Which commutative rigs arise from a distributive category?

Question

A rig is an algebraic object with multiplication and addition, such that multiplication distributes over addition and addition is commutative. However, instead of requiring that the set forms an abelian group under addition, we require only that it forms an abelian monoid. A commutative rig is a rig in which multiplication induces an abelian monoid.

A distributive category is a small category with finite products and coproducts, which I'll denote * and + respectively, such that the canonical morphism X * Y + X * Z \rightarrow X * (Y+Z) is an isomorphism. Essentially, taking products distributes over taking coproducts.

There's an apparent formal similarity between the definitions, and in fact you can get a commutative rig out of a distributive category C by taking the objects of the rig to be isomorphism classes of objects in the category (equivalently, considering the skeletal category) and letting coproducts correspond to addition and products to multiplication. For instance, if you start with the category of finite sets, you can skeletonize to obtain a commutative rig isomorphic to the rig of natural numbers.

So the question is, does every commutative rig arise this way from some distributive category? I suspect the answer is "no," and I can even think of some likely counterexamples (the nonnegative real numbers, Z), but I don't see an obvious way to prove that they are counterexamples.

Supposing that my hunch is correct, is there a nice way to classify the rigs that do arise in this manner? Is there a way to classify the commutative rings that arise when you add negatives to these rigs (analogously to the Grothendieck group)?

Answer 1

No commutative rig with a nontrivial solution to x + y = 0 (in particular, no nonzero ring) can arise from a distributive category. I believe Schanuel observes this near the beginning of one of his papers on Euler characteristics.

The proof is simple. Suppose C is a distributive category and R is its associated commutative rig. Suppose x + y = 0 for some x, y in R. Note that 0 in R is the isomorphism class of the initial object 0 of C, since the latter is obviously an additive unit. Then there are objects X and Y in C and an isomorphism X ∐ Y -> 0. In other words, for every object Z of C, • = Hom(0, Z) = Hom(X ∐ Y, Z) = Hom(X, Z) × Hom(Y, Z), and this can only happen when Hom(X, Z) = Hom(Y, Z) = •, so X and Y are also initial objects and thus x = y = 0.

Of course, group completing the rig you get from a category is one way around this. Alternatively you can work in some kind of homotopical setting, e.g., finite CW complexes, and replace the condition about coproducts with one about homotopy pushouts. Either way, you have to define the ring in such a way that its elements are not isomorphism classes of objects, but in the latter case you can at least find representatives in the category for every ring element.

Answer 2

Reid gave half the answer I was going to give — that for your rig to be arise from a category (what Schanuel calls the "Burnside rig"; by the way, the Schanuel paper in question is MR1173024, and is very good) a necessary condition is that there be no nontrivial solutions to 0 = x + y. Similarly, there can be no nontrivial solutions to 1 = x * y. Indeed, these conditions do not require any distributivity, so it's highly unlikely that they are sufficient conditions.

Since Categorization is popular, as Reid mentions there are constructions to get around these difficulties. Schanuel shows that a number of natural categories have Burnside rig N[x]/(x=2x+1); the cancelative quotient of this rig is Z. Baez and Dolan like the 2-category of finite groupoids as a prototypical example in which division can be introduced. Indeed, they define "groupoid cardinality" to take values in the non-negative rationals. These examples can be combined to some category of finite orbifolds. But all these maps are very lossy.

Let me quote a bit from Schanuel's paper:

The Burnside rig (of isomorphism classes of objects, added by coproduct and multiplied by product) of a distributive category has some special features, the first of which we have already seen.

1. If a + b = 0, then a = b = 0.
2. If Σ a_i = Σ b_j, then there exist c_ij such that Σ_j c_ij = a_i and Σ_i c_ij = b_j.
3. If a is connected (a is not 0, and a = b + c implies b = 0 or c = 0), then a is cancellable (a + x = a + y implies x = y).
4. 1 is cancellable (whether it is connected or not).
5. If ab=1, then a=b=1.

Properties (1) and (3) follow from (2), which follows easily from the observation that coproduct decompositions A = Σ_I A_i correspond to maps A → 1 + 1 + ... + 1 (I terms). I don't know what additional properties characterize Burnside rigs of distributive categories.

I'll conclude by mentioning two homomorphisms out of any rig, which may be of interest in trying to classify Burnside rigs. First, there's the cancelative quotient R/~, where a~b if there is any x with a+x=b+x; because of Schanuel's geometric picture, he calls this "Euler Characteristic". Second there's what Schanuel calls "dimension" R/~ where the relationship ~ is generated by 1+1~1.