The notion of 22-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 22-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 22-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.


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:

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:

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:

Enriched monoidal categories

  1. A 22-rig might be an Ab-enriched category which is enriched monoidal?.

  2. A 22-rig might be an additive category which is enriched monoidal.

  3. A 22-rig might be a distributive monoidal category: a monoidal category with finite coproducts such that the monoidal product distributes over the coproducts.

  4. A 22-rig might be a closed monoidal category with finite coproducts.

  5. Finally, a 22-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) the following is considered:


A 2-rig is a monoidal cocomplete category where the tensor product respects colimits.

One 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 Symm2Rig\mathbf{Symm2Rig} 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.

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).



2Ab2Cat 2 Ab \in 2Cat

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

(CJF, def. 2.1.8)


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


Given an ordinary ring RR, its category of modules Mod RMod_R is presentable, hence may be regarded as a 2-abelian group.

(CJF, example 2.1.5)


The 2-category 2Ab2Ab is a closed? symmetric monoidal 2-category with respect to the tensor product :2Ab×2Ab2Ab\boxtimes \colon 2Ab \times 2Ab \to 2Ab such that for A,B,C2AbA,B, C \in 2Ab, Hom 2Ab(AB,C)Hom_{2Ab}(A \boxtimes B, C) is equivalently the full subcategory of functor category Hom Cat(A×B,C)Hom_{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.


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

Set 𝒞 1Set 𝒞 2Set 𝒞 1×𝒞 2. Set^{\mathcal{C}_1} \boxtimes Set^{\mathcal{C}_2} \simeq Set^{\mathcal{C}_1 \times \mathcal{C}_2} \,.

For RR a ring the category of modules Mod RMod_R is presentable and

Mod R 1Mod R 2Mod R 1R 2, Mod_{R_1} \boxtimes Mod_{R_2} \simeq Mod_{R_1 \otimes R_2} \,,

(CJF, example 2.2.7)


For R 1,R 2R_1, R_2 two rings, the category of 2-abelian group homomorphisms between the categories of modules is naturally equivalent to that of R 1R_1-R 2R_2-bimodules and their intertwiners:

()(): R 1Mod R 2Hom 2Ab(Mod R 1,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 bimodule NN to the functor given by the tensor product over R 1R_1:

()N:Mod R 1Mod R 2. (-) \otimes N \;\colon\; Mod_{R_1} \to Mod_{R_2} \,.

This is the Eilenberg-Watts theorem.



2Ring2Cat 2Ring \in 2Cat

for the 2-category of monoid objects internal to 2Ab2 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)


The category Set with its cartesian product is a 2-ring and it is the initial object in 2Ring2Ring.

(CJF, example 2.3.4)


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


For RR an ordinary commutative ring, Mod RMod_R equipped with its usual tensor product of modules is a commutative 2-ring.


For RR an ordinary ring and Mod RMod_R its ordinary category of modules, regarded as a 2-abelian group by example 1, the structure of a 2-ring on Mod RMod_R is equivalently the structure of a sesquiunital sesquialgebra on RR.

If RR is in addition a commutative ring that Mod RMod_R is a commutative 2-ring and is canonically an AbAb-2-algebra in that

AbMod Mod R. Ab \simeq Mod_{\mathbb{Z}} \to Mod_R \,.

(CJF, example 2.3.7)


For AA a 2-ring, def. 3, write

2Mod A2Cat 2Mod_A \in 2Cat

for the 2-category of module objects over AA in 2Ab2Ab.

This means that a 2-module over AA is a presentable category NN equipped with a functor

ANN A \boxtimes N \to N

which satisfies the evident action property.

(CJF, def. 2.3.3)


Let RR be an ordinary commutative ring and AA an ordinary RR-algebra. Then by example 1 Mod AMod_A is a 2-abelian group and by example 6 Mod RMod_R is a commutative ring. By example 3 Mod RMod_R-2-module structures on Mod AMod_A

Mod RMod AMod A Mod_R \boxtimes \Mod_A \to Mod_A

correspond to colimit-preserving functors

Mod R AMod 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 AR \otimes_{\mathbb{Z}} A-AA bimodules. AA itself is canonically such a bimodule and it exhibits a Mod RMod_R-2-module structure on Mod AMod_A.


Initial object


The analog role in 2-rigs to the role played by the natural numbers among ordinary rigs should be played by the standard categorification of the natural numbers: the category of finite sets. One is therefore inclined to demand that a reasonable definition of 2-rigs should be such that FinSetFinSet is the initial object (in the suitably higher categorical sense) in the 2-category of 2-rigs.

For the notion in def. 1 this was conjectured by John Baez, for the notion in def. 3 this is asserted in (Chirvasitu & Johnson-Freyd, example 2.3.4).

Tannaka duality

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

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

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module


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

  • John Baez, James Dolan, Higher-dimensional algebra III: nn-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

This is related to

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

Revised on March 24, 2014 07:10:57 by Urs Schreiber (