The n-Category Café

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.)

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).

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).

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.

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

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 $V$.

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

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.

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.

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.

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!

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!

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?

I have the feeling, Patrick, that Jim Stasheff was not asking about your comment, but simply asking where — if anywhere — someone had written about ‘partially defined monads’. This sort of misunderstanding can happen when someone clicks on “Reply to this” instead of “Post a New Comment”.

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.

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.

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.

(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.

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.”

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

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

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.