The notion of building, due to Jacques Tits, was invented to give a framework that unites two distinct developments: on the one hand, the structure and geometry of classical Lie groups via their Borel subgroups and maximal tori, and on the other, general forms of synthetic or axiomatic incidence geometry. It ranks as one of the most important concepts in 20th century geometry and group theory.
There are several ways of introducing this concept, none particularly easy. Perhaps the best way to learn about buildings is to play with examples first before embarking on the full-fledged definition which, however one cuts it, involves a certain amount of preface involving the details of Coxeter groups (roughly, symmetry groups generated by reflections). There are a number of good introductory texts available (such as those by Ken Brown and Richard Weiss), each providing several chapters of preparation before giving the actual definition.
I myself began learning about buildings a few years ago through conversation with James Dolan, as part of the larger groupoidification project. Together we developed an alternative way of considering buildings which makes contact with enriched category theory and the theory of quantales, an approach which we felt would be congenial to (or at least readily apprehended by) category theorists. The purpose of these notes is to explain this approach, and tentatively suggest an enriched-category generalization of buildings which might be of interest.
For those who already know something about buildings, we can explain the rough idea as follows. A popular slogan in the so-called “local approach” due to Tits is that a building is like a metric space, except that “distances” between points are measured not by real numbers but by elements in Coxeter groups. To a categorist, this might recall Lawvere’s interpretation of metric spaces as categories enriched in a quantale of extended nonnegative reals, and the question arises whether Tits’s slogan can be similarly interpreted within a quantalic niche. This is indeed the case. To make it come out right, though, one needs to use certain “Coxeter monoids” equipped with a Bruhat ordering – certain quantales attached to a Coxeter diagram – instead of the associated Coxeter groups. (Curiously, there seems to be little mention of these monoids in the literature; they should be better known.) In the end, buildings are certain categories enriched in such Coxeter quantales, satisfying additional isometry conditions which are naturally understood in terms of homomorphisms which preserve both quantale products and internal homs. This forms the basis of a beautiful link between classical geometry and categorical logic.
A Coxeter diagram codes the same information as a Coxeter matrix but in easy-to-read form; we’ll start with Coxeter matrices.
A Coxeter matrix indexed by a set $I$ is a function
such that $m_{i j} = m_{j i}$ and $m_{i j} = 1$ if and only if $i = j$.
The associated Coxeter diagram is a labeled simple graph whose set of vertices is $I$, and where there is an edge between vertices $i$ and $j$ (labeled $m_{i j}$) if and only if $m_{i j} \geq 3$. In the literature it is standard to drop the label if $m_{i j} = 3$, and to draw instead two edges between $i$ and $j$ if $m_{i j} = 4$.
The associated so-called Coxeter group $W$ is a group presentation whose generators $s_i$ are in bijection with elements $i \in I$, and whose relations are given by equations
whenever $m_{i j} \lt \infty$.
In particular, $(s_i)^2 = 1$ for all $i$. This also shows that the monoid presented by such equations is a group. It additionally follows that this monoid can be equivalently presented using the same generators and equations
$\,$
where each side of (2) is an alternating word of length $m_{i j}$.
An important general fact about Coxeter groups is that equality is decidable: there is an algorithm that decides whether two words in the $s_i$ are equivalent modulo the relator subgroup. Suppose given a word $w$ in letters $s_1, \ldots, s_n$ and of length $k$; there are $n^k$ such words. If $w$ contains consecutive letters $s_i s_i$, then by relation (1) we may delete those letters to obtain an shorter word which is equivalent to $w$. We say the word $w$ is reducible. If $w$ is not reducible, we consider all words obtainable from $w$ in finitely many steps by replacing one side of an equation of type (2) with the other. Such replacements again produce a word of length $k$ in the letters $s_1, \ldots, s_n$, and so there are only finitely many words obtainable from $w$ in this way. If none of these words is reducible, then $w$ is said to be reduced. We have the following results:
One can decide in finitely many steps whether a word is reduced.
Every word is equivalent to a reduced word.
Two reduced words are equivalent iff one is obtainable from the other by making replacements of type (2).
Thus whether two words are equivalent takes only finitely many steps to decide.
The Coxeter group carries the obvious involution given by inversion. It is therefore a $*$-monoid.
