nLab rig category

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

$$d_\ell \;\colon\; x \otimes (y \oplus z) \longrightarrow (x \otimes y) \oplus (x \otimes z)$$

$$d_r \;\colon\; (x \oplus y) \otimes z \longrightarrow (x \otimes z) \oplus (y \otimes z)$$

and absorption/annihilation natural isomorphisms

$$a_\ell \;\colon\; x \otimes 0 \longrightarrow 0$$

$$a_r \;\colon\; 0 \otimes x \longrightarrow 0$$

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”(as in rig) 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

A rig category where $\oplus$ is the category-theoretic coproduct and $\otimes$ is the category-theoretic product (Cartesian product) is called a distributive category.

For example,

• the category Set of sets,

• any topos,

• the category Top of topological spaces with respect to forming product topological spaces and disjoint union topological spaces;

are distributive categories, hence rig categories with $\times$ and $+$.

Distributive monoidal categories

In between, we have the notion of distributive monoidal category, where $\oplus$ is the coproduct but $\otimes$ is a possibly non-cartesian monoidal structure.

Examples of this sort include Ab, $R$Mod, Vect and Vect(X):

• Ab, the category of abelian groups equipped with the tensor product of abelian groups

• $R$Mod, the category of modules over a commutative ring $R$, equipped with the tensor product of modules

• $k$Vect = $k$Mod, the category of vector space over some field $k$, equipped with the tensor product of vector spaces,

• $k$Vect(X), the category of (topological) vector bundles for $X$ some (topological) space, equipped with the tensor product of vector bundles.

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

Also:

• the category of pointed sets with respect to forming smash product and wedge sum,

• more generally, the category of pointed topological spaces with respect to forming smash product and wedge sum (e.g. Hatcher, Section 4.F).

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 was established in (Elgueta 2021). See also the proof using sheet diagrams in (Comfort-Delpeuch-Hedges, Sec. 8), and the detailed proof of a restricted version in (Johnson-Yau, Part I Sec. 2.7). Yau has also proved a braided analogue of this conjecture (Johnson-Yau, Part II Thm. 7.3.4).

Related concepts

• distributivity for monoidal structures

  • distributive category, distributive monoidal category

  • If the distributivity morphisms are not invertible, then we have the notion of colax-distributive rig category.

  • duoidal category

  • 2-rig

• ring groupoid

• The notion of a bipermutative category is a strictification of the notion of symmetric rig category. Every symmetric rig category is equivalent to a bipermutative category ([May, prop. VI 3.5]). Similarly, every rig category is equivalent to a strict rig category (Guillou, theorem 1.2).

• K-theory of a bipermutative category

• BDR 2-vector bundle.

References

A textbook treatment is given in

• Niles Johnson, Donald Yau, Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic K-Theory (three volume book; abridged version arXiv:2107.10526).

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

• M. L. Laplaza, Coherence for distributivity, Lecture Notes in Mathematics 281 Springer (1972) 29-72 [pdf]

• G. M. 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:1005.2227)

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

• Cole Comfort, Antonin Delpeuch, Jules Hedges, Sheet diagrams for bimonoidal categories, (arXiv:2010.13361)

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, Journal of Pure and Applied Algebra 225 Issue 11 (2021) 106738, doi:10.1016/j.jpaa.2021.106738, arXiv:2004.08684.

On the idea of a symmetric monoidal category as a module over a commutative rig category:

• John D. Berman, On the commutative algebra of categories, Algebraic and Geometric Topology, 18(5):2963–3012, August 2018, (arXiv:1606.05606).