Tall-Wraith monoid

Tall–Wraith monoids


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.



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:


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 (although see below for some reasonably general constructions).

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.


  • 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!

General results and constructions

We now recapitulate the discussion above in a slightly more general context.

For now our context is that of monads TT on Set. The category of TT-algebras is denoted Set TSet^T, with forgetful functor U:Set TSetU: Set^T \to Set and free functor F:SetSet TF: Set \to Set^T, whose composite is the monad T=UFT = U F, and whose counit is denoted ϵ:FU1 Set T\epsilon: F U \to 1_{Set^T}.

For each TT-algebra RR, there is an adjoint pair of functors

Rhom(R,):Set TSet- \cdot R \dashv \hom(R, -): Set^T \to Set

with associated monad hom(R,R)\hom(R, - \cdot R). The functor R:SetSet T- \cdot R: Set \to Set^T takes a set XX to the TT-algebra XRX \cdot R, an XX-indexed coproduct of copies of RR in Set TSet^T. We define a TT-bialgebra to be a TT-algebra RR equipped with a morphism of monads ϕ:Thom(R,R)\phi: T \to \hom(R, -\cdot R). By the adjunction, the datum ϕ\phi is equivalent to a left TT-algebra structure

α:Thom(R,)hom(R,)\alpha: T \circ \hom(R, -) \to \hom(R, -)

on hom(R,):Set TSet\hom(R, -): Set^T \to Set, thus giving a lifting denoted (by abuse of language) hom(R,):Set TSet T\hom(R, -): Set^T \to Set^T. This datum is also equivalent to a right TT-algebra (aka right TT-module) structure

β:WTW\beta: W T \to W

where W=W RR:SetSet TW = W_R \coloneqq - \cdot R: Set \to Set^T. A TT-bialgebra map is a TT-algebra map f:RSf: R \to S such that the induced map W f:W RW SW_f: W_R \to W_S is a morphism of right TT-modules.


A good case to keep in mind is that of birings, which are TT-bialgebras for the Lawvere theory TT of commutative rings. The monad morphism Thom(R,R)T \to \hom(R, -\cdot R) has components TXhom(R,XR)T X \to \hom(R, X \cdot R) for each set XX. Here XRX \cdot R is an XX-indexed coproduct of copies of RR, where coproduct in the category of commutative rings Set TSet^T is given by tensor product. Thus, for example, 2R2 \cdot R is the ring RRR \otimes R. The component T(2)hom(R,2R)T(2) \to \hom(R, 2\cdot R) therefore “interprets” each element θT(2)\theta \in T(2), i.e., each binary operation in the Lawvere theory, as a binary co-operation RRRR \to R \otimes R. This applies in particular to the elements m,aT(2)m, a \in T(2) which abstractly represent multiplication and addition (seen as natural operations on the category of commutative rings).


Let Ladj(Set T,Set T)Ladj(Set^T, Set^T) (Radj(Set T,Set T)Radj(Set^T, Set^T)) be the category of left (right) adjoint functors Ψ:Set TSet T\Psi: Set^T \to Set^T. The functor TT-BiAlgLadj(Set T,Set T)BiAlg \to Ladj(Set^T, Set^T) that takes (R,ϕ)(R, \phi) to the right TT-module (W R,β)(W_R, \beta) is an equivalence. Or, what is the same, the functor TT-BiAlgRadj(Set T,Set T) opBiAlg \to Radj(Set^T, Set^T)^{op}, taking (R,ϕ)(R, \phi) to the left TT-algebra hom(R,),α)\hom(R, -), \alpha), is an equivalence.

Sketch of proof

The main thing to check is that the functor Rhom(R,)R \mapsto \hom(R, -) to Radj(Set T,Set T)Radj(Set^T, Set^T) is essentially surjective. The essential point is that Φ\Phi has a left adjoint iff UΦU \Phi has a left adjoint iff UΦ:Set TSetU \Phi: Set^T \to Set is representable: UΦhom(R,)U \Phi \cong \hom(R, -) for some TT-algebra RR (in which case the lift Φ\Phi of hom(R,)\hom(R, -) through UU is tantamount to a TT-algebra structure on hom(R,)\hom(R, -)). The only (mildly) tricky part is that Φ\Phi has a left adjoint if UΦU\Phi has a left adjoint W=W RW = W_R. To define the left adjoint Ψ\Psi of Φ\Phi objectwise, we take any TT-algebra SS with its canonical presentation

