# Rig categories

## Idea

Recall that a rig is a ‘ring without negatives’: a monoid object in the category of commutative monoids with the usual tensor product. Categorifying this notion, we obtain various notions of 2-rig, one of which is the notion of ‘rig category’.

A typical example would be the groupoid of finite sets and bijections, with disjoint union playing the role of addition and cartesian product playing the role of multiplication. This rig category can be thought of as a categorification of the set of natural numbers. Note that in this example, disjoint union is not the categorical coproduct, and product of sets is not the categorical product (because we are working in the groupoid of finite sets).

## Definition

A rig category $C$ is a category with a symmetric monoidal structure $\left(C,\oplus ,0\right)$ for addition and a monoidal structure $\left(C,\otimes ,I\right)$ for multiplication, together with left and right distributivity natural isomorphisms

${d}_{\ell }:x\otimes \left(y\oplus z\right)\to \left(x\otimes y\right)\oplus \left(x\otimes z\right)$d_\ell : x \otimes (y \oplus z) \to (x \otimes y) \oplus (x \otimes z)
${d}_{r}:\left(x\oplus y\right)\otimes z\to \left(x\otimes z\right)\oplus \left(y\otimes z\right)$d_r : (x \oplus y) \otimes z \to (x \otimes z) \oplus (y \otimes z)

satisfying a set of coherence laws worked out by Kelly and Laplaza:

• M. Laplaza, Coherence for distributivity, Lecture Notes in Mathematics 281, Springer Verlag, Berlin, 1972, pp. 29-72.

• G. Kelly, Coherence theorems for lax algebras and distributive laws, Lecture Notes in Mathematics 420, Springer Verlag, Berlin, 1974, pp. 281-375.

Note that these authors used the term ‘ring category’. We take the liberty of switching to ‘rig category’ because it is typical for these to lack additive inverses.

While a rig can have the extra property of being commutative, a rig category can have the extra structure of being braided and may then have the further property of being symmetric.

## Baez’s conjecture

###### Conjecture (John Baez)

Using the correct definition of the 2-category of symmetric rig categories, the groupoid ${\mathrm{FinSet}}^{×}$ of finite sets and bijections is the initial symmetric rig category, just as $N$ is the initial commutative rig. Note that a suitably weakened concept of ‘initial’ is needed here; see 2-limit. In other words, given any symmetric rig category $R$, there is a unique symmetric rig morphism ${\mathrm{FinSet}}^{×}\to R$, up to an equivalence which is itself unique up to an isomorphism which is actually unique (up to equality).

Revised on September 12, 2012 02:58:55 by Toby Bartels (98.19.47.153)