nLab
2-rig

Idea

A $2$-rig is a categorification of a rig. There is more than one inequivalent notion; just as a rig is a multiplicative monoid whose underlying set also has a notion of addition, so a $2$-rig is a monoidal category whose underlying category also has a notion of addition, and we can describe this notion of addition in a few different ways.

Note that we don't expect a $2$-rig to have additive inverses; by the same argument as in the Eilenberg swindle, they are unreasonable to expect. However, in a monoidal abelian category, we have as close to additive inverses as is reasonable and so a categorification of a ring.

Compare also the notion of rig category.

Definitions

1. A $2$-rig might be an Ab-enriched category which is enriched monoidal?.

2. A $2$-rig might be an additive category which is enriched monoidal.

3. A $2$-rig might be a monoidal category with finite coproducts such that the monoidal product distributes over the coproducts.

4. A $2$-rig might be a closed monoidal category with finite coproducts.

5. Finally, a $2$-ring is a monoidal abelian category.

Note that (2) is a special case of both (1) and (3), which are independent. (4) is a special case of (3), by the adjoint functor theorem. (5) is a special case of (2), of course.

Yet another definition

The following paper:

• John Baez and James Dolan, Higher-dimensional algebra III: $n$-categories and the algebra of opetopes, Adv. Math. 135 (1998), 145-206. [[arXiv](http://arxiv.org/abs/q-alg/9702014)]

uses the term ‘2-rig’ in yet another way: it defines a 2-rig to be a monoidal cocomplete category where the monoidal product distributes over colimits. We can define braided and symmetric 2-rigs in this sense (and indeed, also in the other senses listed above). In particular, there is a 2-category $\mathrm{Symm}2\mathrm{Rig}$ with:

• symmetric monoidal cocomplete categories where the monoidal product distributes over colimits as objects,

• symmetric monoidal cocontinuous functors as 1-morphisms,

• symmetric monoidal natural transformations as 2-morphisms.

This leads to the following:

Conjecture (John Baez)

The initial symmetric 2-rig is $\mathrm{Set}$, in a suitably weakened sense. Namely, if $R$ is any object of $\mathrm{Symm}2\mathrm{Rig}$, then there is a 1-morphism $\mathrm{Set}\to R$ that is unique up to a 2-isomorphism.

Furthermore, the free symmetric 2-rig on one object is the category of species, $\hat{\mathbb{P}}$ — that is, the category of presheaves on the groupoid of finite sets and bijections, $\mathbb{P}$. This symmetric 2-rig is free on the object $X$ which is the presheaf sending the one-element set to the one-element set, and every other set to the empty set.

More precisely: if $R$ is any symmetric 2-rig and $x\in R$, there exists a 1-morphism $F:\hat{\mathbb{P}}\to R$ with $F(X)=x$, and $F$ is unique up to a 2-isomorphism.