FUFUSFUϵϵFUSFUSϵSF U F U S \stackrel{\overset{\epsilon F U S}{\to}}{\underset{F U \epsilon}{\to}} F U S \stackrel{\epsilon}{\to} S

which is a coequalizer diagram. A left adjoint Ψ\Psi must preserve this coequalizer, and we must have ΨFW\Psi F \cong W since both sides are left adjoint to UΦU \Phi. Thus we define Ψ(S)\Psi (S) to be a coequalizer

W(TUS)WUϵSβUSW(US)Ψ(S)W(T U S) \stackrel{\overset{\beta U S}{\to}}{\underset{W U\epsilon S}{\to}} W(U S) \to \Psi(S)

where β:WTT\beta: W T \to T is the TT-module structure coming from the monad morphism ϕ:Thom(R,R)\phi: T \to \hom(R, -\cdot R). This objectwise definition of Ψ\Psi easily extends to morphisms by universality and provides a left adjoint to Φ\Phi. Remaining details are left to the reader.

The import of this proposition is that left adjoint endofunctors on Set TSet^T compose, i.e., endofunctor composition gives a monoidal structure on Ladj(Set T,Set T)Ladj(Set^T, Set^T), and this monoidal structure transports across the categorical equivalence of the proposition to give a monoidal structure on TT-BiAlgBiAlg. The resultant monoidal product on TT-bialgebras is denoted \odot.

A direct construction of \odot can be extracted by following the proof of the proposition. If R,SR, S are TT-bialgebras, then the underlying TT-algebra of SRS \odot R (corresponding to composition of hom(S,)hom(R,)\hom(S, -) \circ \hom(R, -) of right adjoints Set TSet TSet^T \to Set^T) is computed as a reflexive coequalizer in Set TSet^T:

TUSRβUSϵUSRUSRSR.T U S \cdot R \stackrel{\overset{\epsilon U S \cdot R}{\to}}{\underset{\beta U S}{\to}} U S \cdot R \to S \odot R.

Here ϵUS\epsilon U S is the same as the TT-algebra structure TUStUST U S \t U S on SS. Whereas βX:TXRXR\beta X: T X \cdot R \to X \cdot R is a component of the TT-module structure W RTW RW_R T \to W_R; it is mated by the Rhom(R,)- \cdot R \dashv \hom(R, -) adjunction to the component of the coalgebra structure ϕX:TXhom(R,XR)\phi X: T X \to \hom(R, X \cdot R).

To extract the TT-coalgebra structure on SRS \odot R, let us observe generally that if F:CDF: C \to D is a left adjoint, then for any category EE there is an induced left adjoint [1 E,F]:[E,C][E,D][1_E, F]: [E, C] \to [E, D] and similarly an induced left adjoint Ladj(E,C)Ladj(E,D)Ladj(E, C) \to Ladj(E, D). Applying this to the case where C=D=E=Set TC = D = E = Set^T and where FF is the left adjoint to the lift hom(R,):Set TSet T\hom(R, -): Set^T \to Set^T, we find that

R:Ladj(Set T,Set T)Ladj(Set T,Set T)- \odot R: Ladj(Set^T, Set^T) \to Ladj(Set^T, Set^T)

is a left adjoint, and in particular preserves YY-indexed copowers YSY \cdot S. In other words, for each XX we have canonical isomorphisms

X(SR)(XS)R,TX(SR)(TXS)RX \cdot (S \odot R) \cong (X \cdot S) \odot R, \qquad T X \cdot (S \odot R) \cong (T X \cdot S) \odot R

so that the desired right TT-module structure is given componentwise by a composite

β(SR) X(TX(SR)(TXS)R(βS) XR(XS)RX(SR)).\beta (S \odot R)_X \coloneqq \left(T X \cdot (S \odot R) \cong (T X \cdot S) \odot R \stackrel{(\beta S)_X \odot R}{\to} (X \cdot S) \odot R \cong X \cdot (S \odot R)\right).


  • 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 July 28, 2014 08:55:40 by Todd Trimble (