The purpose of these notes is to study the notions of buildings and BN-pairs, and ultimately define them in a short crisp form which we hope will be congenial to category theorists. A building is sometimes described as a kind of "metric space" in which the distance function takes values in a Coxeter group instead of the real numbers. Here we will take that description seriously, from Lawvere's point of view that the appropriate generalization of metric space is that of a category enriched in a monoidal category. An ulterior motive is that the idea of building as standardly defined seems rather closely tied to the combinatorics of Coxeter groups, and hence does not seem obviously exportable to other enriched category (or other logical) contexts. We correct this impression by reformulating the notion in terms of a natural quantalic logic, in a way which invites much wider generalization within enriched category theory. 1. The Rough Idea From one point of view, buildings are axiomatizations of incidence geometries (such as projective planes). The basic objects of study are "flags" in the geometry (chains of incidence relations); for example, in the case of projective plane geometry, a flag would be an ordered pair (point, flag) where the point is incident to the flag. To specify the geometry, we specify certain types of geometric relations on the set of flags, and a Coxeter group is used to keep track of these relations. For example, in the case of projective planes, two flags f, f' stand in one of the following six types of relation: -- f and f' are identical -- f and f' have the same point -- f and f' have the same line -- the point of f lies on the line of f' -- the point of f' lies on the line of f -- f' is in generic position with respect to f. The six relations here are tracked by the six elements of the Coxeter group S_3. To make this precise, consider a very degenerate example of a "projective plane", where the "points" are the vertices (0), (1), (2) of a triangle. and the "lines" are the edges (01), (02), (12). Let us choose a flag f as our "favorite", say <(0), (01)>. With respect to f, any other flag f' will be best (i.e., most sharply) described by exactly one of the six relations above, and the unique permutation on 0, 1, 2 which sends f to f' is the element of S_3 used to track that relation. Hence: Relation: Element: -- f and f are identical e -- f and f' have the same point (1 2) -- f and f' have the same line (0 1) -- point of f lies on line of f' (0 2 1) -- point of f' lies on line of f (0 1 2) -- f in general position wrt f (0 2) In general, a Coxeter group W is used to define a certain type of incidence geometry, and in fact is considered a simple "degenerate" model for that geometry, with exactly one flag f' to bear witness to each type of geometric relation with respect to a chosen favorite flag. Then, a general model for that geometry, called a W-building, is then given by a set of "flags" F together with a function d: F x F --> W where d(f, f') is the relation which "most sharply" describes how f' stands in relation to f. This "distance" function d must satisfy certain building axioms which it is our purpose to describe. 2. Coxeter groups and "Murphy monoids". The "distance" d should be thought of as measuring "how close two flags are to being identical". When we say "d(f, f') is the relation which *most sharply* describes how f' stands in relation to f", we seem to be suggesting that the relations are ordered in terms of "sharpness", where the sharpest possible relation is the identity relation. In other words, we seem to be suggesting a sharpness poset structure on W, with the identity playing the role of maximal element. However, W is agroup. Unless W is trivial, the group operations on W (multiplication, inversion) do not respect a partial order structure on W. Thus the group structure on W is at odds with this (to-be-defined) sharpness order, and in fact thinking of W as a group, while traditional, may not be the best way to express the sense in which buildings are "metric spaces with W-valued distances". Interestingly, there is another way to think of the underlying set of W, as carrying a monoid structure which James Dolan has taken to calling the "Murphy monoid", which does respect the sharpness order. This monoid is denoted W_{oo}, and buildings will be