symmetric monoidal (∞,1)-category of spectra
Given an algebraic theory , a -algebra is a model of in the category . A Tall–Wraith -monoid is the kind of thing that acts on -algebras.
To see why these are what acts on -algebras one needs to understand what a co--object in actually is. A co--object in some category is a representable covariant functor from to . To give a particular -object, , the structure of a co--object is to give a lift of the -valued -functor to . Thus a co--object in is a representable covariant functor from 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 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 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--algebra objects in , say and , there is a product and a natural isomorphism
for any -algebra, .
A Tall–Wraith -monoid is thus a triple with and (where is the free -algebra on one element — this represents the identity functor), satisfying the obvious coherence diagrams. An action of on a -algebra, say , is then a morphism again satisfying certain coherence diagrams.
Ah, but I have not told you what is! At the moment, one can take the “product” of two co--algebra objects in but now I want to take the product of a co--algebra object with a -algebra. How do I do this? I do this by observing that a -algebra is a co--algebra object in ! That’s a complicated way of saying that represents a covariant functor . Precomposing this with the functor represented by yields again a covariant functor . This is again representable and we write its representing object .
As an aside, we note a consequence. As we’ve seen, the category of co--algebra objects in is a monoidal category, with the tensor product . Now we’re seeing this monoidal category acts on the category of -algebras. Indeed, it acts on the categories of -algebra and co--algebra objects in a reasonably arbitrary base category.
One postscript to this is that although the category of co--algebra objects in is not a variety of algebras, for a specific Tall–Wraith -monoid , the category of -modules is a variety of algebras.
If is the theory of commutative unital rings, a -algebra is a commutative unital ring, a co--algebra object in is a biring and the corresponding sort of Tall–Wraith -monoid is called, in Tall and Wraith’s original paper, a biring triple.
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 (though this picture gets the morphisms the wrong way around; in full abstraction then the category of co-abelian group objects in is the opposite category of the category of abelian group objects in ). Concretely, such a thing is an abelian group together with group homomorphisms
where 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, . Thus is forced to be the map that sends everything to : we have no choice here.
We also have that . That means that for , for some . Now, one of the laws says that is a counit for . This means that and similarly for . Thus and is the diagonal map. So, we have no choice here either.
The diagram for (representing the inverse map) is a little more complicated. As is the initial object in , there is a unique morphism (inclusion of the zero). Composing this with yields a morphism which maps every element to the zero in . Using and we can construct another morphism as
where is the co-diagonal. The relations for abelian groups say that this morphism must be the same as the zero morphism . Using the fact that and that is the diagonal, this says that . Hence, by the uniqueness of inverses for abelian groups, .
Thus if is a co-abelian group object in then is the diagonal, the inverse from abelian groups, and the zero morphism.
However, that is still not quite the same as saying that is a co-abelian group object in . Certainly, is a co-commutative co-monoid object in since is the diagonal, which is automatically co-commutative and co-associative, and the zero map, which is the co-unit for the diagonal. What remains is to fit into the structure.
The first issue is that is not automatically a morphism in . 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 is a morphism of abelian groups and so 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 is the inverse for . The relation that is the inverse for addition (let us write it as, say, ) is that
is the zero map . This is precisely the relation that is the inverse for since we have the following identifications: , , and . Also, and is the initial morphism in .
Thus the fact that is the inverse for the diagonal+zero co-monoidal structure is due to the fact that is the inverse for and is the co-diagonal in and is the unit.
It is part of the general theory that the category of co--objects in is monoidal (though not, in general, symmetric). For details on this see The Hunting of the Hopf Ring, referred to below. 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!
We now recapitulate the discussion above in a slightly more general context.
For each -algebra , there is an adjoint pair of functors
with associated monad . The functor takes a set to the -algebra , an -indexed coproduct of copies of in . We define a -bialgebra to be a -algebra equipped with a morphism of monads . By the adjunction, the datum is equivalent to a left -algebra structure
on , thus giving a lifting denoted (by abuse of language) . This datum is also equivalent to a right -algebra (aka right -module) structure
where . A -bialgebra map is a -algebra map such that the induced map is a morphism of right -modules.