Let $D$ be a Coxeter diagram. The Coxeter monoid $W_\infty = W_\infty(D)$ is the monoid presented using the same generators as the Coxeter group $W$, but subject to the equations
$\,$
The concept of reduced word is the same as for the Coxeter group; the set of reduced words is the same for the Coxeter monoid $W_\infty$ as for the Coxeter group $W$. Hence equality is decidable in the Coxeter monoid.
Identifying the underlying set of $W_\infty$ with a set of equivalence classes of reduced words, $W_\infty$ carries an anti-involution $w \mapsto w^*$ where $w^*$ is the (reduced) word obtained by writing the (reduced) word $w$ in reverse.
The Coxeter monoid $W_\infty$ carries a partial ordering, the smallest reflexive transitive relation for which $W_\infty$ is a monoidal poset and the identity 1 is the bottom element. We usually write this as $x \geq 1$, and we think more in terms of $(W_\infty, \geq)$, just we think of $([0, \infty], \geq)$ as the base of enrichment for metric spaces. The involution $(-)^*$ preserves the partial order. We call this the Bruhat order on $W_\infty$.
Evidently, then, $W_\infty$ is a $*$-monoid in the bicategory $Ord$ of preorders and order-preserving maps. The free sup-lattice functor
(which takes the sup-lattice of downward-closed sets) is a strong monoidal functor which takes the $*$-monoid $W_\infty$ in $Ord$ to a $*$-monoid $P(W_\infty) = [W_{\infty}^{op}, 2]$, also called a $*$-quantale. The quantalic multiplication is given by the usual Day convolution, which here takes the simple form
for $R$, $S$ down-closed subsets of $W_\infty$. Alternatively, if
The functor $P: Ord \to sLat$ factors through the Kleisli bicategory of the associated monad, which is equivalent to the cartesian bicategory of preorders and bimodules between preorders, denoted $OrdMod$.
If $X$ is any set, $X \times X$ may be regarded as a $*$-monoid in the cartesian bicategory $Rel$ of sets and relations (and therefore in $OrdMod$, and therefore in $sLat$). Indeed, $Rel$ is compact, and the dual of any set $X$ is $X$ itself, so we have a monoid structure on
and a $*$-monoid where $f^*$ is the transpose of $f \in \hom(X^*, X^*) = \hom(X, X)$. Thus, $P(X \times X)$ is a $*$-quantale; the monoidal product is just composition of binary relations:
and the transpose is just the taking of the opposite relation.
Any quantale carries a residuated structure defined by operations $/$, $\backslash$ that satisfy the adjunction conditions
$\,$
As examples we have:
In $P(X \times X)$, we have $(T/S)(x, y) = \forall_z S(y, z) \Rightarrow T(x, z)$ and $(R \backslash T)(y, z) = \forall_x R(x, y) \Rightarrow T(x, z)$.
In $P(W_\infty)$, we have $T/S = \{x \in W_\infty: \forall_y y \in S \Rightarrow x y \in T\}$ and $R \backslash T = \{y \in W_\infty: \forall_x x \in R \Rightarrow x y \in T\}$.
Let $D$ be a Coxeter diagram. A $D$-building consists of a set $F$ and a bicontinuous homomorphism
of residuated $*$-quantales.
Some observations that clarify this definition:
Any bicontinuous map $[Y, 2] \to [X, 2]$ is of the form $[f, 2]: [Y, 2] \to [X, 2]$ for some $f: X \to Y$, so the data of a $D$-building is given by a function
A lax homomorphism of $*$-quantales
takes an up-closed set $S \subseteq W_\infty$ to $\{(f, f'): d(f, f') \in S\}$. This is the unique cocontinuous map which is determined by what it does to principal filters, i.e., by relations
and the lax homomorphism property says in part
in $W_{\infty}^{op}$, which by Yoneda simply means
in $W_\infty$. The other lax homomorphism condition is that
in $W_{\infty}^{op}$, or $1 \geq d(f, f)$ in $W$, for all $f \in F$. Since $1$ is the minimal element in the Bruhat order, this means $1 = d(f, f)$ for all $f$. Taken together, these just say $F$ is enriched in the monoidal poset $W_\infty$. The condition that $[2, d]$ preserves the $*$ operation just means that $d(f', f) = d(f, f')^*$ for all $f, f' in F$.
Thus $F$ behaves like a $W$-valued pseudometric space, if $[d, 2]$ is lax $*$-monoidal. To say that $[d, 2]$ is strong $*$-monoidal means additionally that whenever