nLab
Tall-Wraith monoid

Idea

Given an algebraic theory $V$, a $V$-algebra is a model of $V$ in the category $\mathrm{Set}$. A Tall–Wraith $V$-monoid is the kind of thing that acts on $V$-algebras.

Definition

Definition

Let $V$ be an algebraic theory and let $V\mathrm{Alg}$ be the category of models of this theory in $\mathrm{Set}$. Then a Tall–Wraith $V$-monoid is a monoid object in the category of co-$V$-objects in $V\mathrm{Alg}$.

To see why these are what acts on $V$-algebras one needs to understand what a co-$V$-object in $V\mathrm{Alg}$ actually is. A co-$V$-object in some category $D$ is a representable covariant functor from $D$ to $V\mathrm{Alg}$. To give a particular $D$-object, $d$, the structure of a co-$V$-object is to give a lift of the $\mathrm{Set}$-valued $\mathrm{Hom}$-functor $D(d,-)$ to $V\mathrm{Alg}$. Thus a co-$V$-object in $V\mathrm{Alg}$ is a representable covariant functor from $V\mathrm{Alg}$ 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 $V\mathrm{Alg}\to V\mathrm{Alg}$ 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 $V\mathrm{Alg}$ 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-$V$-algebra objects in $V$, say $R_1$ and $R_2$, there is a product $R_1\odot R_2$ and a natural isomorphism

$${\mathrm{Hom}}_V(R_1\odot R_2,A)\cong {\mathrm{Hom}}_V(R_1,{\mathrm{Hom}}_V(R_2,A))$$Hom_V(R_1 \odot R_2,A) \cong Hom_V(R_1, Hom_V(R_2,A))

for any $V$-algebra, $A$.

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

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

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

One postscript to this is that although the category of co-$V$-algebra objects in $V\mathrm{Alg}$ is not a variety of algebras, for a specific Tall–Wraith $V$-monoid $P$, the category of $P$-modules is a variety of algebras.

Examples

• If $V$ is the theory of commutative unital rings, a $V$-algebra is a commutative unital ring, and the corresponding sort of Tall–Wraith $V$-monoid is called a biring.

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

• If $V$ is the theory of abelian groups, than a $V$-algebra is an abelian group, and the corresponding sort of Tall–Wraith $V$-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 ${\mathrm{AbGp}}^{\mathrm{op}}$ (though this picture gets the morphisms the wrong way around; in full abstraction then the category of co-abelian group objects in $\mathrm{AbGrp}$ is the opposite category of the category of abelian group objects in ${\mathrm{AbGrp}}^{\mathrm{op}}$). Concretely, such a thing is an abelian group $A$ together with group homomorphisms

  $$\begin{array}{rl}\mu & :A\to A\coprod A,\\ ϵ& :A\to I\\ \iota & :A\to A\end{array}$$\begin{aligned} \mu &: A \to A \coprod A, \\ \epsilon &: A \to I \\ \iota &: A \to A \end{aligned}

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

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

  The diagram for $\iota$ (representing the inverse map) is a little more complicated. As $I$ is the initial object in $\mathrm{AbGrp}$, there is a unique morphism $I\to A$ (inclusion of the zero). Composing this with $ϵ$ yields a morphism $A\to A$ which maps every element to the zero in $A$. Using $\mu$ and $\iota$ we can construct another morphism $A\to A$ as

  $$A\stackrel{\mu }{\to }A\coprod A\stackrel{1\coprod \iota }{\to }A\coprod A\stackrel{{\Delta }^c}{\to }A$$A \overset{\mu}\rightarrow A \coprod A \overset{1 \coprod \iota}\rightarrow A \coprod A \overset{\Delta^c}\rightarrow A

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

  Thus if $(A,\mu ,\iota ,ϵ)$ is a co-abelian group object in $\mathrm{AbGrp}$ then $\mu$ is the diagonal, $\iota$ the inverse from abelian groups, and $ϵ$ the zero morphism.

