What kind of operations does the Tall-Wraith monoid encode?

Question

According to the nLab page, for an algebraic theory V a Tall–Wraith V-monoid is "the kind of thing that acts on V-algebras". Well, it certainly does act on V-algebras, but in which sense is it "the kind of thing"?

Can one express explicitly (in terms of elements and operations of the V-algebra) what is an action of the kind taken into account by TW-V-monoids?

In general TW-V-monoids do not form a variety, and they probably are not related to the structure of the set of endomorphisms of a V-algebra. But is it true that, for V a commutative theory, TW-V-monoids form a variety again, with operations given by those of V (applied pointwise on endomorphisms) plus the monoid structure coming from composition - i.e. the structure of the set endomorphisms?

(This plethory of questions remained after looking into Stacey/Whitehouse's "Hunting Hopf"-paper and the google-browsable parts of the Bergman/Hausknecht book, are there other references for the general V-algebra case?)

Answer 1

But is it true that, for V a commutative theory, TW-V-monoids form a variety again, with operations given by those of V (applied pointwise on endomorphisms) plus the monoid structure coming from composition - i.e. the structure of the set endomorphisms?

Yes, this is true. And the simplest example of a TW-V-monoid is where V is abelian groups whereupon TW-V-monoids are simply rings (unital, not necessarily commutative).

What led us to write our paper is that we couldn't find an accurate description of what we wanted (the full structure on unstable operations in a "generator-relation" type description). After a paper chase through near rings and composition rings, we finally came across Tall and Wraith's 1970 paper in PLMS and the Borger-Wieland paper. Neither was quite what we wanted, though, since the algebras from cohomology theories are more complicated than "mere" algebras. The Hunting of the Hopf Ring paper (we were sure that a referee was going to complain about the title, by the way) concentrates on those technicalities rather than trying to explain the simple details of TW V-monoids. We're currently writing another paper which tries to lay out the simpler ideas and which will include a proof of the commutative case (not that this is difficult).

The slogan "TW V-monoids are that-which-acts-on-V-monoids" is intended as an adaptation of the slogan "Rings are that-which-acts-on-abelian-groups". The story is more complicated because it is not automatic that Hom(V,V) is a TW V-monoid. One simple case where it is is when V is a finite ring, but in general one needs a Kunneth-type formula to hold. Nonetheless, it is quite easy to find lots of examples of TW V-monoids and the Tall-Wraith paper, and the Borger-Wieland paper contain plenty of examples (as will ours). The point about TW V-monoids being the thing that acts is that they are the representable things that act. The general idea is that whenever you have a set-with-structure acting on a V-algebra then there is an associated Tall-Wraith V-monoid. Of course, one might wonder why bother with that bloated gadget, but then one might wonder why bother with group rings when you already have the group.

I feel that this isn't really a sufficient answer, though. My problem is that we came at this with examples in hand that we already knew about and wanted some algebraic way of encoding the structure that we already had. As we couldn't find such a description, we invented one (and we hope that Tall and Wraith don't object to the approbation of their names!). So for us, the intuition is all in cohomology operations and not in "bare algebra". If you can expand on what you are looking for a little, I may be able to refine my answer somewhat.

Answer 2

I don't know the paper well enough but the notion of plethory in
J. Borger, B. Wieland, Plethystic algebra, Advances in Mathematics 194/2 (2005), pp 246-283
(which is available from Borger's website) is I think pretty much the notion of Tall-Wraith monoid for commutative rings so it might be worth a look to see if why a plethory is "the thing" that acts on commutative rings is better explained there and if it is whether it makes it clear why this should be true in general.

Answer 3

If I read the nLab page correctly, the TW V-monoids turn out to be the monads T on the category of V-algebras such that the underlying functor T: V-Alg -> V-Alg preserves colimits.

You could say that monads on V-Alg the things that act on it; these are just particularly nice actions.

Edited to address Peter's comment: The nLab page does not say what I said. But what I think happens is that the underlying functor T: V-Alg -> V-Alg has a right adjoint C. It is formal that C is a comonad. I believe that in this algebraic setting (edit: no this happens always), T-algebras turn out to be the same thing as C-coalgebras.

The functor C: V-Alg -> V-Alg is the gadget that is "corepresentable"; that is, there is a co-V-object P in V-Alg such that

CA = Hom(P, A)

for any V-algebra A.

This is certainly what happens in the case I understand best: plethories (as mentioned in Greg's answer).