nLab
2-rig

Idea

The notion of $2$-rig is supposed to be a categorification of that of a rig. Several inequivalent formalizations of this idea are in the literature.

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

Since categorification involves some arbitrary choices that will be determined by the precise intended application, there is a bit of flexibility of what exactly what one may want to call a 2-ring. We first list some immediate possibilities of classes of monoidal and enriched categories that one may want to think of as 2-rings:

• Enriched monoidal categories

But a central aspect of an ordinary ring is the distributivity law which says that the product in the ring preserves sums. Since sums in a 2-ring are given by colimits, this suggests that a 2-ring should be a cocomplete category which is compatibly monoidal in that the the tensor product preserves colimits:

• Compatibly monoidal cocomplete categories

But there are still more properties which one may want to enforce, notably that homomorphisms of 2-rings form a 2-abelian group?. This is achieved by demanding the underlying category to be not just cocomplete by presentable:

• Compatibly monoidal presentable categories.

Enriched monoidal categories

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.

Compatibly monoidal cocomplete categories

In (Baez-Dolan) a 2-rig is defined 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.

Compatibly monoidal presentable categories

The following refines the above by demanding the underlying category of a 2-ring to be not just cocomplet but even a presentable category. This was motivated in (CJF, remark 2.1.10).

Definition

Write

$$2\mathrm{Ab}\in 2\mathrm{Cat}$$2 Ab \in 2Cat

for the 2-category of presentable categories and colimit-preserving functors between them.

(CJF, def. 2.1.8)

Remark

By the adjoint functor theorem this is equivalently the 2-category of presentable categories and left adjoint functors between them.

Example

Given an ordinary ring $R$, its category of modules ${\mathrm{Mod}}_R$ is presentable, hence may be regarded as a 2-abelian group.

(CJF, example 2.1.5)

Proposition

The 2-category $2\mathrm{Ab}$ is a closed? symmetric monoidal 2-category with respect to the tensor product $⊠:2\mathrm{Ab}\times 2\mathrm{Ab}\to 2\mathrm{Ab}$ such that for $A,B,C\in 2\mathrm{Ab}$, ${\mathrm{Hom}}_{2\mathrm{Ab}}(A⊠B,C)$ is equivalently the full subcategory of functor category ${\mathrm{Hom}}_{\mathrm{Cat}}(A\times B,C)$ on those that are bilinear in that they preserve colimits in each argument separately.

See also at Pr(∞,1)Cat for more on this.

Example

For $𝒞$ a small category, the category of presheaves ${\mathrm{Set}}^{𝒞}$ is presentable and

$${\mathrm{Set}}^{{𝒞}_1}⊠{\mathrm{Set}}^{{𝒞}_2}\simeq {\mathrm{Set}}^{{𝒞}_1\times {𝒞}_2}.$$Set^{\mathcal{C}_1} \boxtimes Set^{\mathcal{C}_2} \simeq Set^{\mathcal{C}_1 \times \mathcal{C}_2} \,.

Example

For $R$ a ring the category of modules ${\mathrm{Mod}}_R$ is presentable and

$${\mathrm{Mod}}_{R_1}⊠{\mathrm{Mod}}_{R_2}\simeq {\mathrm{Mod}}_{R_1\otimes R_2},$$Mod_{R_1} \boxtimes Mod_{R_2} \simeq Mod_{R_1 \otimes R_2} \,,

(CJF, example 2.2.7)

Proposition

For $R_1,R_2$ two rings, the category of 2-abelian group homomorphisms between the categories of modules is naturally equivalent to that of $R_1$-$R_2$-bimodules and their intertwiners:

$$(-)\otimes (-):{}_{R_1}{\mathrm{Mod}}_{R_2}\stackrel{\simeq }{\to }{\mathrm{Hom}}_{2\mathrm{Ab}}({\mathrm{Mod}}_{R_1},{\mathrm{Mod}}_{R_2}).$$(-)\otimes (-) \;\colon\; {}_{R_1}Mod_{R_2} \stackrel{\simeq}{\to} Hom_{2Ab}(Mod_{R_1}, Mod_{R_2}) \,.

The equivalence sends a bimdoule $N$ to the functor given by the tensor product over $R_1$:

$$(-)\otimes N:{\mathrm{Mod}}_{R_1}\to {\mathrm{Mod}}_{R_2}.$$(-) \otimes N \;\colon\; Mod_{R_1} \to Mod_{R_2} \,.

This is the Eilenberg-Watts theorem.

Definition

Write

$$2\mathrm{Ring}\in 2\mathrm{Cat}$$2Ring \in 2Cat

for the 2-category of monoid objects internal to $2\mathrm{Ab}$. An object of this 2-category we call a 2-ring.

Equivalently, a 2-ring in this sense is a presentable category equipped with the structure of a monoidal category where the tensor product preserves colimits.

(CJF, def. 2.1.8)

Example

The category Set with its cartesian product is a 2-ring and it is the initial object in $2\mathrm{Ring}$.