certain types of metric spaces valued in W_{oo}, or more precisely certain categories enriched in the monoidal poset W_{oo}. But before we carry this out, it is a good idea to recall some basic facts about Coxeter groups, as some of these facts do play a role in Murphy monoids. First, recall that a Coxeter group is really a group presentation, described by an n x n Coxeter matrix M. The generators s_1, ..., s_n correspond to rows of M, and the relations are given by equations (s_i s_j)^{m_{ij}} = 1 where m_{ij} is the entry of M in row i and column j. The axioms on M are that each diagonal entry m_{ii} is 1) (the generators s_i are involutions), and each m_{ij} = m_{ji} is a positive integer greater than 1, possibly infinite. Quite extraordinarily, these simple conditons are enough to ensure that the Coxeter group defined by these relations can be realized as a group of reflections (on a sphere, or on an affine space, or on a hyperbolic space). Some of the finite Coxeter groups fall within well-known infinite series (such as those of types A, B, and D); others are more special (e.g., E_6, E_7, and E_8). All these reflection groups are also equivalently described by well-known Coxeter diagrams. Let us rewrite the defining relations of the group presentation in the equivalent form (si)^2 = 1 [quadratic relations] (si)(sj)(si)... = (sj)(si)(sj)... [generalized braid relations] where each side of the second equation is an alternating word of length m_{ij}. It turns out that the word problem for this presentation of Coxeter groups is decidable: there is an algorithm for deciding whether two words in the alphabet s1, ..., sn are related by applying a sequence of braid relations and quadratic relations. In rough outline, this is a terminating and confluent algorithm for finding a word in "normal form" which represents a given group element. This word will be in particular a "reduced" word, meaning a word of shortest length among all words which represent that group element. This means no sequence of applications of braid relations will deform the word into another in which two si's appear consecutively (which would permit a shortening by replacing (si)^2 by 1). Any reduced word can be brought to normal form by applying a certain set of braid relations; we won't go into the details here. Suffice it to say that the normal form gives a section Normal form: W --> Free monoid on {s1, ..., sn} of the canonical projection Free monoid on {s1, ..., sn} --> W which sends a word to its value in W. Now we turn to the definition of "Murphy monoid" based on a Coxeter matrix M. This is presented as a monoid with generators s1, ..., sn as before, but subject to the defining relations (si)^2 = si [quadratic relations] (si)(sj)(si)... = (sj)(si)(sj)... [generalized braid relations] Thus, the only difference between this and the previous presentation is that the si are idempotents, not involutions. The word problem for the Murphy monoid is also decidable; in fact, one just uses the same normalizing algorithm used for the Coxeter group! The point is that a reduced word for the Coxeter group is the same as a reduced word for the Murphy monoid (or, in other words, if a word is not reduced because it can be shortened by applying (si)^2 = 1 on the Coxeter side, it is not reduced because it can be shortened by applying (si)^2 = si on the Murphy side). Thus, the set of words in normal form is the same in each case: the image of the section Normal form: W_{oo} --> Free monoid on {s1, ..., sn} is the same as the image of the normalizing section of the Coxeter group, and hence there is a definite way of identifying W with W_{oo} as sets of normal words (or of braid-equivalence classes of reduced words). In this way, we can view W_{oo} as a monoid deformation of W in a meaningful way. There is one more piece of algebraic structure we need for the Murphy monoid: a *-operator, whose counterpart for the Coxeter group is inversion. Recall that a *-operator on a monoid is a map x |--> x* such that 1* = 1, (xy)* = y*x*, and x** = x. On the Murphy monoid, there is a unique *-operator such that s* = s for each generator s (observe that if a *-operator is defined in this way on the free monoid, where it writes a word backwards, then this operator clearly respects the quadratic and braid relations, and hence induces a well-defined operator on the Murphy monoid). 