## February 25, 2010

### 3000 and One Things to Think About

#### Posted by Urs Schreiber

In a burst of activity, Zoran Škoda a few minutes ago went beyond the $n$Lab entry of nominal count 3000 with beginning a discussion of crystals.

Other activity we have seen recently:

And then there is notable activity on the personal webs.

And you haven’t even contributed yet! (Unless you have, of course).

Posted at February 25, 2010 9:59 PM UTC

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### Re: 3000 and One Thing to Think about

Erp! I’d better get cracking! :-)

Posted by: Todd Trimble on February 25, 2010 11:20 PM | Permalink | Reply to this

### Re: 3000 and One Thing to Think about

At the $n$Lab Todd wrote

buildings are certain categories enriched in such Coxeter quantales

This gets my magnitude antennae twitching in an automatic reflex. I’m intrigued by what you’re going to say there. Are these finite enriched categories? Is there going to be a way to define the Euler characteristic of a building? This would require some notion of cardinality for the objects of your Coxeter quantales category.

Posted by: Simon Willerton on February 26, 2010 1:18 AM | Permalink | Reply to this

### Re: 3000 and One Thing to Think about

Since you’re here at UCR, Simon, you can get Jim Dolan to explain this stuff. Todd worked it out with Jim.

Posted by: John Baez on February 26, 2010 1:39 AM | Permalink | Reply to this

### Re: 3000 and One Thing to Think about

Most of the examples of buildings Jim and I discussed were in fact finite buildings (enriched in a finite Coxeter monoid). Jim and I did an awful lot of playing around with things like Schubert cell decompositions and connecting it with $q$-mathematics and so forth, but I’d have to have a serious think to connect what we were doing to what you and Tom have worked on; that would be very good to talk about. I’d like to bring Jim into that discussion too.

I was actually planning on writing a blog entry on this, to usher in my new status as host.

Posted by: Todd Trimble on February 26, 2010 1:50 AM | Permalink | Reply to this

### Buildings for category theorists

The “Buildings for category theorists” page does not offer the “edit” option to me - is this an error or on purpose?

All I would like to do is add a few references to Coxeter groups and buildings, like the book by Kenneth Brown and Peter Abramenko. If the page is not supposed to be editable by everyone, maybe we could create a supplemental page that is and that links to it?

(Note that I’m a newbie, since buildings are typically not a topic of physics graduate schools, but I’d like to learn a bit about them from the bottom up - maybe this is not the kind of audience you’d like to attract :-).

Posted by: Tim van Beek on February 26, 2010 9:15 AM | Permalink | Reply to this

### Re: Buildings for category theorists

I will be the first to admit that I am no expert on buildings. I learned about them under Jim Dolan’s tutelage, mainly through playing around with pictures and examples first. Then we moved on to more formal aspects.

The actual definitions of things like buildings and BN-pairs are a bit prickly and formidable – I doubt there’s anyone who could just stare at the definition, without some sort of preparation as to what it was really about, and say, “Ah ha! What a wonderfully clear and beautiful concept!”

The way I learned about them is that they are supposed to encapsulate an idea of an incidence geometry. For example, a projective plane is one kind of building. But instead of talking about points and lines to describe this geometry, you describe the geometry in terms of flags (for projective planes, a flag would be a point-line pair where the point sits on the line), and the ways that flags are related. For instance, a flag $(p, l)$ may be related to a flag $(p', l')$ by having their points $p$, $p'$ be the same but the lines $l$, $l'$ are not. Or, they could be related by the fact that $p$ and $p'$ are not the same and $l$, $l'$ are not the same, but line $l$ happens to pass through point $p'$.

The kinds of possible relationships between flags can be organized as forming the elements of a Coxeter group, which you can more or less think of as a group of reflections of Euclidean space. For projective planes, the relevant Coxeter group would be the symmetric group $S_3$. Maybe I’ll be just a bit cryptic at this point in describing how this works, since I think I’ll want to devote an initial blog post to describing it in more detail, but one way of picturing the Coxeter group in the case of projective planes is by thinking of a triangle (three points, three lines) as forming a degenerate sort of “projective plane” – the projective plane over the field with one element $\mathbb{F}_1$, if you will, $\mathbb{P}^2 \mathbb{F}_1$ – which has six flags. Here, a flag would be a maximal chain of inclusions of subsets of a three-element set $\{1, 2, 3\}$, such as

$\emptyset \subset \{1\} \subset \{1, 2\} \subset \{1, 2, 3\}$

with $\{1\} \subset \{1, 2\}$ being the point-line pair. Now the permutation group $S_3 = Aut(\{1, 2, 3\})$ acts naturally on these flags, and in fact the collection of six flags is a torsor for $S_3$. This means there is a division map

$d: Flag(\mathbb{P}^2\mathbb{F}_1) \times Flag(\mathbb{P}^2\mathbb{F}_1) \to S_3$

so that $S_3$ is the group of possible “ratios” between these flags.

