## July 19, 2009

#### Posted by John Baez

My friend the combinatorist Bill Schmitt breezed through Paris recently, taking a train back from Hopf-in-Lux with Paul-André Melliès, and spending a day here before going home to DC. We had some time to talk, and during the course of it I realized I’d become less scared of certain topics involving monads.

Monads seem to bother a lot of people. There’s even a YouTube video called The Monads Hurt My Head! It’s a long aimless self-indulgent rant, and it’s not exactly focused on monads in the mathematical sense, but you may enjoy listening to it anyway if you start at this particular second: 7:30. Shortly thereafter, the woman speaking exclaims:

What the heck?! How do you even explain what a monad is?

If you’re one of the people who still finds monads terrifying, I’ll just offer you two words of advice — not a complete explanation, just pointers toward what you need to learn:

A monoid is a set $T$ with an associative multiplication

$m : T \times T \to T$

and unit

$i : 1 \to T$

But the idea generalizes from $Set$ to any category with products, or indeed any monoidal category, or indeed any 2-category: you can still write down the definition. And that’s a monad.

On the other hand, a monad is a way of describing and studying algebraic gadgets. This is especially clear for monads in the 2-category Cat. A monad in Cat is a functor $T : C \to C$ going from some category $C$ to itself, equipped with an associative multiplication

$m: T T \Rightarrow T$

and unit

$i: 1_C \Rightarrow T$

An algebra for this monad is an object $a \in C$ together with a morphism $\alpha : T a \to a$ satisfying laws that mimic those of an action. A typical example is to let $C = Set$ and let $T a$ be the underlying set of the free group on the set $a$. The algebras for this particular monad are just groups! Groups are just one of zillions of algebraic gadgets that we can describe this way.

(It was, in fact, looking for these videos that led me to The Monads Hurt My Head! I’m not sure if the author of that video would be helped or further damaged by watching the Catsters.)

Anyway: here are some topics in monad theory that used to hurt my head, which I am now eager to understand:

• If we have a monad on a monoidal category there are various ways these structures can interact. The monad can have a tensorial strength, which is a natural transformation $a \otimes T b \to T(a \otimes b)$ making some diagrams commute. What’s the point of this? I don’t know, but I’m sure there is one. Someone tell me the fundamental theorem that justifies this notion! Do their algebras form a monoidal category, or something?
• We can also talk about monoidal monads, which are just monads in MonCat. This seems like a perfectly nice notion, but I’d still like to know the fundamental theorem about them, if there is one.
• If we have a monad on a cartesian category, we can ask for it to be a cartesian monad. I know this concept comes up in Leinster’s work on higher-dimensional algebra… but again, there should be some really fundamental theorem that justifies this notion.
• If we have two monads on the same category, say $S,T : C \to C$ they can be related by a distributive law, which is a natural transformation $T S \Rightarrow S T$ with just the right properties to make $S T$ into a monad. Distributive laws are a lot like braidings, and you can understand them nicely using string diagrams. I’m perfectly happy with this concept, since I just stated the fundamental theorem about it: a distributive law lets you make $S T$ into a monad. So, I add it to this list just for completeness. If you’re not happy with distributive laws yet, try my description in week257 of Cheng’s paper Iterated distributive laws.
• If we have two monads on different categories, they can be related by a ‘map of monads’ — see section 6.1 of Leinster’s book. I should like these just as much as distributive laws, but I don’t yet. At least I know the fundamental theorem about them: if we have a map of monads, we get a functor sending algebras of the first to algebras of the second.

I’ve linked to the Wikipedia for some definitions, and the $n$Lab for others. Not all the above concepts are defined on the $n$Lab yet. Of course they eventually will be. And if you can help me out here, that’ll be a good start.

Posted at July 19, 2009 12:15 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2018

I was about to have a ‘go’ at you for not linking to nLab’s monad, but you come up with a good excuse.

It does seem a shame we can’t find a reliable server. Perhaps we need to earn some money to pay for one with some relevant advertising – degrees for sale, mathematics project writing for a fee, etc.

Posted by: David Corfield on July 19, 2009 2:02 PM | Permalink | Reply to this

I think Urs is looking for a more reliable server. I don’t think money is a big issue, but if it is, he should let me know.

Posted by: John Baez on July 19, 2009 2:25 PM | Permalink | Reply to this

### migration update

I’d prefer to link to the nLab, but the $n$Lab seems to be down

Yes, it’s a shame. I just restarted it once again.

When you find the $n$Lab not reacting, what you can do is ssh into the machine and type ./x. Then log out again. That’ll restart the server and after that it does react again.

We do that all the time.

I think Urs is looking for a more reliable server.

Yes. Andrew Stacey is mainly taking care of it now, luckily, as he is versed in these issues, which I am not.

He has now subscribed to a different hosting company and is in the process of experimenting with setting up the $n$Lab there. It shouldn’t take much longer until we can migrate.

I don’t think money is a big issue, but if it is, he should let me know.

No, money is not the bottleneck. What would be helpful is dedicated people with time and IT expertise or maybe even with hardware and the like.

Posted by: Urs Schreiber on July 22, 2009 5:04 PM | Permalink | Reply to this

### Re: migration update

When you find the nLab not reacting, what you can do is ssh into the machine and type ./x.

Actually, you should enter

~/x


which is more reliable. (But ./x shouldn't break anything; it just might not work.)

Posted by: Toby Bartels on July 23, 2009 12:42 AM | Permalink | Reply to this

### Re: migration update

No, money is not the bottleneck. What would be helpful is dedicated people with time and IT expertise or maybe even with hardware and the like.

The major issue is simply finding a reliable hosting company. All hosting companies claim to be wonderful but not all are. Also, a hosting company may be fine while there are no problems but come apart in a crisis.

Thus if anyone has a genuine recommendation of a company of the form “When everything went wrong it got sorted out very quickly” then please let me know. Otherwise I’m very reluctant for us to shift the lab from one host to another host without having done a bit of prior testing (which is what I’m doing now with one company, by the way, just so that everyone knows that something is being done about this).

Posted by: Andrew Stacey on July 23, 2009 8:05 AM | Permalink | Reply to this

### Re: migration update

http://www.mythic-beasts.com have always been very quick to respond to my enquiries, though I only have e-mail and DNS hosting with them, not web.

Posted by: Tom on July 23, 2009 10:48 AM | Permalink | Reply to this

### Re: migration update

Come to think of it, I have used them for hosting, and they’re good at that too.

Posted by: Tom Ellis on July 23, 2009 11:03 AM | Permalink | Reply to this

Isn’t a monad exactly the same thing as an algebra (maybe you category-people call it a “monoid object”) in a monoidal category? I always think of monads as the functors

A ⊗ –

associated to an algebra, and thought that you had the converse just by applying all the diagrams to the unit object.

In the language of algebras in monoidal categories, “distributive laws” are called “twisted tensor products” and (shameless self promotion) I did my fair amount of work on them (thesis and everything), including the iterated version, with nice diagrammatic proofs!:
http://arxiv.org/abs/math.QA/0511280.

I had a geometrical focus in mind, but since you are interested in distributive laws you might find some of the results in my thesis insteresting.

Posted by: javier on July 19, 2009 3:02 PM | Permalink | Reply to this

Javier wrote:

Isn’t a monad exactly the same thing as an algebra (maybe you category-people call it a “monoid object”) in a monoidal category?

Yes, but you still might be a little confused — I’m not sure.

A monad on a category $C$ is precisely a monoid object in the monoidal category $End(C)$, which is defined so that

• the objects of $End(C)$ are functors $T : C \to C$,
• the morphisms are natural transformations between these,

and composition of functors gives the tensor product.

By the way, it’s nice to say “monoid object in a monoidal category” instead of “algebra in a monoidal category”, at least in this context, because monads have “algebras”, which are something else entirely!

More generally, a monad on an object $C$ in a 2-category $X$ is precisely a monoid object in the monoidal category $End(C)$, which is defined so that

• the objects of $End(C)$ are morphisms $T : C \to C$,
• the morphisms are 2-morphisms between these,

and composition of functors gives the tensor product.

Note that a monad on a category $C$ is precisely the same as a monad on the object $C$ in the 2-category $Cat$.

I always think of monads as the functors $A\otimes -$ associated to an algebra…

That’s one way to get a monad on a category $C$, which only works if $C$ is monoidal: take a monoid object $A$ in $C$, and let our monad be

$T = A \otimes -$

But there are lots of monads that are not of this form. For example, the monad $T: Set \to Set$ taking each set to the free group on that set.

Indeed, you can have monads on $C$ even when $C$ is not monoidal!

Note how nice and sneaky your example is:

A monad on a category $C$ is a monoid object in the monoidal category $End(C)$. But if $C$ is a monoidal category, any monoid object in $C$ gives a monoid object in $End(C)$.

Say that ten times fast!

In the language of algebras in monoidal categories, “distributive laws” are called “twisted tensor products” …

That’s worth knowing. I don’t love either of these terms, since I think the really good term would have a word like “braid” or “commute” in it somewhere. Maybe “twist” is good. Maybe if I were king of the universe, I’d call a distributive law a “twist”.

and (shameless self promotion) I did my fair amount of work on them (thesis and everything), including the iterated version, with nice diagrammatic proofs!:

http://arxiv.org/abs/math.QA/0511280

I’ll check it out! You might like my summary of Cheng’s paper in week257 — I use string diagrams to explain some stuff about distributive laws. You probably know a lot of this stuff, but you probably talk about it differently.

Posted by: John Baez on July 19, 2009 4:05 PM | Permalink | Reply to this

By the way, it’s nice to say “monoid object in a monoidal category” instead of “algebra in a monoidal category”, at least in this context, because monads have “algebras”, which are something else entirely!

The problem is that monads should never have had ‘algebras’; they should have had ‘modules’ all along. Even given that the terminology be inspired by monads in their guise as algebraic theories, still they should have had ‘models’ (a conveniently similar word).

Posted by: Toby Bartels on July 19, 2009 6:53 PM | Permalink | Reply to this

The problem is that monads should never have had ‘algebras’; they should have had ‘modules’ all along.

Yes!

It’s one of these cases where a simple idea is entirely buried behind weird unsuggestive terminology.

Of course once one gets it its all easy and obvious, and so everybody keeps using it. But so many people new to the subject will be turned off just because of confusing terminology.

We should change it. ;-)

Posted by: Urs Schreiber on July 22, 2009 5:11 PM | Permalink | Reply to this

A monad on a category $C$ is a monoid object in the monoidal category $End(C)$. But if $C$ is a monoidal category, any monoid object in $C$ gives a monoid object in $End(C)$.

So we have a functor (all this is functorial, right?) from monoid objects in $C$ to monads over $C$.

But it is easy to see how to recover the monoid $A$ from the functor $A\otimes -$, we just have to apply it to the unit object in $C$, since $A\otimes \boldsymbol{1}$ is naturally isomorphic to $A$. I haven’t really checked the details, but always believed that for any monad $T$ over $C$ (monoidal) the object $T(\boldsymbol{1})$ in $C$ has a monoid structure induced by the monad structure of $T$. I don´t expect this to be an equivalence, but looks like a nice adjunction nonetheless.