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

  The first issue is that $\iota$ is not automatically a morphism in $\mathrm{AbGrp}$. 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 $\iota$ is a morphism of abelian groups and so $(A,\mu ,\iota ,ϵ)$ 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 $\iota$ is the inverse for $\mu$. The relation that $\iota$ is the inverse for addition (let us write it as, say, $\alpha$) is that

  $$A\stackrel{\Delta }{\to }A\times A\stackrel{1\times \iota }{\to }A\times A\stackrel{\alpha }{\to }A$$A \overset{\Delta}\rightarrow A \times A \overset{1 \times \iota}\rightarrow A \times A \overset{\alpha}\rightarrow A

  is the zero map $A\to I\to A$. This is precisely the relation that $\iota$ is the inverse for $\mu$ since we have the following identifications: $\mu =\Delta$, $A\coprod A=A\times A$, and ${\Delta }^c=\alpha$. Also, $ϵ=0$ and $\eta :I\to A$ is the initial morphism in $\mathrm{AbGrp}$.

  Thus the fact that $\iota$ is the inverse for the diagonal+zero co-monoidal structure is due to the fact that $\iota$ is the inverse for $(\alpha ,\eta )$ and $\alpha :A\oplus A\to A$ is the co-diagonal in $\mathrm{AbGrp}$ and $\eta :I\to A$ is the unit.

  Andrew Stacey: I’ve deleted the original query boxes. The above now contains the all the details, but I suspect that there may be neater ways of putting it and other ways to see it. If so, these are certainly worth discussing but I think that the above is a new launching point for discussion. If anyone is interested in the discussion that led up to that point then they can look in the history but I don’t think that that is necessary to discuss the above.

  And remember what Tony Blair said: Indentation, Indentation, Indentation.

  It is part of the general theory that the category of co-$V$-objects in $V$ 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

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

• Plethystic algebra, J. Borger and B. Wieland, Adv. Math, 2005. web

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

Andrew Stacey: I changed the $V$ to $𝒱$ in line with the notation from The Hunting of the Hopf Ring. Should there be a lab standard on fonts for categories, objects, functors, and the like?

Toby: Based on this discussion, ‘$𝒱$’ is more likely to cause problems for people that haven't installed appropriate fonts. (Not that I'm changing it back, mind you ….)

John Baez: I hate ‘fancy fonts for fancy gizmos’, because they mainly serve to make gizmos seem fancy and intimidating. That’s why I had changed $𝒱$ to $V$. I think I’ll change it back again, just to annoy Andrew. It’s just a category with products, for god’s sake! Surely we can’t insist on calligraphic font whenever we see one of those: must we say $\mathrm{𝒮ℯ𝓉}$? (Yes, I’m being ornery.)

Andrew Stacey: Okay, okay, I surrender! I’ll even change back the ones that you missed.

But I reject the charge of ‘fancy fonts for fancy gizmos’. My intention was to use the change of fonts to represent the relationships between things. Thus $a$ is an element of $A$ which is an object of $𝒜$ which is mapped by the functor $𝔄$. It’s a bit like using $m,n$ for natural numbers. Yes, we can and often do use $a$ for a natural number, but if I scan a page and see $n$ then I automatically think “natural number” (okay, maybe integer). Since that is reasonably widespread, authors can use it to make their work clearer so that the reader doesn’t continually have to go back to the first page to work out whether $A$ was the category or the object. Of course we can write

$$\forall \delta \forall x>0\exists y>0:\forall ϵ,\mid ϵ-\delta \mid <y\Rightarrow \mid f(ϵ)-f(\delta )\mid <x$$\forall \delta \forall x \gt 0 \exists y \gt 0 : \forall \epsilon, |\epsilon - \delta| \lt y \implies |f(\epsilon) - f(\delta)| \lt x

but, frankly, wouldn’t you rather we didn’t?