If you patiently work through what I have just told you, then you can write down, for each element $g \in S_3$, a corresponding kind of flag relationship. I’m being somewhat dishonest here because in the way I’ve set it up there are some choices involved, but if you take $\{1\} \subset \{1, 2\}$ as a standard flag or standard of comparison, then the permutation $(1 2)$ would represent the flag relationship “the lines are equal, $l = l'$, but the points $p$ and $p'$ are different” because $(1 2)$ takes the standard flag $\{1\} \subset \{1, 2\}$ to the flag $\{2\} \subset \{1, 2\}$.

I’ll explain it better some other time, but let me try to wrap it up in a quick summary. So, in this way of doing incidence geometry, a “projective plane” consists of a set of flags $F$ and a map

$d: F \times F \to S_3$

where $d(f, f') = g$ if $g$ corresponds to the kind of relationship that holds between the flags $f$, $f'$. More generally, in the so-called “local approach” to buildings devised by Tits around 1980 or so, a $G$-building for a particular Coxeter group $G$ consists of a set $F$ (whose elements are called “flags”) and a function

$d: F \times F \to G$

where $G$ is, intuitively, a group of flag relationships whatever type of incidence geometry is being controlled by $G$, and the $d$ here is a kind of “distance” function: elements of $G$ are distances which specify how closely related two flags are related, some distances being closer to the identity than others. Indeed, Tits conceived of a $G$-building as being like a metric space with distances valued in $G$ instead of $\mathbb{R}$, and the building axioms reflect the properties this distance function is supposed to have.

What Jim and I did is take this point of view seriously. From Lawvere we know that metric spaces can be viewed as categories enriched in a quantale $[0, \infty]$, and we wanted to find a corresponding story for buildings which would say what they are from an enriched category perspective. And we found such a description, but it takes a bit of reformulating to get there. To get there, it’s better to shift perspective from Coxeter groups to “Coxeter monoids” which form certain types of quantales. I’ll have to explain this another time.

Posted by: Todd Trimble on February 26, 2010 2:12 PM | Permalink | Reply to this

### Re: Buildings for category theorists

Todd wrote:

Indeed, Tits conceived of a G-building as being like a metric space with distances valued in G instead of $\R$ and the building axioms reflect the properties this distance function is supposed to have.

That’s a point of view that Brown and Abramenko added in the second edition of the “buildings” book. They call it “buildings as W-metric spaces”, W being the Coxeter Group (W to honor Weyl as in Weyl group), so this part looks distantly familiar to me.

Todd wrote:

From Lawvere we know that metric spaces can be viewed as categories enriched in a quantale [0, $\infinity$].

Then I suppose I will have to read

Posted by: Tim van Beek on February 26, 2010 9:28 PM | Permalink | Reply to this

### Re: Buildings for category theorists

I haven’t looked at the book by Brown and Abramenko, but I did get a lot out of Brown’s earlier book on buildings.

Yes, that seminal paper by Lawvere was what I was referring to. There the triangle inequality

$d(x, y) + d(y, z) \geq d(x, z)$

is likened to the composition mapping

$\hom(x, y) \otimes \hom(y, z) \to \hom(x, z),$

both fitting into the more general niche of enriched category theory. The paper is a delight to read.

Since I’ve already used the word “quantale”, I may as well recall the definition: a quantale is a cocomplete closed monoidal category whose underlying category is a poset. Lawvere argues that it makes a lot of sense to generalize metric spaces by dropping the symmetry condition

$d(x, y) = d(y, x)$

and the separation condition

$0 = d(x, y) \Rightarrow x = y$

both because many reasonable examples violate one or both of these conditions, and the general theory obtained by dropping those conditions is often much nicer. Thus, Lawvere proposes to redefine “metric space” so that it means precisely a category that is enriched in the quantale $[0, \infty]$, ordered by $\geq$ and taking $+$ to be the monoidal product.

However, it is also fruitful sometimes to partially reinstate the symmetry condition by considering enrichment in a $*$-quantale. A $*$-quantale is a sort of like a $C^*$-algebra: a quantale $Q$ equipped with an additional structure of an involution

$* : Q \to Q$

for which $(x \otimes y)^* = y^* \otimes x^*$ and $1^* = 1$, where $1$ denotes the monoidal unit. (The operator is assumed to be covariant with respect to the poset structure.) Then a $*$-enrichment in a $*$-quantale is an ordinary enrichment

$d: X \times X \to Q$

for which $d(y, x) = d(x, y)^*$.

As you could probably guess if you’ve seen the $W$-metric space approach to buildings, something like this last condition occurs for buildings; here the $*$-operator takes a Coxeter word and writes it backwards. In the Coxeter group, this would be the same as inversion.