Posted by: javier on July 21, 2009 10:20 PM | Permalink | Reply to this

John, right now my brain hurts because I’ve been up all night, but maybe I’ll have a crack at the first one, about tensorial strength, and hopefully others will chime in about some of the others. Or if not, I can come back to it later.

(I ignored your advice and listened to the young lady before 7:30, which I began to regret after a while. I found my attention drifting and wondering what was making the blue ball behind her move around now and then. But I digress.)

I guess the first thing to notice about (covariant) tensorial strengths is that they attach to a functor from a monoidal category to itself, say $T: V \to V$. (The concept doesn’t make much immediate sense if $T$ is a functor between different monoidal categories.)

One possible reason why it may be hard to grasp the notion of strength is because in the case $V = Set$, they’re sort of invisible. Every functor $T: Set \to Set$ has one, in fact a canonical one! This will make more sense in a moment.

I find strengths easier to understand by considering the case where $V$ is closed monoidal. Here $V$ is enriched in itself, and one of the most important aspects of strengths is this:

• A functor $T: V \to V$ with a strength is the same thing as a $V$-enrichment on $T$.

Here’s how that works. An enrichment on $T$ in $V$-enriched category theory consists of a natural family of morphisms in $V$,

$hom(a, b) \to hom(T(a), T(b)),$

satisfying some suitable axioms. Here $hom$ denotes the enrichment of $V$ in itself. Starting from a strength on $T$, you can cook up an enrichment on $T$, roughly as follows: the map above is equivalent by hom-tensor adjunction to a map

$hom(a, b) \otimes T(a) \to T(b)$

and now the strength allows you to slide $hom(a, b)$ inside $T$, and from there you just apply $T$ to the evaluation:

$T(hom(a, b) \otimes a) \overset{T(eval_{a, b})}{\to} T(b)$

and presto, you’re done. The strength axioms ensure that this enrichment structure on $T$ satisfies the axioms you need for a functor to be enriched.

(And going back the other way, from an enrichment to a strength, is also easy – there you have to dualize. That is, instead of using the evaluation which is the counit of the hom-tensor adjunction, you use the coevaluation which is the unit. I’m going to leave that to you to work out yourself.)

After some work, you can convince yourself that the notions of strength and enrichment for endofunctors on closed monoidal categories really are equivalent notions.

And now we can understand why strengths in the case $V = Set$ are so “invisible” – every functor $T: Set \to Set$ is $Set$-enriched; that’s what we mean by an ordinary functor!

There’s rather a lot more one could say about strengths, and I may come back to more of that later, but I would like to say that strengths are kind of a trade secret. The first mathematician I know of who intuitively grasped strength was C.S. Peirce! And particularly in his Alpha graphs, the notion of strength plays an important role.

The insight here can be related back to the enrichment = strength phenomenon. Suppose for instance we’re in the theory of Boolean algebras – say we’re studying the structure of a free Boolean algebra on a set of generators, $B[X]$. Then again we can think of this, or of any Heyting algebra, as enriched in itself. Further, we have definable unary operators

$T: B[X] \to B[X]$

in the theory, such as $T = p \wedge (-)$, or $T = p \Rightarrow (-)$, etc. The great discovery of Peirce is that any definable unary operator in the theory of Boolean algebras carries a strength, or if you prefer is enriched. That to me is the essence of Peirce’s iteration rule for Alpha (if you think about it long enough), and it categorifies right over to a similar statement for the theory of closed categories, $*$-autonomous categories, what have you: all definable unary covariant functors in such theories carry canonical strengths.

(Peirce went a little further, and incorporated notions of contravariant strength as well, which I haven’t discussed.)

It’s striking to me how this insight gets rediscovered from time to time, without awareness of this history.

Posted by: Todd Trimble on July 19, 2009 3:36 PM | Permalink | Reply to this

Todd, you just wrote a bunch of stuff explaining the content of the one link that John made to Wikipedia that he actually could not have made to the nLab instead: tensorial strength (or strong monad. You know what to do … (^_^)

Posted by: Toby Bartels on July 19, 2009 7:01 PM | Permalink | Reply to this

### tensroial strenght

Toby wrote:

Todd, you just wrote a bunch of stuff explaining the content of the one link that John made to Wikipedia that he actually could not have made to the $n$Lab instead: tensorial strength (or strong monad. You know what to do … (^_^)

I did it now, for Todd. Or was it for John? ;-) Anyway:

tensorial strength

Posted by: Urs Schreiber on July 22, 2009 6:08 PM | Permalink | Reply to this

All this stuff you wrote is great, Todd, but you’ve got me yearning for an explanation of strengths for functors $T : V \to V$ where $V$ is a monoidal category that’s not closed. Is this a stupid generalization that people do just because they can, or does it really mean something?

I’m suddenly hoping it means something, maybe something like this:

There’s an obvious sense in which a monoidal category $V$ can ‘act’ on a category $X$. I guess we can efficiently define this by saying we have a monoidal functor

$\alpha: V \to End(X)$

To be precise, this is a ‘left action’ of $V$ on $X$.

And given two monoidal categories on which $V$ acts, say $X$ and $X'$, we can say a functor

$F : X \to X'$

‘laxly preserves the action’ if there’s a natural transformation

$\alpha(v)(F x) \to F(\alpha'(v) x)$

satisfying two coherence laws mimicking two of the three you see in the definition of a lax monoidal functor.

This must surely be standard stuff.

Next, any monoidal category $V$ has a left action on itself by tensor product. So, we can ask for a functor

$T : V \to V$

to laxly preserve this left action. And that means we have a natural transformation

$v \otimes T(w) \to T(v \otimes w)$

for all $v,w\in V$. And this is starting to look like the definition of a tensorial strength!

I think I’m getting just two of the four diagrams in the definition of a tensorial strength for a monad. But that’s okay: all I’m seeking so far is a slick definition of tensorial strength for an endofunctor.

Maybe there’s a slick way to get the other two diagrams when our endofunctor is a monad. A monad is a monoid in monoidal category of endofunctors; maybe a strong monad is a monoid in the monoidal category of ‘strong endofunctors’ — i.e. endofunctors that laxly preserve the left action of $V$ on itself.

(Here I’m presuming there is a monoidal category of strong endofunctors, whose tensor product rides on top of composition of endofunctors.)

Posted by: John Baez on July 20, 2009 1:49 PM | Permalink | Reply to this

I would love it if some expert out there would tell me if I’m on the right track, and if so, whether what I’m saying seems new.

I’m really hoping this is beautiful explanation of the scary-looking diagrams in the definition of strong monads. If my previous explanation was too longwinded, here it is again, more tersely.

I hope that for any monoidal category $V$ there’s a 2-category $V-Act$ where:

• an object is a category $C$ equipped with a left $V$-action, meaning a monoidal functor $\alpha: V \to End(C)$,
• a morphism is a functor laxly preserving the left $V$-action,
• a 2-morphisms is a natural transformations compatible with the left $V$-action.

Yes the second two points are a bit sketchy, but the necesssary diagrams can be stolen from the definition of ‘lax monoidal functor’ and ‘monoidal natural transformation’.

$V$ has a left action on itself thanks to its tensor product, so $V \in V-Act$.

Conjecture: A monad on the object $V$ in the 2-category $V-Act$ is the same as a strong monad on $V$.

Posted by: John Baez on July 21, 2009 7:21 PM | Permalink | Reply to this

I would love it if some expert out there would tell me if I’m on the right track,

For what it’s worth, I can tell you that I think this looks right.

I’d say it as follows:

The 2-category $V\text{-}Act$ is

$[\mathbf{B}V, Cat] := Lax2Funct(\mathbf{B}V, Cat)$

The object $V$ sits in there as a certain canonical 2-functor $\hat V$.

The monad in question is a lax natural transformation $T : \hat V \to \hat V$, which is itself a certain functor in components by the logic of [[(2,1)-transformations]] The monad’s product is a lax modification $\mu$.

Then the diagrams at the Wikipedia entry Strong Monad are:

the first: unitalness of the component functor of $T$;

the second: naturalness of the unit modification

the third: functoriality of the component functor of $T$;

the fourth: naturalness of the product modification

Posted by: Urs Schreiber on July 22, 2009 7:30 PM | Permalink | Reply to this

Urs wrote:

For what it’s worth, I can tell you that I think this looks right.

Okay, thanks! If it’s this close to being right, it must be right: there can’t be two good concepts so close to each other in concept space — especially in this very simple, primordial region of concept space.

I’m still wondering if this is a new discovery or a standard fact. I can’t believe nobody noticed it before — but nobody who tried to explain strong monads to me before ever mentioned it! And that includes people who are very good at this sort of category theory: Martin Hyland and Todd Trimble.

Maybe they just thought I’d be too stupid to understand this. But in fact I was too stupid to comprehend strong monads until I understood this.

Here’s what amuses me: the definition of bialgebra and Frobenius algebra are very similar. A bialgebra is a coalgebra in the category of algebras. A Frobenius algebra $V$ is a coalgebra in the category of $V-V$ bimodules.

Similarly: say we have a monoidal category $V$. Then a lax monoidal monad $T: V \to V$ is a monad in the bicategory of monoidal categories (with lax morphisms). A strong monad $T : V \to V$ is a monad in the bicategory of $V$-actions (with lax morphisms).

We could also look at monads in the category of $V-V$-biactions (with lax morphisms).

Conversely, we could look at coalgebras in the category of $A$-modules. But those are just called coalgebras over $A$.

In a million years, godlike beings will look at these concepts as the equivalents of the simplest chess openings.

By the way, I changed a bunch of Wikipedia links in this blog entry to $n$Lab links. But not all: the diagrams for ‘distributive law’ and ‘strong monad’ still aren’t on the $n$Lab, and I’m too lazy to put them in tonight.

Posted by: John Baez on July 22, 2009 7:52 PM | Permalink | Reply to this

### strong monads on the nLab tonight

By the way, I changed a bunch of Wikipedia links in this blog entry to $n$Lab links.

Thanks!

But not all: the diagrams for ‘distributive law’ and ‘strong monad’ still aren’t on the nLab, and I’m too lazy to put them in tonight.

Here is [[strong monad]] for you. :-)

Posted by: Urs Schreiber on July 22, 2009 9:20 PM | Permalink | Reply to this

### Re: strong monads on the nLab tonight

The page even has fixed plural links now ;)

Posted by: Eric Forgy on July 22, 2009 11:19 PM | Permalink | Reply to this

Let me just quickly corroborate that you are right, John, and sorry for the delay in responding. I’d like to go into more detail on this later.

Posted by: Todd Trimble on July 22, 2009 10:47 PM | Permalink | Reply to this

### comonoids in bimodules

A Frobenius algebra $V$ is a coalgebra in the category of $V\text{-}V$ bimodules.

To be nitpicking, shouldn’t you say: a Frobenius algebra structure on the algebra $V$ is a coalgebra structure on the object $V$ itself (instead of some other object) regarded as an object in the category of $V\text{-}V$-bimodules?

A general coalgebra in $V$-bimodules is something more general.

