# 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 $R$ equipped with ring homomorphisms called coaddition:

$R\to R\otimes R$R \to R \otimes R

cozero:

$R\to ℤ$R \to \mathbb{Z}

$R\to R$R \to R

comultiplication:

$R\to R\otimes R$R \to R \otimes R

and the multiplicative counit:

$R\to ℤ$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 $R$ equipped with a lift of the functor

$\mathrm{hom}\left(R,-\right):\mathrm{CommRing}\to \mathrm{Set}$hom(R, -) : CommRing \to Set

to a functor

$\mathrm{hom}\left(R,-\right):\mathrm{CommRing}\to \mathrm{CommRing}$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 $\Lambda$, the ring of symmetric polynomials. This is actually a plethory.

## Categorified Birings

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

$\left[ℙ,\mathrm{FinVect}\right]$[\mathbb{P}, FinVect]

where $ℙ$ is the permutation groupoid and $\mathrm{FinVect}$ is the category of finite-dimensional vector spaces over a field $k$ of characteristic zero. $\Lambda$ is also the Grothendieck group of

$\left[ℙ,\mathrm{Vect}\right]$[\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 $\Lambda$ may emerge naturally from a ‘categorified biring’ structure on $\left[ℙ,\mathrm{Vect}\right]$. 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 $X$ and $Y$, there is a cocomplete linear category $X\otimes Y$ such that:

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

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

With any luck these two assumptions will let us show that for any categories $A$ and $B$,

(1)$\left[A×Y,\mathrm{Vect}\right]\simeq \left[A,\mathrm{Vect}\right]\otimes \left[B,\mathrm{Vect}\right]$[A \times Y, Vect] \simeq [A,Vect] \otimes [B, Vect]

where we use $\left[-,-\right]$ to denote the functor category.

Assuming all this, we obtain the following operations on the category $\left[ℙ,\mathrm{Vect}\right]$:

1. Addition: form the composite functor

$\left[ℙ,\mathrm{Vect}\right]×\left[ℙ,\mathrm{Vect}\right]\to \left[ℙ,\mathrm{Vect}×\mathrm{Vect}\right]\to \left[ℙ,\mathrm{Vect}\right]$[\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

$\oplus :\mathrm{Vect}×\mathrm{Vect}\to \mathrm{Vect}$\oplus : Vect \times Vect \to Vect

$\oplus :\left[ℙ,\mathrm{Vect}\right]×\left[ℙ,\mathrm{Vect}\right]\to \left[ℙ,\mathrm{Vect}\right]$\oplus : [\mathbb{P}, Vect] \times [\mathbb{P}, Vect] \to [\mathbb{P}, Vect]

It’s really just the coproduct in $\left[ℙ,\mathrm{Vect}\right]$.

2. Multiplication: first form the composite functor

$\left[ℙ,\mathrm{Vect}\right]×\left[ℙ,\mathrm{Vect}\right]\to \left[ℙ,\mathrm{Vect}×\mathrm{Vect}\right]\to \left[ℙ,\mathrm{Vect}\right]$[\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

$\otimes :\mathrm{Vect}×\mathrm{Vect}\to \mathrm{Vect}$\otimes : Vect \times Vect \to Vect

This composite is our multiplication:

$\otimes :\left[ℙ,\mathrm{Vect}\right]×\left[ℙ,\mathrm{Vect}\right]\to \left[ℙ,\mathrm{Vect}\right]$\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

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

$\left[ℙ,\mathrm{Vect}\right]\to \left[ℙ×ℙ,\mathrm{Vect}\right]\to \left[ℙ,\mathrm{Vect}\right]\otimes \left[ℙ,\mathrm{Vect}\right]$[\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 $ℙ$ (a restriction of coproduct in FinSet), and the second comes from our hoped-for relation (1). This is our coaddition:

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

$\left[ℙ,\mathrm{Vect}\right]\to \left[ℙ×ℙ,\mathrm{Vect}\right]\to \left[ℙ,\mathrm{Vect}\right]\otimes \left[ℙ,\mathrm{Vect}\right]$[\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 $ℙ$ (a restriction of product in FinSet), and the second comes from our hoped-for relation (1). This is our comultiplication:

$\mathrm{comult}:\left[ℙ,\mathrm{Vect}\right]\to \left[ℙ,\mathrm{Vect}\right]\otimes \left[ℙ,\mathrm{Vect}\right]$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 $ℙ$ and $\mathrm{Vect}$ here. For example, the category of ordinary non-linear species, $\left[ℙ,\mathrm{Set}\right]$, should also become a categorified biring if there is a tensor product of cocomplete categories with properties analogous to those assumed for cocomplete $k$-linear categories above. But we could also replace $ℙ$ 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)