John Baez: In lots of ordinary math the strategy of ‘bigger fancier fonts for bigger fancier things’ works fine: $a$ for an element of the set $A$ in the category $𝒜$. But the problem is that in $n$-category theory everything is an object in some $n$-category… including every $n$-category, and every functor between them. So, the idea of a clear hierarchy of two or three types of things, with bigger fancier things getting bigger fancier fonts, breaks down. For example, while you might want to say

“Consider the functor $𝔄:𝒜\to 𝒜$…”

you’ll often want to continue that sentence with

”… and think of it as an object $𝔄\in \mathrm{End}(𝒜)$. Now, since $\mathrm{End}(𝒜)\in \mathrm{MonCat}$, it makes sense to equip $𝔄$ with the structure of a monoid object, and we then call it a monad.”

This is just a very simple example of the kind of fancy level-shifting that takes place. And this level-shifting is precisely the style of thinking that makes $n$-category theory so powerful.

For this reason, at some point I decided Jim Dolan was very wise to put everything in the same font: it encourages mental flexibility. I later impressed Ross Street greatly when I wrote on the blackboard “Consider a bicategory $x$.”

Even though I’ll be eternally proud of that moment, I think it can be good to use a couple of different fonts in a given discussion to help ‘set the types’ of the entities in question — hence the term ‘typesetting’. But this sort of distinction is only helpful locally: it’s not so good to demand globally consistent typesetting. Your big fancy algebraic theory $𝒜$ is likely to become someone else’s puny little object $a$ in $\mathrm{FinProdCat}$, and then $\mathrm{FinProdCat}$ will become a puny little object in $\mathrm{Doctrines}$, and so on.

Andrew Stacey: I take your point; I was aware that it would have been hard to continue further. Whilst I have an old Lettraset catalogue to draw on for blackboards, sadly it’s not implemented in the STIX fonts. But I would like to use the fonts we do have, and so Toby’s point is more important that us squabbling over whether or not we should change $𝒱$ to $V$ or not.

John Baez: I agree with that. I guess some group of people should argue about whether most of our readers will have various fancy fonts installed… I think we should not assume this.

Andrew Stacey: If we’re going to take this sort of thing into account then we need it clear somewhere since otherwise those of us fortunate enough to have the correct fonts don’t know what doesn’t display so well on other, more limited, systems!

I did find it getting awkward in writing out the proof that a TW-monoid in abelian groups was a ring to talk of $A$ being an object of $V$. I’d much rather have said $A$ is an object of $𝒜$. So if we’re allowed to use this sort of hierarchy locally, can I change it all back now?

John Baez: If you insist, but I don’t like it: I don’t think most reader will find the letter $V$ more ‘awkward’ than $𝒜$. And more people will be able to read $V$ than $𝒜$.

Andrew Stacey: Linking this to your question above, for something like this where there is a fair amount of notation, what do you think of a “notation used in this page” section for easy reference?

John Baez: I don’t think there’s intrinsically more notation in this subject compared to most subjects on the $n$Lab, or more need for fancy fonts. I think that with this subject, as with almost any, it’s possible to explain the material very nicely with only a little notation. But that’s a kind of pet peeve of mine: I think mathematicians tend to make their work hard to read by introducing more notation than necessary, and not reminding the reader about what it means frequently enough. They like to build towers of thought that are beautiful in principle but not very user-friendly.

Andrew Stacey: firstly, when I was linking to your question above, I meant to the ‘VV^c’ question much higher. It was awkward not to use some notation for this category, but it wasn’t clear to me that it merited a whole definition to itself. An explanatory sentence is reasonable the first time that it is used, and maybe after a long period with no use, but one of the advantages of hyperlinks is that I could put a “Notation used in this page” section at the bottom, stick in:

• ${\mathrm{VV}}^c$: the category of co-$V$-algebraic objects in $V$

and put a little hyperlink whenever it is used, something like ${\mathrm{VV}}^c$$(notation) (okay, that doesn’t work as I haven’t created the notation section); possibly with a little semi-quaver instead of the word “notation”.