In fact, Zoran Škoda recently turned my attention to such comonoids in bimodule categories. They are called corings.

The interesting thing is that those corings that have the property that they contain what’s called a “grouplike element” are by a theorem by Roiter equivalent to quasi-free DGA.

That’s interesting: we talked about quasi-free DGAs a lot here. They are (if in non-negative degree and degreewise dualizable) precisely the dual incarnation of Lie-$\infty$-algebroids.

Now, differential graded algebras are a dime a dozen. I have always wondered how that quasi-freeness condition that makes them into Lie $\infty$-theoretic objects could be understood from general nonsense. Maybe comonoids in bimodules have the answer.

Posted by: Urs Schreiber on July 22, 2009 8:14 PM | Permalink | Reply to this

Urs wrote:

To be nitpicking…

No, that’s not nitpicking at all. I just didn’t say that sentence right: one too many beers tonight, perhaps.

‘Corings’, eh? Jim Dolan uses that term to mean things like the set of functions from a ring to an abelian group: all the ring operations become ‘co-operations’ on this. I guess his sort of coring could be defined as a ‘ring object in $AbGp^op$. But you’re talking about a ‘comonoid in $(A,A)$-Bimod’ when $A$ is some ring.

These sound related when $A = \mathbb{Z}$, but I don’t think this is a good time for me to try to state any theorems.

Posted by: John Baez on July 22, 2009 8:45 PM | Permalink | Reply to this

If you’re coming from computer science and don’t understand category theory yet, but you’re comfortable with type constructors and lists, you might like Functors and Monads.

Posted by: Mike Stay on July 19, 2009 3:53 PM | Permalink | Reply to this

John asks for the point of a tensorial strength. We had plenty of help back here, and especially from here on.

Posted by: David Corfield on July 19, 2009 4:15 PM | Permalink | Reply to this

If you want your head to hurt, but in a good way, try:

Given any 2-category $C$, there’s a 2-category $Mnd(C)$ where:

• the objects are monads in $C$,
• the morphisms are monad maps.

A monad in $Mnd(C)$ is just a pair of monads in $C$ related by a distributive law! That’s already beautiful, and it sort of explains why maps of monads are nice — if you like distributive laws.

But they also consider $Mnd$ as a functor from the category $2Cat$ to itself. And it turns out to be a monad!

(Yes, they know it’s a bit evil to consider $2Cat$ as a mere category.)

Since $Mnd$ is a monad, we get a map

$Mnd(Mnd(C)) \to Mnd(C)$

so we can turn any monad into $Mnd(C)$ — that is, any pair of monads related by a distributive law — into a monad!

For extra credit: guess what’s a monad in $Mnd(Mnd(C))$!

Give up?

Posted by: John Baez on July 19, 2009 4:55 PM | Permalink | Reply to this

John said:

Given any 2-category $C$, there’s a 2-category Mnd$(C)$ where:

• the objects are monads in $C$,
• the morphisms are monad maps.

Hmm. So what’s a ‘monad map’? Is a monad to a monoid as a monad map is to a monoid homomorphism? The problem with a definition like that is that there can never be a map between monads with different underlying objects. This is ‘evil’ at the very least, as equivalent 2-categories would give rise to non-equivalent 2-categories of monads.

I was recently talking about monads to Andrei Akhvlediani, a PhD student in our group here in Oxford who’s currently studying Steve Lack’s work on PROs and PROPs and trying to generalize it in some interesting ways. He pointed out to me that given a 2-category, a nice way to get a 2-category of monads out of it is to consider internal monads, bimodules and bimodule homomorphisms. The usual rules for defining bimodules carry over easily; you just have to replace tensor product with 1-cell composition. I would have thought this is one of the most interesting ways to get a 2-category of monads from a 2-category.

Posted by: Jamie Vicary on July 19, 2009 11:00 PM | Permalink | Reply to this

Jamie Vicary wrote:

Hmm. So what’s a ‘monad map’?

That’s how I used to feel, too. That’s why ‘map of monads’ is on the list of concepts that used to hurt my head. I wrote:

If we have two monads on different categories, they can be related by a ‘map of monads’ — see section 6.1 of Leinster’s book. I should like these just as much as distributive laws, but I don’t yet. At least I know the fundamental theorem about them: if we have a map of monads, we get a functor sending algebras of the first to algebras of the second.

You could probably have fun figuring out the definition from this one clue.

But if you click the link, you’ll be effortlessly whisked to page 148 of Tom’s book — page 178 of the PDF file — where you may gaze at the diagrams defining a lax map of monads. Then your work will start: figuring out what they mean.

I urge you to redraw these diagrams as 2-categorical string diagrams. Then you’ll see that a lax map of monads really is a good generalization of ‘homomorphism between monoid objects in the same monoidal category’.

Briefly, given monads $T : E \to E$ and $T': E' \to E'$, a lax map of monads is a morphism $Q: E \to E'$ together with a 2-morphism

$\psi: T' Q \Rightarrow Q T$

which you should draw as a kind of ‘braiding’.

We require that $\psi$ make two 2-morphisms

$T' T' Q \Rightarrow Q T$

equal: the one where you multiply the two $T'$s and then braid them past $Q$, and then one where you braid them both past $Q$ and the multiply them.

This is the way $Q$ and $\psi$ ‘preserve multiplication’. We also impose a law saying that they ‘preserve the unit’.

If $E = E'$ and $Q$ is the identity, we’re back down to the usual definition of a homomorphism between monoid objects in the same monoidal category!

So, this concept should not be the private reserve of category theorists. Indeed, if we look at the special case of monads in the bicategory of rings, bimodules and bimodule homomorphisms, this concept should reduce to a very nice concept of ‘homomorphism from an $R$-algebra to an $R'$-algebra riding atop a chosen $(R,R')$-bimodule’. Is it already known? It must be!

Posted by: John Baez on July 20, 2009 8:38 AM | Permalink | Reply to this

Jamie wrote:

I was recently talking about monads to Andrei Akhvlediani, a PhD student in our group here in Oxford who’s currently studying Steve Lack’s work on PROs and PROPs and trying to generalize it in some interesting ways. He pointed out to me that given a 2-category, a nice way to get a 2-category of monads out of it is to consider internal monads, bimodules and bimodule homomorphisms. The usual rules for defining bimodules carry over easily; you just have to replace tensor product with 1-cell composition. I would have thought this is one of the most interesting ways to get a 2-category of monads from a 2-category.

It sounds interesting, but I think it’s different than the ‘map of monads’ idea. Given monads $T : E \to E$ and $T' : E' \to E'$, a bimodule would be a morphism $B: E \to E'$ equipped with 2-morphisms

$B T \Rightarrow B$

$T' B \Rightarrow B$

satisfying some equations. That looks pretty different than my sketchy description of a map of monads.

There are probably some fun relationships between the two ideas, though!

Posted by: John Baez on July 20, 2009 9:53 AM | Permalink | Reply to this

I have been studying Leinster’s work on higher categories and operads recently, so I’ll have a go at the cartesian monads question.

I’m sure most people around here are familiar with the fact that ordinary categories can be defined as monads in the bicategory of spans of sets. (If you haven’t seen this before, it’s a fun short exercise.) Multicategories can be defined in a similar way. A multicategory is like an ordinary category where each morphism has a list of objects as its domain, and a single object as its codomain (think vector spaces and multilinear maps).

To see how a multicategory $C$ can be defined as a monad in some appropriate bicategory, let $C_0$ be the set of objects of $C$, and notice that the domain of a morphism of $C$–a finite list of objects–is an element of $T C_0$, where $T$ is the free monoid monad. In this way the data for $C$ can be conveniently organized in the diagram

(1)$\begin{matrix} &&C_1&&\\ &\overset{d}\swarrow& &\overset{c}\searrow&\\ T C_0 &&&& C_0. \end{matrix}$

Leinster’s work is built on the idea of generalized multicategories, where the domain of a morphism can have a more general, higher dimensional shape than just a list. This is accomplished by considering a category $\mathcal{E}$ other than $\mathrm{Set}$, and a monad $T$ on $\mathcal{E}$, and mimicking the above construction. So the data for a $T$-multicategory is a digram in $\mathcal{E}$ like the one above.

To state the structure required on the data for a $T$-multicategory, we want to define a bicategory in which the above span is an endomorphism. Then a $T$-multicategory will be such a span together with structure making it a monad in that bicategory. The bicategory is $\mathcal{E}_{(T)}$, the bicategory of $T$-spans in $\mathcal{E}$. Its objects are the objects of $\mathcal{E}$, and its morphisms are spans

(2)$\begin{matrix} &&M&&\\ &\overset{d}\swarrow& &\overset{c}\searrow&\\ T E &&&& E'. \end{matrix}$

This won’t in general be a bicategory without a few extra assumptions. Identity spans are defined using the unit $\eta: \mathrm{Id}\to T$. Composition of spans is defined using pullbacks and the multiplication $\mu: T^2\to T$, so the category $\mathcal{E}$ must at least have pullbacks–usually it will be cartesian. The associativity and unit 2-cells are defined using the universal property of the pullbacks. However, these 2-cells won’t in general be invertible. In fact, it turns out that requiring the monad $T$ to be cartesian is exactly what is needed to ensure that the coherence 2-cells are isomorphisms, and hence that $T$-spans do in fact form a bicategory. Maybe this should be the “fundamental theorem of cartesian monads”.

Now $T$-multicategories can be defined simply as monads in the bicategory $\mathcal{E}_{(T)}$. Then for $T$ the identity monad on $\mathrm{Set}$, $T$-multicategories are exactly ordinary categories, and for $T$ the free monoid monad on $\mathrm{Set}$, $T$-multicatories are exactly ordinary multicategories.

As an indication of how this theory is useful as a language for higher categories, take $T$ to be the free strict-$\omega$-category monad on the category of globular sets. Then $T$-multicategories with exactly one object are called globular operads, and Leinster defines one such globular operad (the initial “globular operad with contraction”) for which whose algebras are weak $\omega$-categories.

(I know, I really should get around to putting this on the nLab.)

Posted by: Patrick Schultz on July 19, 2009 8:50 PM | Permalink | Reply to this

i.e. TT ⇒ T except only on a subfunctor of TT such that…

Posted by: jim stasheff on July 19, 2009 9:17 PM | Permalink | Reply to this

Sorry, I’m not sure I understand the question. There are two monads arising in the definition of a $T$-multicategory: there is the monad $T$, which is an ordinary monad on a category $\mathcal{E}$, i.e. a monad in the 2-category $\mathrm{End}(\mathcal{E})$, and then a $T$-multicategory is itself defined to be a (abstract) monad in the bicategory $\mathcal{E}_{(T)}$ of $T$-spans in $\mathcal{E}$. Which, if either, are you referring to?

Posted by: Patrick Schultz on July 19, 2009 11:15 PM | Permalink | Reply to this

Personally I would want to treat a ‘partially defined monad’ as a monad in the 2-category of

• categories,
• partially defined functors, and
• partially defined (?) natural transformations

or something like that. Then I’d get for free all the general results that apply to monads in any 2-category, and the work would devolve to understanding the properties of this particular 2-category.

