nLab
biring

The idea

Just as a bimonoid is both a monoid and a comonoid in a compatible way, a ‘biring’ is both a commutative ring and a commutative coring in a compatible way.

Definition

A biring is a commutative ring RR equipped with ring homomorphisms called coaddition:

RRR R \to R \otimes R

cozero:

R R \to \mathbb{Z}

co-additive inverse:

RR R \to R

comultiplication:

RRR R \to R \otimes R

and the multiplicative counit:

R R \to \mathbb{Z}

satisfying the usual axioms of a commutative ring, but ‘turned around’.

More tersely, and also more precisely, a biring is a commutative ring object in the opposite of the category of commutative rings (also known as the category of affine schemes).

Equivalently, a biring is a commutative ring RR equipped with a lift of the functor

hom(R,):CommRingSet hom(R, -) : CommRing \to Set

to a functor

hom(R,):CommRingCommRing hom(R, -) : CommRing \to CommRing

Birings form a monoidal category thanks to the fact that functors of this form are closed under composition. A monoid object in this monoidal category is called a plethory. A plethory is an example of a Tall–Wraith monoid.

The most important example of a biring is Λ\Lambda, the ring of symmetric polynomials. This is actually a plethory.

Categorified Birings

The biring Λ\Lambda is the Grothendieck group of the category of Schur functors, which is equivalent to the functor category

[,FinVect][\mathbb{P}, FinVect]

where \mathbb{P} is the permutation groupoid and FinVectFinVect is the category of finite-dimensional vector spaces over a field kk of characteristic zero. Λ\Lambda is also the Grothendieck group of

[,Vect][\mathbb{P}, Vect ]

where we drop the finite-dimensionality restriction on our vector spaces and work with all of Vect.

This suggests that the biring structure of Λ\Lambda may emerge naturally from a ‘categorified biring’ structure on [,Vect][\mathbb{P}, Vect ]. In this section we sketch how such a categorified biring might be constructed, based on the assumption that there is a tensor product of cocomplete linear categories with good universal properties.

Namely, we assume that given cocomplete linear categories XX and YY, there is a cocomplete linear category XYX \otimes Y such that:

  • There is a linear functor i:X×YXYi: X \times Y \to X \otimes Y which is cocontinuous in each argument.

  • For any cocomplete linear category ZZ, the category of linear functors XYZX \otimes Y \to Z is equivalent to the category of linear functors X×YZX \times Y \to Z that are cocontinuous in each argument, with the equivalence being given by precomposition with ii.

With any luck these two assumptions will let us show that for any categories AA and BB,

(1)[A×Y,Vect][A,Vect][B,Vect] [A \times Y, Vect] \simeq [A,Vect] \otimes [B, Vect]

where we use [,][-,-] to denote the functor category.

Assuming all this, we obtain the following operations on the category [,Vect][\mathbb{P}, Vect]:

  1. Addition: form the composite functor

    [,Vect]×[,Vect][,Vect×Vect][,Vect] [\mathbb{P}, Vect] \times [\mathbb{P}, Vect] \to [\mathbb{P}, Vect \times Vect] \to [\mathbb{P}, Vect]

    where the last arrow comes from postcomposition with

    :Vect×VectVect \oplus : Vect \times Vect \to Vect

    This composite is our addition:

    :[,Vect]×[,Vect][,Vect] \oplus : [\mathbb{P}, Vect] \times [\mathbb{P}, Vect] \to [\mathbb{P}, Vect]

    It’s really just the coproduct in [,Vect][\mathbb{P}, Vect].

  2. Multiplication: first form the composite functor

    [,Vect]×[,Vect][,Vect×Vect][,Vect] [\mathbb{P}, Vect] \times [\mathbb{P}, Vect] \to [\mathbb{P}, Vect \times Vect] \to [\mathbb{P}, Vect]

    where the last arrow comes from postcomposition with

    :Vect×VectVect \otimes : Vect \times Vect \to Vect

    This composite is our multiplication:

    :[,Vect]×[,Vect][,Vect] \otimes : [\mathbb{P}, Vect] \times [\mathbb{P}, Vect] \to [\mathbb{P}, Vect]

    Since this product preserves colimits in each argument, if we use the hoped-for universal property of the tensor product of cocomplete linear categories, we can reinterpret this as a cocontinuous functor

    :[,Vect][,Vect][,Vect] \otimes: [\mathbb{P}, Vect] \otimes [\mathbb{P}, Vect] \to [\mathbb{P}, Vect]
  3. Coaddition: Form the composite functor

    [,Vect][×,Vect][,Vect][,Vect] [\mathbb{P}, Vect] \to [\mathbb{P} \times \mathbb{P} , Vect] \to [\mathbb{P}, Vect] \otimes [\mathbb{P}, Vect]

    where the first arrow comes from precomposition with the addition operation on \mathbb{P} (a restriction of coproduct in FinSet), and the second comes from our hoped-for relation (1). This is our coaddition:

    coadd:[,Vect][,Vect][,Vect] coadd: [\mathbb{P}, Vect] \to [\mathbb{P}, Vect] \otimes [\mathbb{P}, Vect]
  4. Comultiplication: Form the composite functor

    [,Vect][×,Vect][,Vect][,Vect] [\mathbb{P}, Vect] \to [\mathbb{P} \times \mathbb{P} , Vect] \to [\mathbb{P}, Vect] \otimes [\mathbb{P}, Vect]

    where the first arrow comes from precomposition with the multiplication operation on \mathbb{P} (a restriction of product in FinSet), and the second comes from our hoped-for relation (1). This is our comultiplication:

    comult:[,Vect][,Vect][,Vect] comult: [\mathbb{P}, Vect] \to [\mathbb{P}, Vect] \otimes [\mathbb{P}, Vect]

The additive and multiplicative unit and counit may be similarly defined. Note that we are using rather little about \mathbb{P} and VectVect here. For example, the category of ordinary non-linear species, [,Set][\mathbb{P}, Set], should also become a categorified biring if there is a tensor product of cocomplete categories with properties analogous to those assumed for cocomplete kk-linear categories above. But we could also replace \mathbb{P} by any rig category. So, ‘biring categories’, or more precisely ‘birig categories’, should be fairly common.

References

  • D. Tall and G. Wraith, Representable functors and operations on rings, Proc. London Math. Soc. 3 (1970), 619–643.
Revised on July 20, 2012 13:48:22 by Stephan Alexander Spahn (79.219.126.169)