A good case to keep in mind is that of birings, which are -bialgebras for the Lawvere theory of commutative rings. The monad morphism has components for each set . Here is an -indexed coproduct of copies of , where coproduct in the category of commutative rings is given by tensor product. Thus, for example, is the ring . The component therefore “interprets” each element , i.e., each binary operation in the Lawvere theory, as a binary co-operation . This applies in particular to the elements which abstractly represent multiplication and addition (seen as natural operations on the category of commutative rings).
We thus have several perspectives on what a -bialgebra is:
A -algebra equipped with a monad morphism ,
A -algebra equipped with a compatible -coalgebra structure (actually the same as the preceding item, but in different words),
A -algebra for which is provided with a left -algebra structure,
A -algebra for which is provided with a right -algebra/module structure.
The following proposition gives two more useful descriptions:
Let () be the category of left (right) adjoint functors . The functor - that takes to the right -module is an equivalence. Or, what is the same, the functor -, taking to the left -algebra , is an equivalence.
The main thing to check is that the functor to is essentially surjective. The essential point is that has a left adjoint iff has a left adjoint iff is representable: for some -algebra (in which case the lift of through is tantamount to a -algebra structure on ). The only (mildly) tricky part is that has a left adjoint if has a left adjoint . To define the left adjoint of objectwise, we take any -algebra with its canonical presentation
which is a coequalizer diagram. A left adjoint must preserve this coequalizer, and we must have since both sides are left adjoint to . Thus we define to be a coequalizer
where is the -module structure coming from the monad morphism . This objectwise definition of easily extends to morphisms by universality and provides a left adjoint to . Remaining details are left to the reader.
One import of this proposition is that left adjoint endofunctors on compose, i.e., endofunctor composition gives a monoidal structure on , and this monoidal structure transports across the categorical equivalence of the proposition to give a monoidal structure on -. The resultant monoidal product on -bialgebras is denoted .
A second import of this proposition is that the canonical functor induces a functor which is reasonably denoted
realizing an actegory structure over the monoidal category .
A direct construction of the monoidal product can be extracted by following the proof of the proposition. If are -bialgebras, then the underlying -algebra of (corresponding to composition of of right adjoints ) is computed as a reflexive coequalizer in :
Here is the same as the -algebra structure on . Whereas is a component of the -module structure ; it is mated by the adjunction to the component of the coalgebra structure .
To extract the -coalgebra structure on , let us observe generally that if is a left adjoint, then for any category there is an induced left adjoint and similarly an induced left adjoint . Applying this to the case where and where is the left adjoint to the lift , we find that
is a left adjoint, and in particular preserves -indexed copowers . In other words, for each we have canonical isomorphisms
so that the desired right -module structure is given componentwise by a composite
A Tall-Wraith monoid over is of course a monoid in the monoidal category . We note that the unit in his monoidal category is the free -algebra , equipped with its canonical lift . That is, the -coalgebra structure on is given tautologously by
So, multiplication on a Tall-Wraith monoid is a bialgebra map and the unit is a bialgebra map . Such a monoid is tantamount precisely to a monoid in , i.e., to a left adjoint monad on . In particular, for a Tall-Wraith monoid , one has a category of algebras over that monad, giving a monadic functor .
Now, recall that left adjoint monads are canonically mated to right adjoint comonads , in such a way that the category of algebras over the monad is equivalent to the category of coalgebras over the comonad. In short, Tall-Wraith monoids over are essentially the same thing as functors
which are simultaneously monadic and comonadic: the comonadicity means is a left adjoint and has a left adjoint , so that the monad resides in , and such monads are tantamount to Tall-Wraith monoids.
In the important example where is the theory of commutative rings and is the bialgebra , equipped with a Tall-Wraith multiplication given by plethysm (a decategorified product that arises by viewing as the Grothendieck ring of the category of -valued species together with its substitution or plethystic product), the category may be identified with the category of lambda-rings. In this case the monad has right adjoint given by . The right adjoint to the forgetful functor is the big Witt functor, often denoted .
An old and long query-discussion has been archived starting here.