Posted by: John Baez on July 20, 2009 10:26 AM | Permalink | Reply to this

And on the partially defined front, I’m sure it would worth spending time on Lack and Cockett’s three papers on restrictive categories.

Posted by: David Corfield on July 20, 2009 10:38 AM | Permalink | Reply to this

Thanks for the explanation, Patrick! Maybe this is the ‘fundamental theorem of cartesian monads’. But was it known before Tom came along? Somehow I’d imagined cartesian monads were around for quite a while… along with some simpler, older theorem justifying their study. But anyway, maybe this will do.

Hmm, looking at Tom’s book, I see $T$-multicategories were first invented by Burroni in 1971, under the name ‘$T$-category’.

By the way: I only now noticed that Tom’s ‘fc-multicategories’ were the special case of $T$-multicategories where $T$ is the ‘free category’ monad. I’d always thought this obscure abbreviation must refer to some terribly intimidating concept: ‘f***ing-crazy multicategories’, or something like that.

Posted by: John Baez on July 19, 2009 9:26 PM | Permalink | Reply to this

Probably there are other results about cartesian monads fundamental to some other application, such as topos theory. Johnstone would probably be able to answer that, but I don’t have a copy of the Elephant with me. But anyways, that $\mathcal{E}_{(T)}$ is a bicategory if $T$ is cartesian is certainly the fundamental result for how cartesian monads come up in Leinster’s work.

Speaking of fc-multicategories, I find the concept fascinating (and not that intimidating when you have examples in mind, like how rings have not only homomorphisms between them, but also bimodules). I haven’t read much of Mike Shulman’s work on ‘framed bicategories’, but the two ideas are obviously closely related. One of these days, I want to figure out the connection.

Posted by: Patrick Schultz on July 19, 2009 11:06 PM | Permalink | Reply to this

This is just too good an opportunity to pass up to advertise Mike Shulman and I’s recent paper, “A Unified Framework for Generalized Multicategories” (here).

In addition to Leinster’s (and Burroni’s) T-multicategories, there have been a number of other authors who have defined generalized multicategories for a monad T - see, for example, the “(T,V)-algebras” of Clementino et. al, or Hermida’s “From Coherent Structures to Universal Properties”. We show that, by using monads on fc-multicategories (rather than, say, cartesian monads), we can recover all of the examples, and much of the theory, of each of these different authors in a single framework.

The construction Patrick mentions above essentially carries over exactly to our setting (first, change horizontal arrows X -> Y to arrows X -> TY, then take monads in that fc-multicategory). However, our setup has a couple of advantages: first of all, as mentioned above, we cover not only generalized multicategories of Leinster’s type, but also the other types; things like topological spaces, metric spaces, and Lawvere theories. Secondly, the functors between these generalized multicategories fall directly out of our setup, whereas they need to defined in the other contexts.

We are also able to do things like define “representable” generalized multicategories, extending Hermida’s idea. The table on pg. 6 of our paper has a nice comparison of all the different examples we get of generalized multicategories and their “representable” counterparts. For example, the representable multicategories are monoidal categories, the representable functors A -> S are pseudofunctors S^op -> cat, and the representable T1 spaces are compact Hausdorff spaces.

I can explain more, if people are interested.

Posted by: Geoff Cruttwell on July 20, 2009 1:36 AM | Permalink | Reply to this

Thanks for pointing this out, Geoff. The paper looks very interesting, I’m looking forward to reading it. I’m also glad to see fc-multicategories (or virtual double categories, as you call them) being made such essential use of. When I first read about them in Tom’s book, I felt they were an important and natural idea. I think your paper will help me understand them better.

Posted by: Patrick Schultz on July 20, 2009 3:41 AM | Permalink | Reply to this

I agree, they certainly are important. I should also mention the connection to Shulman’s “framed bicategories”, since you commented on those. A framed bicategory is essentially a pseudo-double category in which a horizontal arrow can be restricted, or extended, along vertical arrows.

Mike found a nice extension of this idea to virtual double categories (or fc-multicategories). (For those not familiar, they are essentially pseudo-double categories without composition of horizontal arrows). If we ask that the virtual double category has horizontal units, and that the horizontal arrows can be restricted (not necessarily extended) along the vertical arrows, then we get what we call a virtual equipment (framed bicategories are equivalent to Wood’s equipments, hence the name here).

To get some of the nice results about generalized multicategories, we need to work with a virtual equipment (all the examples, such as spans and matrices, are virtual equipments), rather than the more general fc-multicategories. So, these seem to be an important structure as well.

Posted by: Geoff Cruttwell on July 20, 2009 4:34 AM | Permalink | Reply to this

(I know, I really should get around to putting this on the nLab.)

Your comment is now linked to from [[cartesian monad]]. That's quick and easy to do and means that it won't be lost, even if it takes a long time for you (or somebody else) to actually put it on the Lab.

Posted by: Toby Bartels on July 19, 2009 11:16 PM | Permalink | Reply to this

### one small step to the nLab, one giant step for a comment

(I know, I really should get around to putting this on the nLab.)

Your comment is now linked to from [[cartesian monad]]. That’s quick and easy to do and means that it won’t be lost, even if it takes a long time for you (or somebody else) to actually put it on the Lab.

Yes, and one should think about it:

you go through the trouble of typing a long and detailed technical comment to the blog. All the work is already done and you have itex source code in front of you.

Now you can

- either just post to the blog. The thing will appear. One and a half readers will take due detailed notice of it. Then it drowns under the comment and entry inflow and is forever forgotten.

- or you go through the minimal trouble of just pasting the material into an $n$Lab entry also and include one or two links back and forth on the crucial keywords.

This way the chances of your technical comment being read by somebody genuinely interested in reading it increase drastically, certainly giving a good ratio of outcome over effort.

You should all get into the habit of doing that. Because before long Toby and I will no longer have the energy to do it for you.

Posted by: Urs Schreiber on July 22, 2009 9:31 PM | Permalink | Reply to this

### Monoid enumerations online; Re: The Monads Hurt My Head — But Not Anymore

Trying to reboot my brain from monoids to monads, staring with small finite enumerations…

Number of monoids (semigroups with identity) of order n.
n a(n)
1 1
2 2
3 7
4 35
5 228
6 2237
7 31559
8 1668997

Number of monoids (semigroups with identity) of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Number of labeled monoids of order n.

and so forth…

Posted by: Jonathan Vos Post on July 20, 2009 8:07 PM | Permalink | Reply to this

### Re: Monoid enumerations online

The last item should be listed first. Then we have:

• number of monoid structures, up to equality, on a given set of cardinality $n$,
• number of monoid structures, up to isomorphism, on a given set of cardinality $n$,
• number of monoid structures, up to (anti)isomorphism, on a given set of cardinality $n$.

Then each is a quotient of the previous.

Posted by: Toby Bartels on July 20, 2009 8:53 PM | Permalink | Reply to this

### Daniel S. Freed’s n-category joke in Bull AMS; Re: Monoid enumerations online

Thank you Toby. Very good point.

REMARKS ON CHERN-SIMONS THEORY, by DANIEL S. FREED
BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 46, Number 2, April 2009, Pages 221–254

The paper being illuminating, even though I have to read each page a dozen times to get half the point.

Footnote #7:
“I once joked that every mathematician also has a category number, defined as the largest integer n such that (s)he can think hard about n-categories for a half-hour without contracting a migraine. When I first said that my own category number was one, and in the intervening years it has remained steadfastly constant, whereas that of many around me has climbed precipitously, if not suspiciously.”

Posted by: Jonathan Vos Post on July 21, 2009 10:01 PM | Permalink | Reply to this

### Re: Daniel S. Freed’s n-category joke in Bull AMS; Re: Monoid enumerations online

About that footnote: oh, snap! I can just hear the guffaws among the assembled mathematicians.

After all these years, the words of Barry Mitchell (Theory of Categories, 1965, from the Preface) still carry some truth:

A number of sophisticated people tend to disparage category theory as consistently as others disparage certain kinds of classical music. When obliged to speak of a category they do so in an apologetic tone, similar to the way some say, “It was a gift – I’ve never even played it” when a record of Chopin Nocturnes is discovered in their possession. For this reason I add to the usual prerequisite that the reader have a fair amount of mathematical sophistication, the further prerequisite that he have no other kind.

Posted by: Todd Trimble on July 22, 2009 4:13 AM | Permalink | Reply to this

### Re: Daniel S. Freed’s n-category joke in Bull AMS;

About that footnote: oh, snap! I can just hear the guffaws among the assembled mathematicians.

After all these years, the words of Barry Mitchell (Theory of Categories, 1965, from the Preface) still carry some truth:

That’s a nice quote. I thought that’s worthwhile to include in the entry on [[category theory]], as I now did. See also my corresponding comments at [[latest changes]].

But to be fair to be Dan Freed here, the quote given doesn’t so much complain about category theory as about the intractibility of higher category theory.

And there I think nobody would deny the truth that in particular algebraic definitions of $n$-categories already for low $n$ greater than 1 are such that half-hour occupation with them is not so unlikely to cause migraines in ordinary mortals.

How many people besides you, Todd, can claim to have seriously thought just about the definition of tetracategory, let alone its application to something.

I’d guess that Freed’s remark is motivated from such phenomena.

On the other hand, for most everything he describes in the context in which the quote is given, of course its not general higher categories that are relevant, but just $(\infty,0)$- and $(\infty,1)$-categories, at worst $(\infty,2)$-categories. For these there are less-migraine-causing tools.

I think there is some truth to the statement on page 15 of HTT

The obvious problem with [[tricategories ]] is that an explicit definition is extremely complicated, to the point where it is essentially unusable.

I am guessing that the quote we are talking about has its origin in such observations. Which are only fair, I think, at least as long as there is no proof-of-principle of the contrary.

Posted by: Urs Schreiber on July 22, 2009 11:34 AM | Permalink | Reply to this

### Re: Daniel S. Freed’s n-category joke in Bull AMS;

On the other hand, for most everything he describes in the context in which the quote is given, of course its not general higher categories that are relevant, but just $(\infty,0)$- and $(\infty,1)$-categories, at worst $(\infty,2)$-categories.

Indeed, it's not so much the level of $n$ as the level of $r$ in $(n,r)$-category that matters.

Posted by: Toby Bartels on July 23, 2009 12:38 AM | Permalink | Reply to this

### Re: Daniel S. Freed’s n-category joke in Bull AMS; Re: Monoid enumerations online

Well, but the crack about categories (ordinary 1-categories) giving him a headache is just so old and tired. You could say it was only a joke, but it reminds me of what the comedienne Ellen DeGeneres says about people who kid around in the manner of, “Is that a new hair cut? Hope you didn’t pay for that thing. I’m just kidding” – well, then you don’t know how to kid properly, because I should be laughing too. And then to follow up with other people’s rising tolerance for $n$-categories being suspicious to him – hm.

The support for category theory in the United States is still, as far as I am aware, just wretched, and little jokes like that (or someone popping up a slide of St. Sebastian in his martyrdom to suggest how tortured they are by the theory of arrows) is on the milder end of a continuum of attitudes which slide very easily into outright hostility (and outright defunding) directed against categories. You can’t deny, Urs, that this takes place. Why make jokes about something so beaten down as it is?