But now a funny wrinkle arises: if we consider a Coxeter group $W$ as a $*$-monoid in this way, then there is no way to put a poset structure on it to make it a $*$-quantale, unless $W$ is trivial! The reason is simple: if $x \leq y$ in a partially ordered group, then $y^{-1} \leq x^{-1}$, hence $y^* \leq x^*$. But also $x^* \leq y^*$, by covariance of the $*$-operator. So now by antisymmetry, $x \leq y$ implies $x^* = y^*$, which implies $x = y$ since $*$ is involutory. But, in a quantale there’s a bottom element $0$, where $0 \leq x$ for all $x$. We then conclude that every element is the bottom element $0$!

For now I’m going to leave this as a puzzle to think about…

Posted by: Todd Trimble on February 27, 2010 1:52 AM | Permalink | Reply to this

### Re: Buildings for category theorists

I’ll try to add some supplementary material to the nLab, the kind of material that would end up in the appendix should James and you decide to write a paper, for as long as I can keep up with you. (That may not be for long, so don’t expect too much). I started with the buildings page, as Urs advised.

I also noted that there is a nLab entry about Lavwere’s viewpoint of metric spaces here.

Posted by: Tim van Beek on February 27, 2010 9:21 AM | Permalink | Reply to this

### Re: Buildings for category theorists

Thanks, that’s very useful, Tim!

Judging from your contributions here at the Café, you’ll be able to follow us just fine. I do want to include some exposition in the accounts and make it generally accessible. Maybe I’ll start with a worked example in the nLab buildings article (or, that might work better on a separate page).

Just a quick comment for now: the notion of building, such as the one you’ve begun to put down in building under “combinatorial approach”, can be made to look short, and the groundwork I’ll have to lay for our approach may wind up looking long-winded by comparison. But, the short definitions tend to look pretty specifically tied to the structure of Coxeter groups, and not particularly exportable to more general contexts. Maybe I can put it like this: if an enriched category is a generalized metric space, then what would a generalized building be??

I’ll add that as of now, I don’t know what all this might be good for, aside from possible aesthetic appeal to those of a categorical bent.

Posted by: Todd Trimble on February 27, 2010 3:40 PM | Permalink | Reply to this

### Re: Buildings for category theorists

I also noted that there is a nLab entry about Lavwere’s viewpoint of metric spaces here.

Todd has in some entry much more details. But I forget in which one! Somehow there seems to be a link missing.

Posted by: Urs Schreiber on February 27, 2010 9:50 AM | Permalink | Reply to this

### Re: Buildings for category theorists

At some point I wrote a bunch of stuff about Cauchy completion of enriched categories, starting with the example of metric spaces. That could be what you’re thinking of.

There is a heck of a lot going on in Lawvere’s metric spaces paper, but the idea of general Cauchy completion is one of the more important contributions it makes.

Posted by: Todd Trimble on February 27, 2010 3:06 PM | Permalink | Reply to this

### Re: Buildings for category theorists

I just inserted the link from the nLab article on metric spaces to the article on Cauchy completion.

Posted by: Todd Trimble on February 27, 2010 3:18 PM | Permalink | Reply to this

### Re: Buildings for category theorists

Enlightening! Thanks

Posted by: jim stasheff on February 27, 2010 1:07 PM | Permalink | Reply to this

### Re: 3000 and One Thing to Think about

The “Buildings for category theorists” page does not offer the “edit”-option to me - is this an error or on purpose?

That page is on Todd Trimble’s personal area of the $n$Lab, which I think he has chosen to be editable only with password access.

All I would like to do is add a few references to Coxeter groups and buildings, like the book by Kenneth Brown and Peter Abramenko.

If you want to add general information, you would do us all a favor if you did that on the main $n$Lab. Then Todd and whoever else has special-purpose entries can link to that.

You could start by expaning the stub page building.

Posted by: Urs Schreiber on February 26, 2010 9:40 AM | Permalink | Reply to this

### Re: 3000 and One Thing to Think about

Thanx for the info, sorry for the dumb questions, I’m a bit unfamiliar with the whole setup. I’ll see what I can do.

Posted by: Tim van Beek on February 26, 2010 10:08 AM | Permalink | Reply to this

### Re: 3000 and One Thing to Think about

sorry for the dumb questions, I’m a bit unfamiliar with the whole setup.

No problem. Better you ask and join in.

I’ll see what I can do.

Okay, thanks!

Posted by: Urs Schreiber on February 26, 2010 10:36 AM | Permalink | Reply to this

### Re: 3000 and One Things to Think About

And you haven’t even contributed yet! (Unless you have, of course).

