# nLab rig category

Rig categories

### Context

#### Monoidal categories

monoidal categories

# Rig categories

## Idea

Recall that a rig is a ‘ring without negatives’: a monoid object in the monoidal category of commutative monoids with the usual tensor product. Categorifying this notion, we obtain various notions of 2-rig. One of these, in which both “addition” and “multiplication” are represented by abstract monoidal structures, is the notion of rig category, also known as a bimonoidal 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, or bimonoidal category, $C$ is a category with a symmetric monoidal structure $(C,\oplus,0)$ for addition and a monoidal structure $(C, \otimes, I)$ for multiplication, together with left and right distributivity natural isomorphisms

$d_\ell : x \otimes (y \oplus z) \to (x \otimes y) \oplus (x \otimes z)$
$d_r : (x \oplus y) \otimes z \to (x \otimes z) \oplus (y \otimes z)$

and absorption/annihilation isomorphisms

$a_\ell : x \otimes 0 \to 0$
$a_r : 0 \otimes x \to 0$

satisfying a set of coherence laws worked out in (Laplaza 72) and (Kelly74).

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 (i.e. of its multiplication being commutative), a rig category can have the extra structure of (its monoidal structure $\otimes$) being braided (compatibly with the distributive laws) and may then have the further property of being symmetric.

## Examples

Rig categories are part of the hierarchy of distributivity for monoidal structures.

### Distributive categories

If $\oplus$ is the categorical coproduct and $\otimes$ is the categorical product, then we have the notion of a distributive category, which is a special case of a rig category. For example, the category Set of sets, as well as any topos, is a distributive category, hence a rig category with $\times$ and $+$.

### Distributive monoidal categories

In between, we have the notion of distributive monoidal category, where $\oplus$ is the coproduct but $\otimes$ is an abstract monoidal structure. Examples of this sort include Ab, $R$Mod, Vect and Vect(X):

In all these cases the coproduct is the respective direct sum (e.g. direct sum of vector bundles in the last case).

## Baez’s conjecture

###### Conjecture (John Baez)

Using the correct definition of the 2-category of symmetric rig categories, the groupoid $FinSet^{\times}$ 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 $FinSet^{\times} \to R$, up to an equivalence which is itself unique up to an isomorphism which is actually unique (up to equality).

This conjecture has been established in (Elgueta 20). See also (Comfort-Delpeuch-Hedges, Sec. 8).

The coherence for the distributivity law in bimonoidal categories has been given in

• 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.

where these categories are called ring categories. Discussion with an eye towards the K-theory of a bipermutative category is in

• Peter May, $E_\infty$ Ring Spaces and $E_\infty$ Ring spectra, Springer lectures notes in mathematics, Vol. 533, (1977) (pdf) chaper VI
• Bertrand Guillou, Strictification of categories weakly enriched in symmetric monoidal categories, arXiv:0909.5270
• Angélica Osorno, Spectra associated to symmetric monoidal bicategories (arXiv)

A string diagram treatment of rig categories via sheet diagrams is in

Biinitiality of the groupoid of finite sets is shown in

• Josep Elgueta, The groupoid of finite sets is biinitial in the 2-category of rig categories, (arXiv:2004.08684).