I can understand the criticism you make about algebraic $n$-categories in particular. The situation might improve if there were more than a handful of people working on it.

Posted by: Todd Trimble on July 22, 2009 12:55 PM | Permalink | Reply to this

### Re: Daniel S. Freed’s n-category joke in Bull AMS; Re: Monoid enumerations online

Maybe I may ask you to re-read the quote carefully again. My impression is that it’s actually not aimed the way that you take it. Maybe I am wrong. But please have another look and let me know.

Notice that the author of the quote says that 1-catgeories are a topic that he can think hard about without getting a headache!

But that for 2-categories he used to develop a migraine after half an hour.

I really just read this as saying that higher categories quickly tend to become a thorny issue, which is undeniable and doesn’t say that one is making fun of the subject.

Just as I would say: I can think about solving elliptic integrals of the second kind only for half an hour without getting a headache.

That’s probably even true. And saying it is far from asserting that somehow the very notion of elliptic integral of the second kind is problematic. Far from it. It just says that higher elliptic integrals are a thorny subject that gets harder and harder for humans to tackle as the degree increases.

If you look at the last part of the quote, I think it is clear that the joke expressed here is an expression of the surprise that there are suddenly so many young people talking about higher categories, where previously it used to be regarded already as quite sophisticated to just talk about categories.

Posted by: Urs Schreiber on July 22, 2009 2:15 PM | Permalink | Reply to this

### Re: Daniel S. Freed’s n-category joke in Bull AMS; Re: Monoid enumerations online

My reading matched yours, Urs. I took the last part of the quote as being self-deprecating.

Posted by: Tim Silverman on July 22, 2009 3:08 PM | Permalink | Reply to this

### Re: Daniel S. Freed’s n-category joke in Bull AMS; Re: Monoid enumerations online

Whoops!! You’re right. I’ve got egg all over my face. I totally misread, and should have spotted the ‘without’.

Ugh, I am so embarrassed. I guess the only lame excuse I have is that I’ve heard stupid (and unkindly meant) jokes about categories for an awfully long time, so that I’ve become conditioned, Pavlov-like, to just thinking, “yeah, yeah, yeah, I’ve heard all that before…” and no longer pay as careful attention as I should.

I’m going to crawl into a hole now… anyway, on rereading, you’re absolutely right, and I’m sorry to Dan Freed for what I said. Not that he’s necessarily reading this.

Posted by: Todd Trimble on July 22, 2009 3:33 PM | Permalink | Reply to this

### Re: Daniel S. Freed’s n-category joke in Bull AMS; Re: Monoid enumerations online

Todd explained:

I’ve heard stupid (and unkindly meant) jokes about categories for an awfully long time, so that I’ve become conditioned, Pavlov-like …

That’s a pretty natural reaction, Todd. I was a little surprised when I read your comment, but after a couple of moments’ reflection I suspected something like this might be going on—not least because I’ve been known to do the same thing myself on some topics, and with more acid and at greater length, so I know just how you felt. I doubt I’m the only one.

Posted by: Tim Silverman on July 23, 2009 12:35 AM | Permalink | Reply to this

### Re: Daniel S. Freed’s n-category joke in Bull AMS; Re: Monoid enumerations online

I can think about solving elliptic integrals of the second kind only for half an hour without getting a headache.

This made me laugh out loud. It’s a great name, “elliptic integral of the second kind”. Something like a close encounter of the third kind. (By the way, I’m holding thumbs John is right).

Posted by: Bruce Bartlett on July 22, 2009 7:27 PM | Permalink | Reply to this

### Posting to the n-Lab for dummies

or you go through the minimal trouble of just pasting the material into an nLab entry also and include one or two links back and forth on the crucial keywords.

could you break that down into baby steps?

just pasting the material into an ?existing? nLab entry

and especially how to
include one or two links back and forth on the crucial keywords.

Posted by: jim stasheff on July 22, 2009 9:53 PM | Permalink | Reply to this

### Re: Posting to the n-Lab for dummies

Posted by: Eric Forgy on July 22, 2009 10:59 PM | Permalink | Reply to this

### Re: Posting to the n-Lab for dummies

could you break that down into baby steps?

Eric Forgy provided

Give this a try:

How to Copy and Paste Material from the n-Cafe and Include Links Back and Forth

Wow. Great. I was on the train thinking about creating an entry HowTo contribute to the $n$Lab for $n$Caféists and now it already exists in a much better way than I could ever have done.

Thanks, Eric!.

Maybe while you are at it, becuase I can’t easily do it in the fashion that you started doing it, I’d have, if I may, the following additional requests:

- a comment along the lines: if you don’t know where to put your material, google for “$n$Lab [your keywords]” to see which entries exist

- if the entry with the title that you are looking for does not yet exist, find some entry that is somehow related, go to the edit window as described above and add a line “see also [[your keyword]]”. Then after hitting submit the [[your keyword]] will appear with a clickable question mark. Click on that question mark to create the desired entry.

- OPTIONAL if you want to be sophisticated: add links back and forth between other nLab entries: enclose important keyword sin your material in double brackets. That equips them with links to the corresponding other nLab entries. Conversely, go to related nLab entries and add pointers to the entry you just created.

- general remark: don’t worry if things don’t come out quite the way you would like them at the first go – the important point is that we have your material in some form – others will eventially go over the entry and polish it, if necessary

Posted by: Urs Schreiber on July 22, 2009 11:17 PM | Permalink | Reply to this

### Re: Posting to the n-Lab for dummies

And I also vote for:

- let’s move this entry from your personal web to the nLab web proper

- let’s call it something slightly more general as “HowTo get started quickly” or something

- and make it such that it also addresses readers who may not come from the Cafe.

just a suggestion, while you are at it :-)

Posted by: Urs Schreiber on July 22, 2009 11:32 PM | Permalink | Reply to this

### Re: Posting to the n-Lab for dummies

This last suggestion I assign to someone new who has yet to contribute to the nLab. Come on lurkers! :) You know who you are! The change can be done anonymously if you want, so no worries (and no excuses!).

Instructions:

1. Create a link on the main nLab grid somewhere by adding [[How To Get Started]] (or something) to any existing page, e.g. [[HowTo]]
2. Follow the link that you created. It will take you to a new Edit Box
3. Hit the “Edit” button on the page I created for Jim
4. Copy all the content and paste it into your new page on the main nLab grid
5. Hit the “Cancel” button on my page since you probably do not want to make changes to it
6. Watch Urs dance in delight
Posted by: Eric Forgy on July 22, 2009 11:51 PM | Permalink | Reply to this

### Re: Posting to the n-Lab for dummies

Done—but now the attribution makes it look as though I created the content rather than just the page. :-(

Posted by: Tim Silverman on July 23, 2009 12:44 AM | Permalink | Reply to this

### Re: Posting to the n-Lab for dummies

Excellent! Thanks :)

To atone for the attribution, you could create ANOTHER page ;)

Posted by: Eric Forgy on July 23, 2009 2:04 AM | Permalink | Reply to this

### Re: Posting to the n-Lab for dummies

And I must confess, I put it on my personal page on accident :)

Posted by: Eric Forgy on July 22, 2009 11:56 PM | Permalink | Reply to this

### Monads and the field of one element

This is slightly off-topic, but a beautiful and deep use of monads that should appeal to many cafe regulars appears in the recent paper Lambda-rings and the field with one element by the n-category cafe’s own James (Borger). Here James gives what to me is the first satisfying definition of the field of one element, by pointing out it has actually been around for decades in the form of the theory of $\Lambda$-rings (aka rings with exterior power operations, as they appear in the theory of Chern classes or symmetric functions - they give the formal structure behind plethysm in symmetric function theory or the splitting principle for characteristic classes… a POV on the theory very much in the flavor of this cafe is in James’ paper with Ben Wieland, Plethystic Algebra).

James observes here that the monadic structure of Lambda-rings (the functor that takes a ring to the free $\Lambda$-ring on it is a comonad, or equivalently a monad on affine schemes) can really be interpreted as DESCENT from the integers (or other coefficients) down to the field with one element. This gives a non-formal definition of what algebraic geometry over $F_1$ is, and he already proves deep arithmetic results about them here (giving as consequences some of the properties people like to take as definitions of $F_1$, but hinting at a much richer underlying theory).

Posted by: David Ben-Zvi on July 23, 2009 2:29 AM | Permalink | Reply to this

### Re: Monads and the field of one element

On the strength of this I started a page for Lambda rings which needs much help. $\lambda$-ring and $\Lambda$-ring are used interchangeably? In Plethystic Algebra it says

It is an exercise in definitions to show that in this way, a $\Lambda$-ring structure on a ring $R$ is the same as a $\lambda$-ring structure. (p. 11)

The $\Lambda$ comes from the symbol for the ring of symmetric functions in countably many variables. The point is that $\Lambda$ is also generated by elementary symmetric functions on these variables?

Posted by: David Corfield on July 23, 2009 9:37 AM | Permalink | Reply to this

### Re: Monads and the field of one element

Yes, a $\Lambda$-ring is the same thing as a $\lambda$-ring. This was a bit of a joke, but there was a point to it. The joke is that the $\lambda$-operations in $\lambda$-ring theory, generate the ring of all symmetric functions, which happens to often be denoted $\Lambda$. (This is essentially Newton’s theorem that every symmetric function is a polynomial in the elementary symmetric functions.) So in an amazing coincidence of notation, a $\Lambda$-ring (i.e. a $P$-ring in the sense of plethystic algebra in the case $P=\Lambda$) is the same thing as a $\lambda$-ring.

Here is an analogue. Let $M$ be a module over the ring $C$ of smooth functions on the real numbers. Let $D$ be the ring of differential operators on $C$ generated (over $C$) by the usual derivative operator $d$. Then a $D$-module structure can be described simply as a linear map $d:M\to M$ satisfying $d(fm)=d(f)m+fd(m)$ for all $f\in C$. This you might call a $d$-module structure on $M$. The point is that a $D$-module structure is the same as a $d$-module structure. In the first case, we take the maximal approach and use all operators, and in the second, we take the minimal approach and just use a generating set for $D$, i.e. $\{d\}$.

I prefer the notation $\Lambda$-ring because it’s an instance of a general point of view, whereas the definition of $\lambda$-ring looks at a formal level like it’s from Mars (at least it did to me).

By the way, since we’re talking about cafe regulars, I should mention that this point of view was also noticed (presumably first) by Tall and Wraith in 1970 or so.

Posted by: James on July 23, 2009 10:12 AM | Permalink | Reply to this

### Re: Monads and the field of one element

And this $\Lambda$ is Hazewinkel’s Symm:

Symm is an object with an enormous amount of compatible structure: Hopf algebra, inner product, selfdual (as a Hopf algebra), PSH, coring object in the category of rings, ring object in the category of corings (up to a little bit of unit trouble), Frobenius and Verschiebung endomorphisms, free algebra on the cofree coalgebra over Z (and the dual of this: cofree coalgebra over the free algebra on one element), several levels of lambda ring structure, …

So is there a snappy description of what a $P$-ring is?