I completely agree with your peeve, but no notation is as horrible as too much notation. Notation should help and guide the reader through a complicated argument, never intruding but always easing the path.

If you take the Dolan principle to its extreme, we should just start the document at the letter $a$ and increase as we proceed. Then the only thing that we have to keep track of is whether or not we have referred to something before or not. Actually, if we combine this with the slogan “Do no evil” then we should always use a new symbol and note where things are isomorphic. Thus

$$\forall a\forall b>c\exists d>e:\forall f,\mid g-h\mid <i\Rightarrow \mid j(k)-l(m)\mid <n$$\forall a \forall b \gt c \exists d \gt e: \forall f, |g - h| \lt i \implies |j(k) - l(m)| \lt n

where

$$c\cong 0,e\cong 0,g\cong a,h\cong f,i\cong d,k\cong a,l\cong j,m\cong h,n\cong b$$c \cong 0, e \cong 0, g \cong a, h \cong f, i \cong d, k \cong a, l \cong j, m \cong h, n \cong b

(I think!)

John Baez: I never take any principle to extremes, not even this one. I mainly just want to make life easy on people who read my stuff or hear my talks. I think links to “notation used on this page” would be great when used in addition to on-the-spot explanations of the notation when it’s introduced, and occasional reminders. I’m afraid however that most mathematicians would use these hyperlinks as an excuse to use more notation, when 99% of them should spend time figuring out how to use less.

Anyway, I think I’m starting to repeat myself, so I’ll stop.

David Roberts: I find myself writing $X$ for a topological groupoid as well as a space, as the latter is one of the former, and I want to replace spaces with groupoids anyway. Ditto with functors (and eventually anafunctors, but that is a little trickier). The move to replace spaces by stacks, which is a good thing, seems to have that psychological barrier of people obeying the urge to call a stack a symbol in an unreadable font. Many-folds were once tricky…

That’s my rant over

Toby: Just stepping in to clear the air about what Jim Dolan actually does. Here is an example, not as he would write it, but using symbols as he would write them on a blackboard:

Let $C$ be a category. Let $C_0$ and $C_1$ be objects of $C$. Let $C_2$ be a morphism from $C_0$ to $C_1$. …

Andrew Stacey: John, I completely agree with your points but I think that you’re being a bit pessimistic. Yes, 99% of mathematicians will misuse a new feature to make their presentations or articles dire. But they would have been dire without those new features. The remaining 1% should be encouraged to use the new features to make their already-excellent stuff even better. The great thing about the wiki-format is that someone else can go along later putting in those notational footnotes. Indeed, it is better if it is someone else because it is clearer to the non-author where they should be. After all, I know what ${\mathrm{VV}}^c$ means when I write it so I don’t need reminding. You, or someone else, aren’t so familiar with the notation so can - and should - put in the little reminders. What makes this easiest is if I have already put a little notational summary at the bottom of the page so that all you have to do is add the correct link wherever you think it necessary.

Of course none of this excuses me explaining the terms correctly and thinking carefully about what notation I ought to use. But just because it has the potential to be horribly misused doesn’t mean that we should shun it if it also has the potential to make stuff much better.

The fact that this stuff is on the web should make a difference. Otherwise we may as well just have a load of PDFs of LaTeX documents. The big addition of the web is the ability to link documents and to navigate back and forth between those links. We should be experimenting with this and trying out different ways to implement it.

Which has gotten us a long way from the argument as to whether or not $𝒱$ is a category or not!

My principle, which I’m quite happy to take to any extreme, is: whatever aids comprehension is Good, whatever hinders it is Bad. Careful choice of notation is Good, “fancy fonts for fancy gizmos” is by itself Bad.

On the issue as to whether or not we should use $𝒜$ and so forth, I’m less inclined to be kind to our Dear Readers. We already insist that they be able to read MathML. That puts a fairly high barrier on entry, I don’t think that requiring the fonts puts that much higher. If you do want to impose this rule, then someone needs to do some extensive testing to find out what is and isn’t allowed.