explicit isomorphism. In other words, they would line up both
herds and try to match each sheep in one herd with a sheep in the other. But
one day, along came a shepherd who invented decategorification. She realized one could take each herd and ‘count’ it, setting up an isomorphism between
it and some set of ‘numbers’, which were nonsense words like ‘one, two,
three, . . . ’ specially designed for this purpose. By comparing the resulting
numbers, she could show that two herds were isomorphic without explicitly
establishing an isomorphism! In short, by decategorifying the category of
finite sets, the set of natural numbers was invented.
They go on to give more details. The idea is that decategorification consists of lumping the members of each isomorphism class together into a single object. This yields a set of objects without any useful category structure. However, as will generally be the case for any 'non-evil' operation, in $\mathbf{FinSet}$ the isomorphism class of $A\times B$ is a function of the isomorphism classes of $A$ and $B$, and similarly for the coproduct. When we decategorify we keep a record of these functions, and this gives us the natural numbers with the operations of multiplication and addition.
My question is, if decategorifying $\mathbf{FinSet}$ in this way yields the natural numbers, is there some other category that we can decategorify in a similar way to get all of the integers, including the negative numbers? I'm imagining this would be a category where the product "behaves like multiplication" and the coproduct "behaves like addition" as they do in $\mathbf{FinSet}$, but where there are objects that behave like they have "negative size."
While I'm here, I may as well also ask about the rationals, reals, nonnegative reals, complex numbers and so on. Are there categories that are known to decategorify into those sets with their usual addition and multiplication operations?
Disclaimer: it may be that Baez and Dolan address my question about integers later in their paper. I don't have a strong background in category theory and intend to come back to the rest of the paper when I've learnt a bit more. If this question seems like a stupid one in the context of that paper please forgive me. In that case, a simple, low-brow answer would be very helpful.
Best Answer
This question is the topic of Stephen Schanuel's paper Negative sets have Euler characteristic and dimension. In particular, Schanuel asks whether there is a category $\mathcal E$ such that (intuitively) the following diagram commutes.
He points out that we can't find a category satisfying exactly the properties we would expect (e.g. in any distributive category, $A + B \cong 0$ implies that $A \cong B \cong 0$). However, though we can't expect (the isomorphism classes of) $\mathcal E$ to be a group under addition, we can expect it to be a rig (that is, a ring without additive inverses) under addition and multiplication.
He goes on to suggest using a generalised notion of Euler characteristic for cardinality: for finite sets the notion of cardinality and Euler characteristic coincide, but for other categories, we may have non-natural "cardinalities". For instance, for an object representing the open interval (with 0 vertices and 1 edge), the Euler characteristic $V - E + F = -1$. This generalised Euler characteristic shares many nice properties with ordinary cardinality, such as its behaviour under coproducts, cartesian products and exponentiation. With this in mind, the category of polyhedral sets $\mathbf{PolySet}$ behaves in the way we wanted our category $\mathcal E$ to behave, with respect to the functor taking each object to its Euler characteristic in $\mathbb Z$.
A nice introduction and further exploration can be found in James Propp's Euler measure as generalized cardinality, where Propp also explores "fractional cardinalities".