Posted by: David Corfield on July 23, 2009 11:05 AM | Permalink | Reply to this

### Re: Monads and the field of one element

Yes! $P$ is a commutative ring together with two types of structure on the set-valued functor $Hom(P,-)$ it represents: (i) a commutative ring structure, which means we can view it as an endofunctor on the category of commutative rings (so $\mathrm{Spec} P$ is a commutative ring scheme), and (ii) a comonad structure on this endofunctor. Then a $P$-ring is simply a commutative ring equipped with a co-action of the this comonad.

This is similar to the situation for operators on abelian groups. Let $A$ be an abelian group, then $Hom(A,-)$ is already abelian-group-valued, so we don’t need to do step (i). Then a comonad structure on the endofunctor $Hom(A,-)$ is the same as a (possibly non-commutative) ring structure on $A$.

The moral of the story is that operators on abelian groups naturally form rings, and operations on commutative rings naturally form $P$’s. What you call such a $P$ is another matter. Ben and I called such a thing a plethory, with a wink at plethysm, which is another name for composition of symmetric functions. Other people take a historical approach and use “Tall-Wraith” combined with some other words. Probably the most descriptive and generalizable name would be something like “composition object in the category of commutative rings”. Since I like the word plethysm, I’m also partial to the general naming scheme “plethystic object in the category $C$”, where $C$ could be the category of commutative rings, the category of abelian groups, and so on.

Posted by: James on July 23, 2009 1:01 PM | Permalink | Reply to this

### Re: Monads and the field of one element

On the very strong recommendation of David Ben-Zvi that James’s approach to $\mathbb{F}_1$ is the best and most satisfying one given so far, I’ve been trying to do some reading here. I’ve found this other article by Hazewinkel extremely helpful in providing background material for plethystic algebra and for James’s field with one element paper, so I thought I’d pass it along.

It’s apparent that $\lambda$-rings are at the center of so many things (representation theory of symmetric groups, Grothendieck-Hirzebruch-Riemann-Roch, K-theory). Hopefully this will be brought out in the nLab by and by.

Posted by: Todd Trimble on July 25, 2009 1:58 PM | Permalink | Reply to this

### Re: Monads and the field of one element

Thanks for pointing out that article by Hazewinkel, Todd. David Corfield has also recommended it to me.

Here are some things I believe to be true, which would go a certain ways towards explaining the ubiquity of $\lambda$-rings. I could be a bit mixed up about some of the nuances, so I’d appreciate corrections.

(The last time I made some of these statements, James called them the orthodox view. On the one hand this was reassuring, but on the other hand it made me feel like an old fogy who is missing out on the real fun, like the stuff about Witt vectors that Hazewinkel explains. The following remarks are still entirely Wittless.)

The Grothendieck group of any symmetric monoidal abelian category is a $\lambda$-ring. A closely related fact: the Grothendieck group of the free symmetric monoidal abelian category on one generator is the free $\lambda$-ring on one generator. Objects in the free symmetric monoidal abelian category on one generator are called ‘Schur functors’, because for obvious reasons they act as functors on any symmetric monoidal abelian category. The irreducible objects in this category are called ‘Young diagrams’. Elements of the free $\lambda$-ring on one generator are also called ‘symmetric functions in countably many variables’.

All of this is part of a bigger picture I was struggling to understand while trying to answer a question by Allen Knutson. I got as far as making a big conjecture, but not as far as proving it. I really should come back to that.

Anyway, here are some more speculations… and here I must apologize to James for tromping ignorantly and with must seem like overweening ambition on turf he knows far better than I do.

I’ve said that you can get a $\lambda$-ring by taking the Grothendieck group of a symmetric monoidal abelian category. I would like to even say that a $\lambda$-ring is precisely this sort of thing: that generically, this $\lambda$-ring structure is all you get on the Grothendieck group of such a category. It would take some work to formalize that and see if it’s true… and I should add that to my to-do list.

But never mind: for now let me just say very vaguely that we can think of the concept of ‘$\lambda$-ring’ as a decategorified version of the concept of ‘symmetric monoidal abelian category’.

This suggests that if someone does something with $\lambda$-rings, someone else should try to do it using symmetric monoidal abelian categories.

But ‘symmetric monoidal abelian category’ is also a plausible candidate for a categorified notion of ‘commutative ring’ or ‘commutative rig’.

So, I can easily imagine someone wanting to redo algebraic geometry with symmetric monoidal abelian categories replacing commutative rings. And this might explain why someone else would want to redo algebraic geometry with $\lambda$-rings replacing commutative rings.

Posted by: John Baez on July 25, 2009 4:53 PM | Permalink | Reply to this

### Re: Monads and the field of one element

Ah, okay, thanks for these suggestions; they are helping me put some things into their proper categorified contexts!

So if I follow, part of what you’re saying is that classical plethysm (on the free $\lambda$-ring on one variable, $\Lambda$) can be seen as the Grothendieck group decategorification of the substitution product on the category of linear species, $Vect^{\mathbb{P}}$, where the exponent is the category of finite sets and bijections (or permutations), and the base consists of finite-dimensional (say complex) vector spaces. In fact, if I’m getting it right, that could be taken as an obvious definition of (classical) plethysm.

That would be just the kind of hint I need to get a grip on things.

One thing I don’t know, that I feel I should better understand, is whether (1) the ring of representations of the general linear group $GL(\mathbb{F}_q)$ (i.e., the groupoid of isomorphisms of finite-dimensional vector spaces over $\mathbb{F}_q$) also admits a structure of plethory, where the ring structure comes from Green convolution, and (2) whether there’s a nice neat categorified explanation of this fact which would be in some sense parallel to the case “$q = 1$”, where we would use $\mathbb{P}$ instead of $GL(\mathbb{F}_q)$. Presumably the categorified explanation would start by saying that $Vect^{GL(\mathbb{F}_q)}$ is the free something-or-other $\mathbb{C}$-linear enriched category on one generator? (But I really don’t know what the “something-or-other” would be.)

Or, perhaps I’m barking up the wrong tree.

Posted by: Todd Trimble on July 26, 2009 12:43 AM | Permalink | Reply to this

### Re: Monads and the field of one element

From what I remember, the plethysm operation on $\Lambda$ comes from the representation theory of the symmetric groups $S_n$ by way of wreath products, specifically a map $S_n \ltimes S_m^n \to S_{m n}$. Changing all the $S$’s to $GL(F_q)$’s here doesn’t look possible, but it might be possible to change all but the first one. So it could be true that the $GL(F_q)$ analogue would be something like a $\Lambda$-$\Lambda$-biring.

Orthodox topics like this have occurred to me before, but I don’t think about them much (however interesting they might be!), so please don’t bet the farm on any of this.

Plethories seem kind of hard to come by. It would not surprise me if there were a classification theorem.

Posted by: James on July 26, 2009 6:26 AM | Permalink | Reply to this

### Re: Monads and the field of one element

James wrote:

Plethories seem kind of hard to come by. It would not surprise me if there were a classification theorem.

I would like to prove you wrong, just so I can write a paper entitled ‘A plethora of plethories’.

Posted by: John Baez on July 26, 2009 3:26 PM | Permalink | Reply to this

### Re: Monads and the field of one element

A few words on the orthodox/heterodox approaches to $\Lambda$-rings…

By the orthodox approach I mean everything involving symmetric functions, including symmetric groups, decategorifying linear tensor categories, K-theory, and so on. Everything mentioned above or pretty much anywhere takes this point of view. Indeed Grothendieck invented $\Lambda$-rings to understand the extra structure on the Grothendieck group inherited from tensor operations (exterior powers etc) on tensor categories.

By the heterodox point of view, I mean everything involving Frobenius lifts. A Frobenius lift at a prime $p$ on a commutative ring $A$ is an endomorphism which reduces to the $p$-th power ‘Frobenius’ map $x\mapsto x^p$ on the quotient ring $A/pA$. If the orthodox point of view is closely related to $K$-theory, the heterodox point of view is closely related to cohomology (e.g crystalline cohomology).

The connection between the two is as follows. Given any $\Lambda$-ring structure in the orthodox sense, the $p$-th Adams operation $\psi_p$ is a Frobenius lift, and the $\psi_p$ for different primes $p$ all commute with each other. (You can see the split forming already: the Adams operations don’t exist on tensor categories until you decategorify.) Wilkerson’s theorem gives the converse, at least in the absence of torsion. It says this: Let $A$ be a commutative ring which is torsion free (in the additive sense), and let $\psi_p$ be a family of commuting Frobenius lifts on $A$ indexed by the prime numbers $p$. Then there is a unique $\lambda$-ring structure on $A$ whose Adams operations are the given Frobenius lifts $\psi_p$. The proof is an unenlightening (IMHO) exercise in composition of symmetric functions. Further, a ring map between two torsion-free $\lambda$-rings is a map of $\lambda$-rings if and only if it commutes with the Adams operations. Thus we have an explicit equivalence between the category of torsion-free $\lambda$-rings and the category of torsion-free commutative rings equipped with commuting Frobenius lifts.

[In fact, we can get around the torsion-free restriction with a dash of category theory. The category of $\lambda$-rings is both monadic and comonadic over the category of rings (commutative, which I will now stop writing). The point is that that the comonad (and hence the monad) is determined by what happens in the torsion-free setting, where it can be described in terms of Frobenius lifts. How does this work? The category of torsion-free $\lambda$-rings is also comonadic over the category of torsion-free rings. You can show this from either orthodox point of view or the heterodox point of view (where your definition of torsion-free $\lambda$-ring would be the one involving Frobenius lifts). Let $W'$ be this comonad, on the category of torsion-free rings. Now let $W$ be the left Kan extension of this functor (or more precisely the composition of $W'$ with the embedding of torsion-free rings into all rings) to the category of all rings. Then $W$ is a comonad on the category of all rings, and the category of its coalgebras is equivalent to the category of $\lambda$-rings, torsion free or not. By the way, this comonad is precisely the ‘big’ Witt vector functor.]

The odd thing is that pretty much no one cares that the two approaches are equivalent, including me. Everything most people want to do is on the orthodox side, and everything I want to do is on the heterodox side. The deeper meaning of connection between the two is completely unclear to me. It could be that it should be viewed as an accident. Indeed the heterodox point of view generalizes to families of Frobenius lifts on other Dedekind domains with finite residue fields in a way that perhaps the orthodox point of view doesn’t. For instance over $F_p[x]$ (instead of $\mathbf{Z}$), we would look at families of $\psi$-operators indexed by the irreducible monic polynomials $f(x)$, and each $\psi_{f(x)}$ would have to be congruent to the $q$-th power map modulo $f(x)$, where $q$ is the size of $F_p[x]/(f(x))$. What is the analogue of such a structure in the world of symmetric functions? Is there a function-field version of symmetric functions? A number field version? It would be great if the answer to these were Yes. But the fact that such analogues are (apparently) unknown suggests that there is something to the split between the two points of view.

Posted by: James on July 26, 2009 6:11 AM | Permalink | Reply to this

### Re: Monads and the field of one element

James wrote:

By the heterodox point of view, I mean everything involving Frobenius lifts.