3. The Murphy monoid as monoidal poset. Murphy actions. We define a "sharpness" order on W_{oo} by stipulating that the identity e is the sharpest element of all (i.e., is the top element in the sharpness order), and then defining sharpness to be the smallest reflexive transitive antisymmetric relation which is respected by the Murphy monoid operations. We will write w => w' or w' <= w to say that w' is sharper than w. (Note: the order <= opposite to the sharpness order is usually called the *Bruhat order*. Thus W_{oo} becomes a monoidal poset. Moreover, it is obvious that the *-operator preserves the order, so we can say that W_{oo} is a *-monoidal poset. Let us illustrate this structure in the case of projective planes, where W = S_3. We take as one generator of W the transposition l = (0 <--> 1), which changes the flag <(0), (01)> in the degenerate model W to <(1), (01)>, thus changing the point but keeping the line. We may call l "the point-changer". For the other generator, we take the transposition p = <1 <--> 2>, which changes the flag <(0), (01)> to <(0), (02)>, thus changing the line but keeping the point. We call p "the line-changer". Now we regard l and p as generators of the W_{oo}, and write out the elements (as reduced words) together with the poset structure. One element a is sharper than another element b if there is a path running up from a to b. e / \ l p |\ /| |/ \| lp pl \ / lpl=plp We now describe the geometric meaning of W_{oo}, using again the example of projective planes to illustrate. There are several (closely related) pictures for how we might think of an element w in W_{oo}. We discuss the case w = p for concreteness, but the general definitions are clear: (a) Bruhat cell picture: d(f, f') = p. This says f and f' have the same point, but that is the most we can say (i.e. f and f' are not identical). The set of f' bearing this relationship to a fixed flag f forms a Bruhat cell B_p of the space of flags. There are six Bruhat cells B_w, one for each element w of W_{oo}; they partition the flag space. (b) Schubert cell picture: d(f, f') <= p. This says f and f' have the same point. They may be equal. The set of f' bearing this relationship to a fixed f forms a Schubert cell S_p of the space of flags. There are six Schubert cells S_w, with an inclusion S_w <= S_w' if w <= w'. (c) Bruhat operator picture: consider p as an operator assigning to each flag f the set of flags f' satisfying d(f, f') = p [i.e., f and f' have different lines, but nothing worse than that]. In other words, we think of the line-changer p as a relation [p]: F -|-> F (a function F --> PF). Each w defines an operator [w]: F -|-> F. (d) Schubert operator picture: consider w as an operator assigning to each flag f the set of flags f' such that d(f, f') => w, i.e., as a relation [w]: F -|-> F. Here are some crucial facts, which may be checked by hand for projective planes, and which form vital ingredients for the general notion of building. (1) The Schubert operators define a homomorphism (W_{oo}, <=) --> P(F x F) of *-monoidal posets (where the multiplication on P(F x F) is relational composition, the *-operator is the taking of relational opposites, and the ordering is by inclusion). We will call this the Schubert condition. (2) If s is a generator of the Murphy monoid, then the Bruhat operator [s] satisfies [s] = [s]*. Moreover, if w is a reduced word (x1)(x2)...(xk) [where the xi are drawn from the set of generators], then in the Bruhat operator picture, we have [(x1)(x2)...(xk)] = [x1].[x2]...[xk]. where the left side refers to the value of the word w in the Murphy monoid (or Coxeter group). We will call (2) the Bruhat-Tits condition. Notice that (2) is manifestly untrue if w is unreduced. Take for example the word w = pp (whose value is p in the Murphy monoid). We have that f'' belongs to ([p].