symmetric monoidal (∞,1)-category of spectra
Just as a rig is a multiplicative monoid whose underlying set also has a notion of addition, so a -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 -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 -rig might be an additive category which is enriched monoidal.
A -rig might be a closed monoidal category with finite coproducts.
Finally, a -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:
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 2-category is a closed? symmetric monoidal 2-category with respect to the tensor product such that for , is equivalently the full subcategory of functor category on those that are bilinear in that they preserve colimits in each argument separately.
See also at Pr(∞,1)Cat for more on this.
The equivalence sends a bimodule to the functor given by the tensor product over :
This is the Eilenberg-Watts theorem.
For a 2-ring, def. 3, write
This means that a 2-module over is a presentable category equipped with a functor
which satisfies the evident action property.
correspond to colimit-preserving functors
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 is the initial object (in the suitably higher categorical sense) in the 2-category of 2-rigs.
|monoid/associative algebra||category of modules|
|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|
|supercommutative Hopf algebra (supergroup)||rigid symmetric monoidal category with fiber functor and Schur smallness|
|form Drinfeld double||form Drinfeld center|
|trialgebra||Hopf monoidal category|
|monoidal category||2-category of module categories|
|Hopf monoidal category||monoidal 2-category (with some duality and strictness structure)|
|monoidal 2-category||3-category of module 2-categories|
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
Another, more algebraic, notion of a categorical ring is introduced in