Thanks for trying to explain this to me! I know you already tried back when I last saw you, but ‘Frobenius lifts’ are still new to me, and nothing I have a feeling for: I can vaguely imagine number theorists performing Frobenius lifts off in some fancy academy somewhere, but I’ve actually seen it done, so it all feels quite ethereal. Someone else might get the same feeling when they hear the words ‘quantizing a gauge theory’.

But now what you said is coming back, and maybe it’ll start to stick.

I’m finding this post of yours to be very helpful taken in combination with Hazewinkel’s survey article on Witt vectors. So let me give a little reader’s guide to Hazewinkel for People Who Want to Understand Borger:

• Hazewinkel talks about both the ‘orthodox’ and ‘heterodox’ approaches to $\lambda$-rings. Indeed he starts out with a lot of material on Witt vectors and their relation to $p$-adics. The $\lambda$-rings only make their debut on page 87, where the operation of ‘taking the Witt vectors’ of a commutative ring is revealed to be the right adjoint to the forgetful functor from $\lambda$-rings to commutative rings.
• He then goes ahead and defines $\lambda$-rings on page 88 — I mention this because David was having trouble finding the definition. And let me warn David: at first this definition looks a bit frustrating, because Hazewinkel defines ‘$\lambda$-ring’ using the concept of ‘morphism of $\lambda$-rings’! But it’s not actually circular; it’s really just a trick to spare us certain ugly equations that appear in the usual definition.
• Just for fun, note the unusual remark in footnote 62 on page 88: warning the reader to ‘steer clear’ of a certain book by two famous authors.
• On page 92, Hazewinkel proves the Wilkerson theorem getting $\lambda$-rings from rings equipped with Adams operations $\psi_p$. And then, at the bottom of page 94, he goes heterodox and defines ‘$\psi$-rings’ to be commutative rings equipped with Adams operations — and notes that over a field of characteristic zero, $\lambda$-rings are the same as $\psi$-rings.
• On page 97, he describes ‘taking the Witt vectors’ as a comonad on the category of commutative rings. This is just another way of talking about the right adjointness property he mentioned on page 87; now he’s proving it.
• On page 98 he shows that Symm, the ring of symmetric functions in countably many variables, is the free $\lambda$-ring on one generator.
• On page 102 he goes orthodox again, and starts talking about ‘plethysm’.

Anyway:

The odd thing is that pretty much no one cares that the two approaches are equivalent, including me. Everything most people want to do is on the orthodox side, and everything I want to do is on the heterodox side. The deeper meaning of connection between the two is completely unclear to me. It could be that it should be viewed as an accident.

Wow! Whenever I hear about a big connection that nobody cares about or might be ‘just a coincidence’, I get intrigued. I also get ‘irritated’: not in the sense of being annoyed, more like having an itch that I can’t help but scratch. So, I’ll have to keep thinking about this, on and off.

But let me ask now: how sharp is the split between ‘orthodoxy’ (tensor operations on tensor categories) and ‘heterodoxy’ (Frobenius lifts) when it comes to algebraic topology?

After all, the $\lambda^n$ operations come from the ‘orthodox’ side, but I know algebraic topologists love working in characteristic $p$, and I know they love the Adams operations $\psi_p$, so I can easily imagine them getting pulled over to the heterodox side. In fact I remember Jack Morava saying something like ‘class field theory has been incorporated into algebraic topology, thanks to the Adams operations; now we need to understand the consequences of the Langlands program’. Or something like that — it was over my head.

I’m just wondering if there’s an almost impenetrable firewall — except for both sides using $\lambda$-rings — or whether there’s some cross-talk that could serve as a clue.

Posted by: John Baez on July 26, 2009 10:34 AM | Permalink | Reply to this

### Re: Monads and the field of one element

The story about the orthodox and heterodox points of view is just my little way of trying to make sense of things, and I shouldn’t make too much of it. It just feels like it’s two different subjects. It could very well be that algebraic topologists do use nontrivial things on both sides. But they’re a slippery lot, always homotoping themselves out of the way when I try to pin them down on some point.

But Morava’s comment interests me, because class field theory = Adams operations has a lot to do with what I’m doing. Was it recent?

Actually have algebraic topologists ever actually used class field theory for anything? (Preferably something real, rather than ‘Check it out! I can combine class field theory and topology!’?) That would be interesting. Maybe someone can fill us in.

Posted by: James on July 26, 2009 1:12 PM | Permalink | Reply to this

### Re: Monads and the field of one element

Morava was talking about the work of Sullivan and Quillen on the Adams conjecture, I believe - Sullivan’s MIT notes and 1970 ICM address some of this as does Adams’ book Infinite Loop Spaces. (I completely don’t understand the connection but just repeating what Morava told me.)

A relation with Langlands has to do with the central role of the moduli of one-dimensional formal groups in homotopy theory, since some pieces of this are modeled by Shimura varieties.. this is the basis of Behrens and Lawson’s theory of topological automorphic forms, which uses the same moduli spaces exploited by Drinfeld, Carayol, Taylor-Harris etc. But beyond the same moduli (and symmetries, eg Hecke correspondences) appearing in both I don’t know of a connection there (the two groups ask very different questions, Langlands ones involving things like etale or L2 cohomology and homotopy theorists involving something like coherent cohomology of a derived thickening of the space).

Posted by: David Ben-Zvi on July 26, 2009 2:13 PM | Permalink | Reply to this

### Re: Monads and the field of one element

Before the experts step in I can just add that the Adams conjecture is related to the image of the J-homomorphism, which is what K-theory, or the multiplicative formal group (or first chromatic level) sees of the stable homotopy groups of spheres. Topological modular forms (Hopkins-Miller) and top.automorphic forms are meant to capture some of the higher chromatic analogues of this, i.e., information associated to formal groups of heights 2 or more.

In any case there’s a very deep arithmetic structure to all this, of which class field theory is just the beginning..

Posted by: David Ben-Zvi on July 26, 2009 2:37 PM | Permalink | Reply to this

### Re: Monads and the field of one element

This comment has now been linked to from [[Lambda-ring]] on the Lab. It would be great if somebody with the source code would just copy it over there. (The language should probably also be depersonalised a bit, or possibly placed in a query box or something, but I can do that myself later.)

This is a good place to remind people how to get started on the Lab!

Posted by: Toby Bartels on July 26, 2009 7:23 PM | Permalink | Reply to this

### Re: Monads and the field of one element

Toby wrote:

This comment has now been linked to from [[Lambda-ring]] on the Lab. It would be great if somebody with the source code would just copy it over there.

I copied one article over there last night. I’ll do more transplantation soon. It would be nice if you could adapt and polish it a bit.

Posted by: John Baez on July 27, 2009 10:22 AM | Permalink | Reply to this

### Re: Monads and the field of one element

I’m trying to understand Adams operations, which I’ve never thought about before. As Jim notes, they don’t live at the level of a symmetric monoidal abelian category $C$, just its Grothendieck ring $K(C)$. But here are two very suggestive examples:

First: take $C$ to be a category of vector bundles. Then if $L \in C$ is a line bundle, the Adams operation $\psi_k$ satisfies

$\psi_k([L]) = [L^{\otimes k}]$

Second: take $C$ to be the category of representations of a finite group $G$ Then $R(C)$ is the representation ring of $G$, which we may identify with the ring of class functions on $G$ — functions that are constant on conjugacy classes. If we think of $\chi \in R(C)$ as a class function, then

$(\psi_k (\chi))(g) = \chi(g^k)$

These suggest that the Adams operations always make perfect sense when applied to the ‘line objects’ $L \in C$ — i.e., objects with tensor inverses. Namely, we should always have

$\psi_k([L]) = [L^{\otimes k}]$

When $C$ is a category of vector bundles, this is the first formula. When $C$ is a category of representations of finite abelian group, it gives the second formula.

Surely this is old news, but I thought I’d record it here. I got a nice email from Morava that suggests to me that this line of thought connects the ‘orthodox’ and ‘heterodox’ approaches to $\lambda$-rings — more on that later, I hope.

Posted by: John Baez on July 27, 2009 5:31 PM | Permalink | Reply to this

### Re: Monads and the field of one element

The best quick & classical link between class field theory and homotopy theory is via Serre’s old book on representation theory of finite groups [cf. II § 12.3 prop 22 of the original edition; there’s more in the 2nd (English) edition]:

In the theory of finite groups, the Adams operations are the endomorphisms of the representation ring functor. On the other hand, an old theorem of Brauer asserts that the values of a character of a finite group are cyclotomic integers, so the Galois group of the maximal abelian extension of the rationals (i.e. the cyclotomic field) acts naturally on the character ring. Serre’s argument shows that these two actions are essentially the same.

[More precisely: the Adams operations define an action of the multiplicative monoid $\mathbb{Z}^\times$ of integers on the representation ring, while Artin reciprocity (via Kronecker–Weber) identifies $Gal(\mathbb{Q}^{ab}/\mathbb{Q})$ with the multiplicative monoid $\widehat{\mathbb{Z}}^\times$ of the profinite completion of the integers. The natural embedding

$\mathbb{Z} \to \widehat{\mathbb{Z}}$

Posted by: Jack Morava on July 28, 2009 5:23 PM | Permalink | Reply to this

### Re: Monads and the field of one element

Thanks, Jack. This does help me get closer to putting my finger on the distinction between the two points of view. Perhaps my real question is Do algebraic topologists care about $\lambda$-rings which do not arise as Grothendieck rings of linear tensor categories? For instance, do you ever come up with a torsion-free ring which a priori only has a commuting family of Frobenius lifts $\psi_p$ and *then* conclude that your rings have $\lambda$-structure by invoking Wilkerson’s theorem? (Since Wilkerson is a topologist, I guess the answer is probably yes!)

Also, while what you write is certainly close to class field theory, I think it’s more precisely just cyclotomic Galois theory (not that you said otherwise). In other words, I think you’ve just used the fact that the Galois group of the maximal cyclotomic extension of $\mathbf{Q}$ is $\hat{\mathbf{Z}}^*$. While there is a little content in that (the irreducibility over $\mathbf{Q}$ of the cyclotomic polynomials), it’s at the level of undergraduate Galois theory. To identify the maximal cyclotomic extension with the maximal abelian extension is much harder, and I don’t think you’re actually using that. So it would still be nice to have an example where you really use a theorem in class field theory, rather than just some of the objects used in class field theory.

Posted by: James on July 29, 2009 3:08 AM | Permalink | Reply to this

### Re: Monads and the field of one element

Thanks, James:

…do you ever come up with a torsion-free ring which a priori only has a commuting family of Frobenius lifts $\psi_p$ and then conclude that your rings have $\lambda$-structure by invoking Wilkerson’s theorem?’

Charles Rezk’s recent [arXiv:0902.2499] posting ‘The congruence criterion for power operations …’ is (I think) a beautiful example of this, though he’s interested in a slight generalization of the classical notion of $\lambda$-rings.

