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 Λ\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 Λ\Lambda into is a monoid object. Borger and Wieland have dubbed a monoid object in the category of birings a ‘plethory’.

But what is Λ\Lambda, and why is it a plethory? While Λ\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 SchurSchur This fact not only allows us to efficiently obtain the plethory structure on Λ\Lambda, it allows us to ‘categorify’ all this structure: that is, to see it as present in the category SchurSchur. That is our aim here.

In the usual treatment of Schur functors, five monoidal structures are studied on the category SchurSchur. Four of these come from the fact that SchurSchur is equivalent to the functor category

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

where \mathbb{P} is the ‘permutation groupoid’ (a skeleton of the groupoid of finite sets) and FinVectFin\Vect is the category of finite-dimensional vector space over a field kk of characteristic zero. Direct sum and tensor product in FinVect\Fin\Vect thus give SchurSchur two monoidal structures: given functors F,G:FinVectF,G : \mathbb{P} \to \Fin\Vect we may define

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

and

(FG)(V)=F(V)G(V) (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 SchurSchur into a rig category.

But SchurSchur 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 SchurSchur 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 SchurSchur has the structure of a biring category.

On the other hand, the category SchurSchur is equivalent to a certain subcategory of endofunctors F:FinVectFinVectF: \Fin\Vect \to \Fin\Vect, the so-called ‘Schur functors’. The fact that Schur functors are closed under composition gives SchurSchur a fifth monoidal structure: the ‘plethystic tensor product’. This makes SchurSchur into a categorified plethory, which in turn makes Λ\Lambda into a plethory.

Now, our use of finite-dimensional vector spaces above is somewhat arbitrary: 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 VectVect. This suggests that the plethory structure of Λ\Lambda may emerge naturally from a categorified plethor structure on [,Vect][\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 RR equipped with ring homomorphisms called coaddition:

coadd:RRR coadd: R \to R \otimes R

cozero:

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

co-additive inverse:

cominus:RR cominus: R \to R

comultiplication:

comult:RRR comult: R \to R \otimes R

and the multiplicative counit:

counit:R 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 CommRing op\Comm\Ring^{op}, 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

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 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×B,Vect][A,Vect][B,Vect] [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 [,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 the coproduct in FinSet\Fin\Set), 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 the product in FinSet\Fin\Set), 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 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.

The subtler features of [,Vect][\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 Λ\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:SetSetT: Set \to Set be a monad, with category of algebras denoted Set TSet^T. Let U:Set TSetU: Set^T \to Set be the forgetful functor. Then the following conditions on a functor G:Set TSet TG: Set^T \to Set^T are equivalent:

  1. GG has a left adjoint,

  2. UG:Set TSetU G: Set^T \to Set has a left adjoint,

  3. UG:Set TSetU G: Set^T \to Set is representable.

If UGU G is presented by RR, or more precisely, if we are given an isomorphism hom(R,)UG\hom(R, -) \cong U G, then GG considered as a lift of hom(R,)\hom(R, -) through UU amounts precisely to a TT-bialgebra structure on RR. Hence TT-bialgebras are equivalent to right adjoint functors G:Set TSet TG: Set^T \to Set^T.

Proof

Of course U:Set TSetU: Set^T \to Set has a left adjoint F:SetSet TF: Set \to Set^T (the free functor), so if GG has a left adjoint HH, then UGU G has left adjoint $HFH F. Thus (1) implies (2). Next, if UGU G has a left adjoint KK, then UGU G is represented by K(1)K(1) since

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

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

hom(XR,Y)hom(R,Y) XUG(Y) Xhom(X,UG(Y))\hom(X \cdot R, Y) \cong \hom(R, Y)^X \cong U G(Y)^X \cong \hom(X, U G(Y))

so that R- \cdot R is left adjoint to UGU G. Finally, to show (2) implies (1), suppose UGU G has a left adjoint W=W R=RW = W_R = - \cdot R. We must show that each functor hom(S,G)\hom(S, G-) is representable. Each object SS of Set TSet^T has a canonical presentation as a coequalizer

FUFUSFUϵSϵFUSFUSstackrelϵSS.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 TUGUGT U G \to U G is mated to a map β:WTW\beta: W T \to W. Consider the coequalizer

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

This YY represents hom(S,G)\hom(S, G-): there is an isomorphism hom(Y,)hom(S,G)\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.

Revised on August 15, 2014 02:28:48 by Todd Trimble? (67.81.95.215)