John Baez Schur functors II

Preface

This is a paper that John Baez and Todd Trimble are writing, a kind of continuation of Schur functors I.

Introduction

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

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

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

$$(F \oplus G)(V) = F(V) \oplus G(V)$$

and

$$(F \otimes G)(V) = F(V) \otimes G(V)$$

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

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

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 plethory 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.

Birings and plethories

First, recall that a biring is a commutative ring $R$ equipped with ring homomorphisms called coaddition:

$$coadd: R \to R \otimes R$$

cozero:

$$cozero: R \to \mathbb{Z}$$

co-additive inverse:

$$cominus: R \to R$$

comultiplication:

$$comult: R \to R \otimes R$$

and the multiplicative counit:

$$counit: R \to \mathbb{Z}$$

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

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

to a functor

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

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.

Categorified birings

A first attempt

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 $B$,

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

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

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

1. Addition: form the composite functor

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

   where the last arrow comes from postcomposition with

   $$\oplus : Vect \times Vect \to Vect$$

   This composite is our addition:

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

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

2. Multiplication: first form the composite functor

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

   where the last arrow comes from postcomposition with

   $$\otimes : Vect \times Vect \to Vect$$

   This composite is our multiplication:

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

   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

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

3. Coaddition: Form the composite functor

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

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

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

4. Comultiplication: Form the composite functor

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

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

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

The additive and multiplicative unit and counit may be similarly defined. Note that we are using rather little about $\mathbb{P}$ and $Vect$ 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.

(…)

A second attempt

Just getting it down for now; a nice exposition can come later…

There are several ways of describing the notion of biring.

Proposition

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:

1. $G$ has a left adjoint,

2. $U G: Set^T \to Set$ has a left adjoint,

3. $U G: Set^T \to Set$ 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 $G: Set^T \to Set^T$.

Proof

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

$$\hom(K(1), -) \cong \hom(1, U G-) \cong U G.$$

Thus (2) implies (3). Conversely, (3) implies (2) because given an isomorphism $\theta: \hom(R, -) \to U G$, we have natural isomorphisms

$$\hom(X \cdot R, Y) \cong \hom(R, Y)^X \cong U G(Y)^X \cong \hom(X, U G(Y))$$

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

$$F U F U S \stackrel{\overset{\epsilon F U S}{\to}}{\underset{F U \epsilon S}{\to}} F U S stackrel{\epsilon S}{\to} S.$$

The canonical structure $T U G \to U G$ is mated to a map $\beta: W T \to W$. Consider the coequalizer

$$W T U S \stackrel{\overset{\beta U S}{\to}}{\underset{W\alpha S}{\to}} W U S \to Y.$$

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.)

References

• 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.