This is a paper that John Baez and Todd Trimble are writing, a kind of continuation of Schur functors I.
The ring of symmetric functions, commonly called $\Lambda $, has a wealth of fascinating structure. It is not only a commutative ring, but also a ‘cocommutative coring’, and these structures are compatible in way that makes it into a ‘biring’. As noted by Tall and Wraith (ref), the category of birings has an interesting monoidal structure that makes $\Lambda $ into is a monoid object. Borger and Wieland have dubbed a monoid object in the category of birings a ‘plethory’.
But what is $\Lambda $, and why is it a plethory? While $\Lambda $ may be defined using generators and relations, a more elegant definition is also possible: it is the Grothendieck group of the category of ‘Schur functors’, which we denote by $\mathrm{Schur}$ This fact not only allows us to efficiently obtain the plethory structure on $\Lambda $, it allows us to ‘categorify’ all this structure: that is, to see it as present in the category $\mathrm{Schur}$. That is our aim here.
In the usual treatment of Schur functors, five monoidal structures are studied on the category $\mathrm{Schur}$. Four of these come from the fact that $\mathrm{Schur}$ is equivalent to the functor category
where $\mathbb{P}$ is the ‘permutation groupoid’ (a skeleton of the groupoid of finite sets) and $\mathrm{Fin}Vect$ is the category of finite-dimensional vector space over a field $k$ of characteristic zero. Direct sum and tensor product in $FinVect$ thus give $\mathrm{Schur}$ two monoidal structures: given functors $F,G:\mathbb{P}\to FinVect$ we may define
and
The first is usually called the ‘direct sum’ of Schur functors, while the second has been called the ‘Hadamard product’ (ref). Together, they make $\mathrm{Schur}$ into a rig category.
But $\mathrm{Schur}$ acquires two more monoidal structures coming from $+$ and $\times $ in $\mathbb{P}$, with the help of a device known as ‘Day convolution’ (ref). (Here $+$ means the monoidal structure coming from coproduct of finite sets, while $\times $ means the monoidal structure coming from product; of course these are not the product and coproduct in the groupoid $\mathbb{P}$.) The structure coming from $+$ is usually called ‘multiplication’ of linear species, or ‘Cauchy product’, while the structure coming from $\times $ appears to have no standard name. Together, these additional monoidal structure make $\mathrm{Schur}$ into a rig in a second way.
One of our innovations is to treat the latter two monoidal structures as ‘co-operations’ rather than operations. This reveals that $\mathrm{Schur}$ has the structure of a biring category.
On the other hand, the category $\mathrm{Schur}$ is equivalent to a certain subcategory of endofunctors $F:FinVect\to FinVect$, the so-called ‘Schur functors’. The fact that Schur functors are closed under composition gives $\mathrm{Schur}$ a fifth monoidal structure: the ‘plethystic tensor product’. This makes $\mathrm{Schur}$ into a categorified plethory, which in turn makes $\Lambda $ into a plethory.
Now, our use of finite-dimensional vector spaces above is somewhat arbitrary: $\Lambda $ is also the Grothendieck group of
where we drop the finite-dimensionality restriction on our vector spaces and work with all of $\mathrm{Vect}$. This suggests that the plethory structure of $\Lambda $ may emerge naturally from a categorified plethor structure on $[\mathbb{P},\mathrm{Vect}]$. In the following sections 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.
First, recall that a biring is a commutative ring $R$ equipped with ring homomorphisms called coaddition:
cozero:
co-additive inverse:
comultiplication:
and the multiplicative counit:
satisfying the usual axioms of a commutative ring, but ‘turned around’.
More tersely, but also more precisely, a biring is a commutative ring object in the category $Comm{Ring}^{\mathrm{op}}$, also known as the category of ‘affine schemes’.
Equivalently, a biring is a commutative ring $R$ equipped with a lift of the functor
to a functor
As noted by Tall and Wraith (ref), 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.
Let us 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 $i:X\times Y\to X\otimes Y$ 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 $X\times Y\to Z$ 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 $B$,
where we use $[-,-]$ to denote the functor category.
Assuming all this, we obtain the following operations on the category $[\mathbb{P},\mathrm{Vect}]$:
Addition: form the composite functor
where the last arrow comes from postcomposition with
This composite is our addition:
It’s really just the coproduct in $[\mathbb{P},\mathrm{Vect}]$.
Multiplication: first form the composite functor
where the last arrow comes from postcomposition with
This composite is our multiplication:
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
Coaddition: Form the composite functor
where the first arrow comes from precomposition with the addition operation on $\mathbb{P}$ (a restriction of the coproduct in $FinSet$), and the second comes from our hoped-for relation (1). This is our coaddition:
Comultiplication: Form the composite functor
where the first arrow comes from precomposition with the multiplication operation on $\mathbb{P}$ (a restriction of the product in $FinSet$), and the second comes from our hoped-for relation (1). This is our comultiplication:
The additive and multiplicative unit and counit may be similarly defined. Note that we are using rather little about $\mathbb{P}$ and $\mathrm{Vect}$ here. For example, the category of ordinary species, $[\mathbb{P},\mathrm{Set}]$, 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 $\mathbb{P}$ by any rig category. So, ‘biring categories’, or more precisely ‘birig categories’, should be fairly common.
The subtler features of $[\mathbb{P},\mathrm{Vect}]$ arise from special features of $\mathbb{P}$ that allow us to define the ‘plethystic tensor product’ on this category. This, in turn, is what makes $\Lambda $ into a plethory.
J. Borger, B. Wieland, Plethystic algebra, Advances in Mathematics 194 (2005), 246–283. (web)
S. Joni and G. Rota, Coalgebras and bialgebras in combinatorics, Studies in Applied Mathematics 61 (1979), 93-139.
G. Rota, Hopf algebras in combinatorics, in Gian-Carlo Rota on Combinatorics: Introductory Papers and Commentaries, ed. J. P. S. Kung, Birkhauser, Boston, 1995.
D. Tall and G. Wraith, Representable functors and operations on rings, Proc. London Math. Soc. 3 (1970), 619–643.