[p])(f) iff f'' can be obtained by changing the line of f twice (once to an intermediate flag f', and then again to get f''). But then it is possible for f'' to be identical to f! Hence we have for the Bruhat operators the equation [p][p] = [p] + 1. hence [pp] = [p] is not equal to [p][p]. But in the case of reduced words, the Bruhat-Tits condition gives a nice way of picturing the Bruhat cell classification based on Murphy words. Take for example the Bruhat cell B_{pl} (with respect to a favored flag f). According to the Bruhat-Tits condition, this is the set of flags obtainable from f by first applying the operator p (change a line of f to get some new flag f') followed by the operator l (change a point of f' to get a new flag f''). [Note: we are composing left to right in the Murphy monoid.] This means precisely that the line of f'' passes through a point of f, and the Bruhat cell B_{pl} consists precisely of all such f''. It is clear that the Bruhat-Tits condition (2) sharpens the Schubert condition (1) in the case where the products occurring in (1) are reduced, and in fact (2) is the standard definition of building! However, by making reference to reduced words, it seems somewhat specific to Coxeter groups: it is not immediately clear how the idea of building might generalize to other contexts. We repair this defect in the next section. Now a word about the terminology "Murphy monoid": who is Murphy? This is actually something of a joke. Looking at the equation (1), it claims that the Murphy product x1...xk gives a kind of worst-case scenario for which Bruhat cell a flag f', obtained as an element or possible outcome of applying the operators [x1], [x2], ..., [xk] to f, can belong to: "worst" in the sense of being as far from identical to f as possible. Thus, if we side with the pessimist Murphy who said, "whatever can go wrong, will go wrong", the prediction is that the Murphy product x1...xk does give the Bruhat cell that f' winds up living in. In fact, the pessimist is generally (I mean generically) right. For example, when we consider the case of an infinite projective plane, and change the line of a flag twice, the chances are nil that, by some miracle, we wind up back at the original flag! This observation also helps explain the notation W_{oo}. If for example we consider the projective plane over a finite field with q elements, and apply the Bruhat line-changer twice, the probability is 1/q that we wind back at the original flag. Thus, interpreting the Bruhat line-changer in a statistical way as a random variable which changes the line of a flag at random, we arrive at the equation [p]^2 = (1 - 1/q)[p] + (1/q)[1] where the coefficients are probability weights. This is the quadratic relation for the Hecke algebra attached to the symmetric group S_n (our case was S_3, but the relation holds for general n; the group S_n is the Coxeter group for projective space in n-1 dimensions). In the q --> oo limit, we arrive at the Murphy relation [p]^2 = [p], hence the notation W_{oo} for this limiting case. 4. The enriched category picture Under the Schubert condition (1) which we posit for buildings of type W, a building F with distance function d: F x F --> W_{oo} becomes an honest-to-goodness W_{oo}-valued metric space, meaning (following Lawvere) that it is a category enriched in the monoidal poset W_{oo}. The composition law gives the triangle inequality d(x, z) <= d(x, y)d(y, z) Roughly speaking, this says that for any y, the Murphy monoid element d(x,y)d(y,z) is a "worst-case scenario" for where z may live with respect to x. In fact, the Schubert condition is stronger than mere enrichment. In general, if V is a monoidal category, then a category enriched in V consists of a set X and a function d: X x X --> V together with a lax monoidal functor V^{op} --> Set^{X x X} v |-----> (x, y) |-> V(v, d(x, y)) into the monoidal category of endospans on X. If V is a monoidal poset, the lax monoidal functor is valued in the monoidal poset of binary relations: V^{op} --> 2^{X x X} and the lax monoidality is not an extra structure, but a property. Now, the Schubert condition is a stronger property on a pair (F, d: F x F --> W_{oo}), with the lax monoidality above replaced with strong (*-)monoidality. Equivalently, the Schubert condition says that the map 2^d: 2^{W_{oo}} --> 2^{F x F} is a (strong) homomorphism of *-quantales. Here the domain is the quantale of upward-closed sets of W_{oo} with respect to the sharpness order =>, i.e., subsets S of W_{oo} such that (s in S and w <= s) implies (w in S). The quantale multiplication is the convolution product, defined by ST = {x: x <= st for some s in S and t in T}, and the *-operator is defined by S* = {s*: s in S}. In fact, 2^{W_{oo}} is the free *-quantale generated by the *-monoid W_{oo}^{op}. The