nLab
Tall-Wraith monoid

Tall–Wraith monoids

Idea

Given an algebraic theory VV, a VV-algebra is a model of VV in the category SetSet. A Tall–Wraith VV-monoid is the kind of thing that acts on VV-algebras.

Definition

Definition

Let VV be an algebraic theory and let VAlgV Alg be the category of models of this theory in SetSet. Then a Tall–Wraith VV-monoid is a monoid object in the category of co-VV-objects in VAlgV Alg.

To see why these are what acts on VV-algebras one needs to understand what a co-VV-object in VAlgV Alg actually is. A co-VV-object in some category DD is a representable covariant functor from DD to VAlgV Alg. To give a particular DD-object, dd, the structure of a co-VV-object is to give a lift of the SetSet-valued HomHom-functor D(d,)D(d,-) to VAlgV Alg. Thus a co-VV-object in VAlgV Alg is a representable covariant functor from VAlgV Alg to itself.

One can therefore consider composition of such representable covariant functors. The main result of this can be simply stated:

Proposition

The composition of representable covariant functors VAlgVAlgV Alg \to V Alg is again representable.

This is a basic result in general algebra, and is not stated here in its full generality.

An almost corollary of this is that the category of representable covariant functors from VAlgV Alg to itself is monoidal (the “almost” refers to the fact that you have to show that the identity functor is representable, but this is not hard).

Thus for two co-VV-algebra objects in VV, say R 1R_1 and R 2R_2, there is a product R 1R 2R_1 \odot R_2 and a natural isomorphism

Hom V(R 1R 2,A)Hom V(R 1,Hom V(R 2,A)) Hom_V(R_1 \odot R_2,A) \cong Hom_V(R_1, Hom_V(R_2,A))

for any VV-algebra, AA.

A Tall–Wraith VV-monoid is thus a triple (P,μ,η)(P,\mu,\eta) with μ:PPP\mu : P \odot P \to P and η:IP\eta : I \to P (where II is the free VV-algebra on one element — this represents the identity functor), satisfying the obvious coherence diagrams. An action of PP on a VV-algebra, say AA, is then a morphism ρ:PAA\rho : P \odot A \to A again satisfying certain coherence diagrams.

Ah, but I have not told you what PAP \odot A is! At the moment, one can take the “product” of two co-VV-algebra objects in VAlgV Alg but now I want to take the product of a co-VV-algebra object with a VV-algebra. How do I do this? I do this by observing that a VV-algebra is a co-SetSet-algebra object in VAlgV Alg! That’s a complicated way of saying that VV represents a covariant functor VAlgSetV Alg \to Set. Precomposing this with the functor represented by PP yields again a covariant functor VAlgSetV Alg \to Set. This is again representable and we write its representing object PAP \odot A.

As an aside, we note a consequence. As we’ve seen, the category of co-VV-algebra objects in VV is a monoidal category, with the tensor product \odot. Now we’re seeing this monoidal category acts on the category of VV-algebras. Indeed, it acts on the categories of VV-algebra and co-VV-algebra objects in a reasonably arbitrary base category.

One postscript to this is that although the category of co-VV-algebra objects in VAlgV Alg is not a variety of algebras, for a specific Tall–Wraith VV-monoid PP, the category of PP-modules is a variety of algebras.

Examples

  • If VV is the theory of commutative unital rings, a VV-algebra is a commutative unital ring, a co-VV-algebra object in VV is a biring and the corresponding sort of Tall–Wraith VV-monoid is called, in Tall and Wraith’s original paper, a biring triple.

  • If VV is the theory of commutative associative algebras over a field kk, then a VV-algebra is a commutative associative algebra over kk, and the corresponding sort of Tall–Wraith VV-monoid is called a plethory.

  • If VV is the theory of abelian groups, than a VV-algebra is an abelian group, and the corresponding sort of Tall–Wraith VV-monoid is a ring.

    To understand the last example, we need to think about co-abelian group objects in the category of abelian groups. Abstractly, such a thing is an abelian group object internal to AbGp opAbGp^{op} (though this picture gets the morphisms the wrong way around; in full abstraction then the category of co-abelian group objects in AbGrpAbGrp is the opposite category of the category of abelian group objects in AbGrp opAbGrp^{op}). Concretely, such a thing is an abelian group AA together with group homomorphisms

    μ :AAA, ϵ :AI ι :AA \begin{aligned} \mu &: A \to A \coprod A, \\ \epsilon &: A \to I \\ \iota &: A \to A \end{aligned}

    where II is the initial object in the category of abelian groups. These homomorphisms must satisfy certain laws: just the abelian group axioms written out diagrammatically, with all the arrows turned around.

    In fact, I={0}I = \{0\}. Thus ϵ\epsilon is forced to be the map that sends everything to 00: we have no choice here.

