The n-Category Café

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.

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.

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!