symmetric monoidal (∞,1)-category of spectra
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.
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:
A $2$-rig might be an Ab-enriched category which is enriched monoidal?.
A $2$-rig might be an additive category which is enriched monoidal.
A $2$-rig might be a distributive monoidal category: a monoidal category with finite coproducts such that the monoidal product distributes over the coproducts.
A $2$-rig might be a closed monoidal category with finite coproducts.
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.
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 $\mathbf{Symm2Rig}$ 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.
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).
Write
for the 2-category of presentable categories and colimit-preserving functors between them.
By the adjoint functor theorem this is equivalently the 2-category of presentable categories and left adjoint functors between them.
Given an ordinary ring $R$, its category of modules $Mod_R$ is presentable, hence may be regarded as a 2-abelian group.
The 2-category $2Ab$ is a closed? symmetric monoidal 2-category with respect to the tensor product $\boxtimes \colon 2Ab \times 2Ab \to 2Ab$ such that for $A,B, C \in 2Ab$, $Hom_{2Ab}(A \boxtimes B, C)$ is equivalently the full subcategory of functor category $Hom_{Cat}(A \times B, C)$ 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 $\mathcal{C}$ a small category, the category of presheaves $Set^{\mathcal{C}}$ is presentable and
For $R$ a ring the category of modules $Mod_R$ is presentable and
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:
The equivalence sends a bimodule $N$ to the functor given by the tensor product over $R_1$:
This is the Eilenberg-Watts theorem.
Write
for the 2-category of monoid objects internal to $2 Ab$. 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.
The category Set with its cartesian product is a 2-ring and it is the initial object in $2Ring$.
The category Ab of abelian groups with its standard tensor product of abelian groups is a 2-ring.
For $R$ an ordinary commutative ring, $Mod_R$ equipped with its usual tensor product of modules is a commutative 2-ring.
For $R$ an ordinary ring and $Mod_R$ its ordinary category of modules, regarded as a 2-abelian group by example 1, the structure of a 2-ring on $Mod_R$ is equivalently the structure of a sesquiunital sesquialgebra on $R$.
If $R$ is in addition a commutative ring that $Mod_R$ is a commutative 2-ring and is canonically an $Ab$-2-algebra in that
For $A$ a 2-ring, def. 3, write
for the 2-category of module objects over $A$ in $2Ab$.
This means that a 2-module over $A$ is a presentable category $N$ equipped with a functor
which satisfies the evident action property.
Let $R$ be an ordinary commutative ring and $A$ an ordinary $R$-algebra. Then by example 1 $Mod_A$ is a 2-abelian group and by example 6 $Mod_R$ is a commutative ring. By example 3 $Mod_R$-2-module structures on $Mod_A$
correspond to colimit-preserving functors
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 $Mod_R$-2-module structure on $Mod_A$.
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 $FinSet$ 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 for categories of modules over monoids/associative algebras
monoid/associative algebra | category of modules |
---|---|
$A$ | $Mod_A$ |
$R$-algebra | $Mod_R$-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 |
---|---|
$A$ | $Mod_A$ |
$R$-2-algebra | $Mod_R$-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 |
---|---|
$A$ | $Mod_A$ |
$R$-3-algebra | $Mod_R$-4-module |
A further slight variant of compatibly monoidal cocomplete categories is that of monoidal vectoids.
The proposal that a 2-ring should be a compatibly monoidal cocomplete category is due to
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
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