This is a paper that John Baez and Todd Trimble are writing, a kind of continuation of Schur functors I.
Note: this is to be rewritten, pending further investigation into the ideas summarized in the first two paragraphs.
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 $Schur$ 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 $Schur$. That is our aim here.
In the usual treatment of Schur functors, five monoidal structures are studied on the category $Schur$. Four of these come from the fact that $Schur$ is equivalent to the functor category
where $\mathbb{P}$ is the ‘permutation groupoid’ (a skeleton of the groupoid of finite sets) and $Fin\Vect$ is the category of finite-dimensional vector space over a field $k$ of characteristic zero. Direct sum and tensor product in $\Fin\Vect$ thus give $Schur$ two monoidal structures: given functors $F,G : \mathbb{P} \to \Fin\Vect$ 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 $Schur$ into a rig category.
But $Schur$ 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 $Schur$ 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 $Schur$ has the structure of a biring category.
On the other hand, the category $Schur$ is equivalent to a certain subcategory of endofunctors $F: \Fin\Vect \to \Fin\Vect$, the so-called ‘Schur functors’. The fact that Schur functors are closed under composition gives $Schur$ a fifth monoidal structure: the ‘plethystic tensor product’. This makes $Schur$ 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 $Vect$. This suggests that the plethory structure of $\Lambda$ may emerge naturally from a categorified plethor structure on $[\mathbb{P}, Vect ]$. 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 $\Comm\Ring^{op}$, 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 $[\mathbb{P}, Vect]$:
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 $[\mathbb{P}, Vect]$.
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 $\Fin\Set$), 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 $\Fin\Set$), 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 $Vect$ here. For example, the category of ordinary species, $[\mathbb{P}, Set]$, should also become a categorified biring if there is a tensor product of cocomplete categories with properties analogous to those assumed for cocomplete $k$-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 $[\mathbb{P}, Vect]$ 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.
(…)
Just getting it down for now; a nice exposition can come later…
There are several ways of describing the notion of biring.
Let $T: Set \to Set$ be a monad, with category of algebras denoted $Set^T$. Let $U: Set^T \to Set$ be the forgetful functor. Then the following conditions on a functor $G: Set^T \to Set^T$ are equivalent:
$G$ has a left adjoint,
$U G: Set^T \to Set$ has a left adjoint,
$U G: Set^T \to Set$ is representable.
If $U G$ is presented by $R$, or more precisely, if we are given an isomorphism $\hom(R, -) \cong U G$, then $G$ considered as a lift of $\hom(R, -)$ through $U$ amounts precisely to a $T$-bialgebra structure on $R$. Hence $T$-bialgebras are equivalent to right adjoint functors $G: Set^T \to Set^T$.
Of course $U: Set^T \to Set$ has a left adjoint $F: Set \to Set^T$ (the free functor), so if $G$ has a left adjoint $H$, then $U G$ has left adjoint $$H F$. Thus (1) implies (2). Next, if $U G$ has a left adjoint $K$, then $U G$ is represented by $K(1)$ since
Thus (2) implies (3). Conversely, (3) implies (2) because given an isomorphism $\theta: \hom(R, -) \to U G$, we have natural isomorphisms
so that $- \cdot R$ is left adjoint to $U G$. Finally, to show (2) implies (1), suppose $U G$ has a left adjoint $W = W_R = - \cdot R$. We must show that each functor $\hom(S, G-)$ is representable. Each object $S$ of $Set^T$ has a canonical presentation as a coequalizer
The canonical structure $T U G \to U G$ is mated to a map $\beta: W T \to W$. Consider the coequalizer
This $Y$ represents $\hom(S, G-)$: there is an isomorphism $\hom(Y, -) \cong \hom(S, G-)$. This is a special case of the adjoint lifting theorem. (I’ll come back to this later.)
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.