… To identify the maximal cyclotomic extension with the maximal abelian extension is much harder, and I don’t think you’re actually using that. So it would still be nice to have an example where you really use a theorem in class field theory…’

In the Quillen–Sullivan work on Adams’ conjectures, what’s really used is the fact that the action of $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$ on the (cohomology of the) étale homotopy type of a Grassmannian factors through its maximal abelian quotient, just as you say. However in his big Annals paper Quillen used Brauer’s theorem to avoid any explicit mention of étale cohomology (and more or less provided a direct proof of the assertion about abelianization).

More recent work in homotopy theory [e.g. Henn, Mahowald and Rezk in the Annals in 2005, or Behrens and Lawson’s monograph [arXiv:math/0702719] on topological automorphic forms] uses everything anybody ever proved about classfield theory (cf e.g. Iwasawa’s book) — and more…

Posted by: Jack Morava on July 29, 2009 7:13 PM | Permalink | Reply to this

### Re: Monads and the field of one element

To close out this topic, here are the slightly edited highlights of an email exchange between me and Jack Morava.

***

Me: It seems that Rezk is generalizing Wilkerson’s criterion, rather than applying it.

Jack: You’re absolutely correct. What I had in mind is that Rezk’s generalization leads us to recognize lambda-LIKE rings, in contexts where, before, we probably wouldn’t have seen them. To the stricter question, I’d have to say no: in fact in homotopy theory lambda-rings are pretty much synonymous with K-theory, so one recognizes them from the beginning. [The stable homotopy ring itself has unstable operations, which contain analogs of a lambda-ring structure; but the associated Adams operations are not at all well-understood (and are in general additive but not multiplicative). I’ve been spinning my wheels a little, trying to get some traction there…]

.
.
.

Me: I should have made it clear that for the lambda-rings discussion, it was global class field theory that I was interested in.

Jack: The stable homotopy category more or less decomposes into p-local complements, so it’s very tempting to work at a single fixed prime — and a lot of the deeper work is concerned with how individual primes differ in their behavior (cf. eg $p=2$ vs $p=$odd).

Where there is real GLOBAL progress is in topics related to modular/automorphic forms, starting with Hopkins and co (cf eg his 2002 ICM talk, arXiv:math/0212397), segueing into Behrnes-Lawson USW. That work is intimately related to local Langlands for $GL_2$, and is deeply connected to non-abelian class field theory (cf Carayol, Harris-Taylor, …), but that’s a developing story and your guess going to be as good as anybody else’s at this point:

By which I guess I’m trying to suggest that lambda-rings proper may really be too restricted a canvas for this subject: I suspect that the generalized lambda-rings implicit in Rezk will eventually be just as important as the ‘classical’ ones. I have spent a fair amount of time wondering about the field with $1^q$ elements, $q = p^n$, and how that might be related to the $K(n)$-localized sphere spectrum, without getting much of anywhere; but I continue to suspect there may be nontrivial connections…

[By the way, when trying to think about global class field theory at this level of generality, what jumps out at me is our lack of a deeper understanding of what’s going on at the archimedean primes (which is what got me interested in $F_1$ in the first place, via Deninger, Consani-Connes, …); this is connected to the old question of the connected component of the id’ele-class group, which I keep picking at like a scab…]

***

Posted by: James on August 2, 2009 9:14 AM | Permalink | Reply to this

### Re: Monads and the field of one element

Todd wrote:

So if I follow, part of what you’re saying is that classical plethysm (on the free $\lambda$-ring on one variable, $\Lambda$) can be seen as the Grothendieck group decategorification of the substitution product on the category of linear species, $Vect^{\mathbb{P}}$, where the exponent is the category of finite sets and bijections (or permutations), and the base consists of finite-dimensional (say complex) vector spaces. In fact, if I’m getting it right, that could be taken as an obvious definition of (classical) plethysm.

I think that’s right. I avoided mentioning ‘linear species’ because I didn’t want to bring the complex numbers — or any particular field $k$ — into my story. So, I wanted to construct the free $\lambda$-ring as the Grothendieck group of an abelian category $X$, not the abelian $k$-linear category $Vect_k^{\mathbb{P}}$. The idea is that $Vect_k^{\mathbb{P}}$ should be $X$ with its homsets changed from abelian groups to vector spaces by tensoring with $k$ — at least when $k$ is the complex numbers. I want this $X$ to act as endofunctors — called ‘Schur functors’ — on any symmetric monoidal abelian category $C$:

$\alpha: X \to End(C)$

And then this $X$ will become a monoidal category with a tensor product — called the ‘plethysm’ tensor product — that gets sent to composition of endofunctors:

$\alpha(x \otimes y) \cong \alpha(x) \alpha(y)$

making $\alpha$ into a monoidal functor.

The interesting and slightly frustrating challenge for me is getting a simple conceptual description of $X$. Certain things depend in a tricky way on the field $k$, or at least its characteristic. My current best way of trying to ‘erase the dependence on $k$’ is the sort of thing only a 2-category theorist would love. I love it, at least if it works, but there may also be something easier.

(I think there’s a pretty simple ‘concrete’ description of $X$ in terms of Young diagrams, but it’s the conceptual description I want, so we can understand plethysm and other things in a nice way.)

One thing I don’t know, that I feel I should better understand, is whether (1) the ring of representations of the general linear group $GL(\mathbb{F}_q)$ (i.e., the groupoid of isomorphisms of finite-dimensional vector spaces over $\mathbb{F}_q$) also admits a structure of plethory, where the ring structure comes from Green convolution, […]

I haven’t yet absorbed the precise definition of ‘plethory’, but I sure see why you’re wondering about this.

Posted by: John Baez on July 26, 2009 9:05 AM | Permalink | Reply to this

### Re: Monads and the field of one element

… at which time Dave Tall was at Sussex and I was his PhD student.

Another fairly recent contribution from a café regular (when he is not contributing to the Lab) is at

arXiv:0711.3722

which shows the relevance of the Tall–Wraith monoid to the understanding of generalised cohomology theories.

Posted by: Tim Porter on July 23, 2009 12:03 PM | Permalink | Reply to this

### Re: Monads and the field of one element

I was wondering whether or not I should bring up that paper, but being the shy retiring type that I am was not sure exactly how to do so!

(Actually, I will confess that I had to look it up as it’s not an arXiv number I remember all that well. I find some are more memorable than others.)

That paper is partly to blame for my involvement here as lots of the background was new to me and I kept finding useful little comments tucked away under the tables of the cafe. It was also responsible for me learning a little more about TeX than I feel comfortable knowing (\expandafter still makes me scream a little).

To - briefly - explain the terminology that Tim brought up (and which Sarah and I invented). A Tall-Wraith monoid is to a variety of algebras (in the sense of general algebra) as a biring is to a ring (from Tall and Wraith’s original paper) or as a plethory is to an algebra (from the Borger and Wieland paper). Technically, it’s a monoid in the category of (take a deep breath) co-$\mathcal{V}$-objects in the variety of $\mathcal{V}$-algebras (it not being too hard to show that this has a monoidal structure).

Thus a Tall-Wraith $\mathcal{V}$-monoid is that which acts on $\mathcal{V}$-algebras. The connection to cohomology operations should be obvious.

Except … cohomology algebras are graded. Okay, not too many problems there, just a bit more bookkeeping (great word). And cohomology algebras are filtered/topologised. Ah, little more work to do there. Hence the length of the paper (that and the fact that our main audience was algebraic topologists and not general algebraists).

And of course, once we were aware of the antecedents of the idea in Tall and Wraith’s paper, the name ‘Tall-Wraith monoid’ was just too good to pass up (in our defence, and also in defence of the title of the paper, we did expect a referee to want them both changed, but whoever it was was happy so we got away with them both. I used to have the first few verses of the obvious poem lying around but I think they got lost in moving to Trondheim. Sadly, this happened before I moved to a document revision system so this mathematical poetry is, in all likelihood, lost to posterity.)

Posted by: Andrew Stacey on July 23, 2009 1:59 PM | Permalink | Reply to this

### Re: Monads and the field of one element

To - briefly - explain the terminology […] A Tall-Wraith monoid is…

Did you mean to say: “A Tall–Wraith monoid is…”?

Posted by: Urs Schreiber on July 23, 2009 2:03 PM | Permalink | Reply to this

### Re: Monads and the field of one element

It’s also important to mention Bergman–Hausknecht. They wrote a whole book on this stuff, although only a small part of it is about the composition/plethystic/Tall-Wraith structures. (If I remember, they call them “Tall-Wraith monad objects”.) There’s a ton of universal algebra in it, but even if you’re like me and not interested in every variety of algebra that universal algebra allows, they still say a lot about some varieties you probably are interested in. For example, from what I remember, they classify composition/plethystic/Tall-Wraith objects in the category of (possibly non-commutative) rings! It turns out they’re much less interesting than in the commutative case. Myself, I think it would be really interesting to look at these things in the category of commutative semi-rings = rigs.

I wrote a bit more about these things a while ago in a comment to this post (which I would link directly to if I knew how).

Posted by: James on July 23, 2009 2:35 PM | Permalink | Reply to this

### Re: Monads and the field of one element

You click on ‘Permalink’ at the bottom of your comment and it reveals http://golem.ph.utexas.edu/category/2007/05/this_weeks_finds_in_mathematic_13.html#c010023.

Posted by: David Corfield on July 23, 2009 2:50 PM | Permalink | Reply to this

### Re: Monads and the field of one element

Thanks! I wish I could say I always wondered what that was for…

Posted by: James on July 23, 2009 10:49 PM | Permalink | Reply to this

### Re: Monads and the field of one element

Where would we be without Permalink? I use it at least once a day.

Posted by: John Baez on July 24, 2009 8:37 AM | Permalink | Reply to this

### Re: Monads and the field of one element

Yes, when we were writing our paper then we had your paper, the Tall and Wraith paper, and the Bergman and Hausknecht book in close reach. They do quite a lot on the structure of algebraic theories which don’t have many operations or relations. I forgot to add them to the reference list in the n-lab …

Posted by: Andrew Stacey on July 23, 2009 8:21 PM | Permalink | Reply to this

### Re: Monads and the field of one element

Just to goad Andrew, James, and David B.-Z. into action, I took Urs’ hint and put a really lousy entry about Tall–Wraith monoids on the $n$Lab, which they will not be able to resist improving.

I’m pretty sure that algebraic theories are a manageable formalism for treating the universal algebra that Andrew does above with ‘varieties’. But I have gotten the rest wrong.

Posted by: John Baez on July 23, 2009 2:13 PM | Permalink | Reply to this
Read the post nLab -- How to get started
Weblog: The n-Category Café
Excerpt: Quick easy information on how to make use of the nLab.n
Tracked: July 23, 2009 12:51 PM

### Carchedi’s Compactly Generated Stacks; Re: The Monads Hurt My Head — But Not Anymore

Is the computation of lax limits of stacks useful to us, regardless of nonstandard terminology, in today’s:

Appendix A

“… This notion of prestack is non-standard. Typically, the name prestack is reserved for those weak presheaves of groupoids which are separated (See Definition A.1.) See for instance [FGIKNV]. We choose to call all weak presheaves of
groupoids prestacks to emphasize the analogy with presheaves of sets….”

Posted by: Jonathan Vos Post on July 25, 2009 1:16 AM | Permalink | Reply to this

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