(CJF, example 2.3.4)

Example

The category Ab of abelian groups with its standard tensor product of abelian groups is a 2-ring.

Example

For $R$ an ordinary commutative ring, ${\mathrm{Mod}}_R$ equipped with its usual tensor product of modules is a commutative 2-ring.

Example

For $R$ an ordinary ring and ${\mathrm{Mod}}_R$ its ordinary category of modules, regarded as a 2-abelian group by example 1, the structure of a 2-ring on ${\mathrm{Mod}}_R$ is equivalently the structure of a sesquiunital sesquialgebra on $R$.

If $R$ is in addition a commutative ring that ${\mathrm{Mod}}_R$ is a commutative 2-ring and is canonically an $\mathrm{Ab}$-2-algebra in that

$$\mathrm{Ab}\simeq {\mathrm{Mod}}_{\mathbb{Z}}\to {\mathrm{Mod}}_R.$$Ab \simeq Mod_{\mathbb{Z}} \to Mod_R \,.

(CJF, example 2.3.7)

Definition

For $A$ a 2-ring, def. 2, write

$$2{\mathrm{Mod}}_A\in 2\mathrm{Cat}$$2Mod_A \in 2Cat

for the 2-category of module objects over $A$ in $2\mathrm{Ab}$.

This means that a 2-module over $A$ is a presentable category $N$ equipped with a functor

$$A⊠N\to N$$A \boxtimes N \to N

which satisfies the evident action property.

(CJF, def. 2.3.3)

Example

Let $R$ be an ordinary commutative ring and $A$ an ordinary $R$-algebra. Then by example 1 ${\mathrm{Mod}}_A$ is a 2-abelian group and by example 6 ${\mathrm{Mod}}_R$ is a commutative ring. By example 3 ${\mathrm{Mod}}_R$-2-module structures on ${\mathrm{Mod}}_A$

$${\mathrm{Mod}}_R⊠{Mod}_A\to {\mathrm{Mod}}_A$$Mod_R \boxtimes \Mod_A \to Mod_A

correspond to colimit-preserving functors

$${\mathrm{Mod}}_{R{\otimes }_{\mathbb{Z}}A}\to {\mathrm{Mod}}_A$$Mod_{R \otimes_{\mathbb{Z}} A} \to Mod_{A}

that satisfy the action property. Such as presented under the Eilenberg-Watts theorem, prop. 2, by $R{\otimes }_{\mathbb{Z}}A$-$A$ bimodules. $A$ itself is canonically such a bimodule and it exhibits a ${\mathrm{Mod}}_R$-2-module structure on ${\mathrm{Mod}}_A$.

Properties

Tannaka duality

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebra	category of modules
$A$	${\mathrm{Mod}}_A$
$R$-algebra	${\mathrm{Mod}}_R$-2-module
sesquialgebra	2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebra	strict 2-ring: monoidal category with fiber functor
Hopf algebra	rigid monoidal category with fiber functor
hopfish algebra (correct version)	rigid monoidal category (without fiber functor)
weak Hopf algebra	fusion category with generalized fiber functor
quasitriangular bialgebra	braided monoidal category with fiber functor
triangular bialgebra	symmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)	rigid braided monoidal category with fiber functor
triangular Hopf algebra	rigid symmetric monoidal category with fiber functor
form Drinfeld double	form Drinfeld center
trialgebra	Hopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category	2-category of module categories
$A$	${\mathrm{Mod}}_A$
$R$-2-algebra	${\mathrm{Mod}}_R$-3-module
Hopf monoidal category	monoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category	3-category of module 2-categories
$A$	${\mathrm{Mod}}_A$
$R$-3-algebra	${\mathrm{Mod}}_R$-4-module

Related concepts

A further sligth variant of compatibly monoidal comcomplete categories is that of monoidal vectoids.

References

The proposal that a 2-ring should be a compatibly monoidal cocomplete category is due to

• John Baez, James Dolan, Higher-dimensional algebra III: $n$-categories and the algebra of opetopes, Adv. Math. 135 (1998), 145-206. (arXiv)

The proposal that a 2-ring should be a compatibly monoidal presentable category is due to

• Alexandru Chirvasitu, Theo Johnson-Freyd, The fundamental pro-groupoid of an affine 2-scheme (arXiv:1105.3104)

This is related to

• Jacob Lurie, Tannaka duality for geometric stacks.

A similar notion is that of “monoidal vectoid” due to

• Nikolai Durov, Classifying vectoids and generalisations of operads, Proc. of Steklov Inst. of Math. 273:1, 48-63 (2011) arxiv/1105.3114), the translation of “Классифицирующие вектоиды и классы операд”, Trudy MIAN, vol. 273

The role of presentable categories as higher analogs abelian groups in the context of (infinity,1)-categories have been made by Jacob Lurie, see at Pr(infinity,1)Cat.

Another, more algebraic, notion of a categorical ring is introduced in

• M. Jibladze , T. Pirashvili, Third Mac Lane cohomology via categorical rings, J. of homotopy and related structures, 2(2), 2007, 187–221 pdf math.KT/0608519