    We also have that AA=AAA \coprod A = A \oplus A. That means that for aAa \in A, μ(a)=(a 1,a 2)\mu(a) = (a_1,a_2) for some a 1,a 2Aa_1, a_2 \in A. Now, one of the laws says that ϵ\epsilon is a counit for μ\mu. This means that (ϵ1)μ=1(\epsilon \oplus 1) \mu = 1 and similarly for 1ϵ1 \oplus \epsilon. Thus a 1=a 2=aa_1 = a_2 = a and μ\mu is the diagonal map. So, we have no choice here either.

    The diagram for ι\iota (representing the inverse map) is a little more complicated. As II is the initial object in AbGrpAbGrp, there is a unique morphism IAI \to A (inclusion of the zero). Composing this with ϵ\epsilon yields a morphism AAA \to A which maps every element to the zero in AA. Using μ\mu and ι\iota we can construct another morphism AAA \to A as

    AμAA1ιAAΔ cA A \overset{\mu}\rightarrow A \coprod A \overset{1 \coprod \iota}\rightarrow A \coprod A \overset{\Delta^c}\rightarrow A

    where Δ c\Delta^c is the co-diagonal. The relations for abelian groups say that this morphism must be the same as the zero morphism AAA \to A. Using the fact that AAAAA \coprod A \cong A \oplus A and that μ\mu is the diagonal, this says that a+ι(a)=0a + \iota(a) = 0. Hence, by the uniqueness of inverses for abelian groups, ι(a)=a\iota(a) = -a.

    Thus if (A,μ,ι,ϵ)(A, \mu, \iota, \epsilon) is a co-abelian group object in AbGrpAbGrp then μ\mu is the diagonal, ι\iota the inverse from abelian groups, and ϵ\epsilon the zero morphism.

    However, that is still not quite the same as saying that (A,μ,ι,ϵ)(A, \mu, \iota, \epsilon) is a co-abelian group object in AbGrpAbGrp. Certainly, (A,μ,ϵ)(A, \mu, \epsilon) is a co-commutative co-monoid object in AbGrpAbGrp since μ\mu is the diagonal, which is automatically co-commutative and co-associative, and ϵ\epsilon the zero map, which is the co-unit for the diagonal. What remains is to fit ι\iota into the structure.

    The first issue is that ι\iota is not automatically a morphism in AbGrpAbGrp. That is to say, when defining an algebraic theory then the operations are defined on the underlying objects. It is a consequence of the relations of abelian groups that the operations lift to morphisms of abelian groups (algebraic theories where this happens for all operations are sometimes called commutative). Thus ι\iota is a morphism of abelian groups and so (A,μ,ι,ϵ)(A, \mu, \iota, \epsilon) is a co-commutative co-monoid with an extra unary co-operation. In fact, it is an involution from the relations for abelian groups.

    The final relation is that ι\iota is the inverse for μ\mu. The relation that ι\iota is the inverse for addition (let us write it as, say, α\alpha) is that

    AΔA×A1×ιA×AαA A \overset{\Delta}\rightarrow A \times A \overset{1 \times \iota}\rightarrow A \times A \overset{\alpha}\rightarrow A

    is the zero map AIAA \to I \to A. This is precisely the relation that ι\iota is the inverse for μ\mu since we have the following identifications: μ=Δ\mu = \Delta, AA=A×AA \coprod A = A \times A, and Δ c=α\Delta^c = \alpha. Also, ϵ=0\epsilon = 0 and η:IA\eta : I \to A is the initial morphism in AbGrpAbGrp.

    Thus the fact that ι\iota is the inverse for the diagonal+zero co-monoidal structure is due to the fact that ι\iota is the inverse for (α,η)(\alpha,\eta) and α:AAA\alpha : A \oplus A \to A is the co-diagonal in AbGrpAbGrp and η:IA\eta : I \to A is the unit.

    It is part of the general theory that the category of co-VV-objects in VV is monoidal (though not, in general, symmetric). For details on this see The Hunting of the Hopf Ring, referred to belelow. This monoidal structure for abelian groups turns out to be the tensor product.

    Thus a Tall–Wraith monoid for abelian groups is actually an ordinary monoid in the category of abelian groups: in other words, a ring!

References

  • D. Tall, G. Wraith, Representable functors and operations on rings, Proc. London Math. Soc. (3), 1970, 619–643, MR265348, doi

  • J. Borger, B. Wieland, Plethystic algebra, Adv. Math. 194 (2005), no. 2, 246–283, doi, pdf, MR2006i:13044

  • A. Stacey and S. Whitehouse, The Hunting of the Hopf Ring, Homology, Homotopy and Applications 11(2), 2009, 75–132, online, arXiv/0711.3722.

An old and long query-discussion has been archived starting here.

Revised on December 9, 2011 00:39:33 by Andrew Stacey (80.203.115.55)