I've been maintaining the linked page of contributors pretty much all by myself, and it had fallen behind, but it should be caught up now. It's helpful if people sign their contributions with their full name (diacriticals optional) and create a page for themselves by clicking on that name. (Then lab elves can add it to the list of contributors and the category of pages about people and give it redirects to or from alternate spellings.)

Posted by: Toby Bartels on February 26, 2010 8:58 AM | Permalink | Reply to this

### Updating the nLab contributor page

Tobey wrote:

I’ve been maintaining the linked page of contributors pretty much all by myself…

Let me create a minor paradox by thanking you for doing this unthankful task!

I’m going to ask a very unwise question: Is it not possible to get the software that runs the nLab update the page of contributors by itself? Or rather: Which part does it already, what has to be done manually and how can contributors help, beside using full names consistently and creating a page for themselves? (How does the latter part help, anyway?).

Posted by: Tim van Beek on February 26, 2010 9:37 AM | Permalink | Reply to this

### Re: Updating the nLab contributor page

Is it not possible to get the software that runs the nLab update the page of contributors by itself? Or rather: Which part does it already, what has to be done manually

It takes the name that you put in when you make an edit and adds it (if it's not already there) to the list of authors; it also takes the name of the page that you edited and adds it (if it's not already there) to the list of pages edited by that author.

But because this is automatic, it is not always correct. It includes the fake names given by occasional successful spammers as well as anonymous names such as AnonymousCoward and its many variations. (Look at the redirects listed at the bottom of the source of that page to see the variations.) And if anyone uses a short name, or a name with odd capitalisation, or even misspells their name, then it shows up on that list twice.

So the list of contributors is the clean version of the list, with fake and anonymous names removed, and each person's real name (or best approximation thereof) listed once. (Also it doesn't list the pages that people have edited, just their names.)

and how can contributors help, beside using full names consistently and creating a page for themselves?

You could add yourself to the list of contributors, of course. (That's a generic ‘you’, Tim; you're already on the list.) However, since Urs is encouraging people to contribute, I didn't want to ask anything that creates extra work for new contributors. Of course, regulars who notice a new contributor can also add the new person to the list. Every once in a while, I go through the automatic author list, run a script to list only the names and remove the known fake and anonymous names and redirects from alternate spellings, then compare this to the contributor list; that works as long as our contributors are numbered in the dozens instead of in the hundreds (141 right now).

(How does the latter part help, anyway?).

Mostly it's nice just for itself, but it also gives something for the list to link to. Also, some of the names on the list of contributors go to more than nonexistent page; this is for when someone has varied the spelling of their signature and I don't know which spelling (if any) they might eventually create a page at. It's cleaner to put those alternate spellings as redirects on their own page rather than as separate links in the list of contributors.

Posted by: Toby Bartels on February 26, 2010 8:39 PM | Permalink | Reply to this

### Re: 3000 and One Things to Think About

Is it not possible to get the software that runs the nLab update the page of contributors by itself?

Sure. As it says at the page I linked to, the automatic author page is here. We just felt that we should supplement by a hand-maintained page.

Posted by: Urs Schreiber on February 26, 2010 9:45 AM | Permalink | Reply to this

### Re: 3000 and One Things to Think About

clicking on here
brought me to my name but also to
Jim Stasheff \greaterthan history
which is empty
??

Posted by: jim stasheff on February 26, 2010 1:45 PM | Permalink | Reply to this

### Re: 3000 and One Things to Think About

clicking on here brought me to my name but also to Jim Stasheff \greaterthan history which is empty ??

This page dates from the days before the software could handle redirects, so both Jim Stasheff and jim stasheff existed as separate pages. Now that we have redirects, both spellings go to Jim Stasheff, while the other page's edit history rests in peace at jim stasheff > history. But you should not have found that last page in any list of authors as such, but instead only in the list of pages edited by some particular author (probably me).

Posted by: Toby Bartels on February 26, 2010 8:32 PM | Permalink | Reply to this

### Re: 3000 and One Things to Think About

create a page for themselves by clicking on that name

click on it where?

Posted by: jim stasheff on February 26, 2010 1:41 PM | Permalink | Reply to this

### Re: 3000 and One Things to Think About

click on it where?

After you make an edit, your signature appears towards the bottom of the page. (Or click on ‘History’ at the bottom to see the signatures of everybody who's ever edited the page.) If there is a ‘?’ link after your name, then click on the question mark to create a page about yourself. Then your signature should lose the question mark (although sometimes a cache bug interferes) and become a link to your new page itself. (This is a generic ‘you’, Jim; you already have a page, so you'll never see a question mark after your signature unless you change the spelling.)

It's always optional for contributors to make such pages, but it's friendly, and I find that people usually don't mind talking about themselves. (^_^)

Posted by: Toby Bartels on February 26, 2010 8:42 PM | Permalink | Reply to this

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