Schubert condition is still not as strong as the Bruhat-Tits condition. But, we will show that the Bruhat-Tits condition is equivalent to the property that 2^d: 2^{W_{oo}} --> 2^{F x F} is a morphism of *-quantales which preserves the biclosed structure. It is *this* condition which is our desired reformulation of the notion of building: one which is clearly generalizable or exportable to other enriched category contexts. Let us step back and abstract a little from what we have done. Let M beany*-monoidal poset (we could even drop the *-operator, but let's keep it for now), and let X be any set equipped with an "M-classification map" d: X x X --> M, generalizing the Bruhat classification. This induces a map 2^d: 2^M --> 2^{X x X} which interprets each upward-closed set of M as a binary relation on X. There is a natural *-quantale structure on 2^M, and, as with any quantale, 2^M is biclosed (in the same way that every locale is a Heyting algebra in a unique way). We can think of the biclosed *-quantale structure as specifying a theory, and we define a pair (X, d) as above to be a *model* of the theory if 2^d: 2^M --> 2^{X x X} of biclosed *-quantales. Thus, our claim is that a W-building is precisely the same as a model of the theory of the Murphy monoid (in this quantalized sense). In the rest of this section, we analyze more concretely what preservation of the biclosed structure means. This analysis works generally for any *-monoidal poset M. Then it becomes a simple matter to show that the Bruhat-Tits condition implies this preservation property in the case of buildings. Lemma: Let X be a set and let d: X x X --> M be a function into a *-monoidal poset M (with partial order denoted =>) such that 2^d: 2^M --> 2^{X x X} is a map of *-quantales. Then 2^d preserves the biclosed structure iff it satisfies the extension property (forall x, y in X) (forall w in M) (exists z in X) d(x, y)w = d(x, z) and d(y, z) <= w. Proof: Define the biclosure operators / and \ in a *-quantale by the adjunction formulas ST --> U TS --> U -------- -------- S --> U/T S --> T\U Then (T\U)* = U*/T*. Consequently it suffices to consider preservation of just one biclosure operator. Because 2^d is a *-quantale map, there is an automatic inclusion arrow 2^d(U/T) --> 2^d(U) / 2^d(T) corresponding to [2^d(U/T)][2^d(T)] = 2^d((U/T)T) --> 2^d(U) and hence it suffices to analyze the case that there is an arrow in the other direction: 2^d(U) / 2^d(T) --> 2^d(U/T). Now, each of the functors 2^d, -/T, and --/2^d(T) has a left adjoint. Thus, letting E_d denote the adjoint to 2^d, this class of arrows is mated to a class of arrows [E_d(S)]T --> E_d[S 2^d(T)] (S in 2^{X x X}, T in 2^M). Since both sides of this arrow are cocontinuous in the separate arguments S and T, it is necessary and sufficient to analyze the case where S is a singleton (x, y) in X x X, and T is the upward-closed subset U_w generated by a single element w of M (i.e., U_w = {v in M: w => v}, which we also write as {v in M: v <= w}). We calculate E_d((x, y)) = {u in M: d(x, y) => u}, and further that the left-hand side [E_d(S)]T becomes simply [E_d((x, y))]U_w = {v in M: d(x, y)w => v} which is just the upwards-closed set generated by d(x, y)w. Since the right-hand side of the previous display is also upwards-closed, it becomes necessary and sufficient to analyze the condition that d(x, y)w belongs to it. As for the right-hand side E_d[S 2^d(T)], the relational composite inside the brackets becomes (x, y).2^d(U_w) = {(x, z): w => d(y, z)} Thus the right-hand side becomes {v in M: d(x, z) => v for some z such that w => d(y, z)} and the condition that this contains d(x, y)w is just exists_z d(x, z) => wd(x, y) and w => d(y, z) which by the triangle inequality is equivalent to exists_z d(x, z) = wd(x, y) and w => d(y, z) as was to be shown. QED 5. Thick Buildings 7. Proofs of the theorems Lemma 2: Let F be a set and let d: F x F --> W_{oo} be a function. The Bruhat-Tits condition on (F, d) implies the Schubert condition. Proof: The Schubert condition says that d(x, x') <= s_{i(1)}...s_{i(k)} iff there exists a chain x = x_0, x_1, ..., x_k = x' such that d(x_{j-1}, x_j) <= s_{i(j)} for j = 1, ..., k. Lemma 3: The Bruhat-Tits condition on (F, d) implies that 2^d preserves biclosed structure. Proof: By Lemma 1 in section 4... [unfinished]

© 2007 Todd H. Trimble