Representing top oi by top ological group oids Carsten Butz and Ieke Mo erdijk, Utrecht Abstract It is shown that every top os with enough p oints is equivalent to the clas- sifying top os of a top ological group oid. 1 De nitions and statement of the result We recall some standard de nitions [1, 5 , 9]. A top os is a category E whichis equivalent to the category of sheaves of sets on a small site. Equivalently, E is a top os i it satis es the Giraud axioms [1], p. 303. The category of sets S is a top os, and plays a role analogous to that of the one{p oint space in top ology. In particular, a point of a top os E is a top os morphism p: S!E. It is given bya functor p : E!Swhich commutes with colimits and nite limits. For an ob ject sheaf E of E , the set p E is also denoted E , and called the stalk of E at p.The p top os E is said to have enough points if these functors p , for all p oints p, are jointly conservative see [9], p. 521, [1]. Almost all top oi arising in practice have enough ^ p oints. This applies in particular to the presheaf top os C on an arbitrary small category C , and to the etale top os asso ciated to a scheme. In fact, any \coherent" top os has enough p oints see Deligne, App endix to Exp os eVIin[1]. We describ e a particular kind of top os with enough p oints. Recall that a groupoid is a category in whicheach arrow is an isomorphism. Such a group oid is thus given by a set X of ob jects, and a set G of arrows, together with structure maps s / u m o / 1 G G G X: X ? / 071234 t i Here s and t denote the source and the target, ux 2 G is the identityat x 2 X , 1 ig =g is the inverse, and mg; h=g h is the comp osition. A topological groupoid is such a group oid in which X and G are each equipp ed with a top ology, for which all the structure maps in 1 are continuous. Given such a top ological group oid, a G{sheaf is a sheaf on X equipp ed with a continuous G{action. Thus a G{sheaf consists of a lo cal homeomorphism p: E ! X together with a continuous action map E G ! E , de ned for all e 2 E and X x g : y ! x in G, and denoted e; g 7! e g ; this map should satisfy the usual identities for an action. 1 The category Sh X ofallsuch G{sheaves, and action preserving maps b etween G them, is a top os. It is called the classifying topos of the group oid G X .Such a classifying top os always has enough p oints. In fact, any ordinary p oint x 2 X de nes a p ointx: S!Sh X , by G 1 x E =E = p x: x The collection of all these p ointsx is jointly conservative. Our main aim is to provethatevery top os with enough p oints is, up to equiva- lence, the classifying top os of some top ological group oid: Theorem 1.1 Let E be any topos with enough points. There exists a topological groupoid G X for which thereisanequivalence of topoi Sh X : E = G We end this intro ductory section with some comments on related work. Repre- sentations of categories of sheaves by group oids go back to Grothendieck's Galois theory [4]. In [8], a general theorem was proved, which is similar to our result, and which states that for every top os E not necessarily with enough p oints there is a group oid G X in the category of lo cales \p ointless spaces" for whichthere Sh X . This theorem was sharp ened, again in the context is an equivalence E = G of lo cales, in [7]. The basic idea for our construction comes from the latter pap er. We wish to p ointout,however, that our result for top oi with enough p oints is not a formal consequence of any of these theorems. Moreover, our pro of is di erent. The pro ofs in [8] and [7] dep end essentially on change{of{base techniques, the internal logic of a top os, and the b ehaviour of lo cales in this context. These techniques cannot b e applied to the present situation. In fact, we b elieve that the pro of of our theorem is much more accessible and direct. 2 Description of the group oid Let E b e a top os with enough p oints. We recall the de nition of the space X = X E from [2], x2, and showthatitispartofagroupoid G X . First, although the collection of all p oints of E is in general a prop er class, there will always b e a set of p oints p for which the functors p are already jointly conservative [5], Corollary 7.17. Fix such a set, and call its memb ers smal l points of E . Next, let S be an ob ject n of E with the prop erty that the sub ob jects of p owers of S , i.e., all sheaves B S for n 0, together generate E .For example, S can b e the disjointsumofallthe ob jects in some small site for E . Let I b e an in nite set, with cardinalitysolarge that cardS cardI p for all small p oints p of E . 2 In general, if A is any set with card A cardI , wecallan enumeration of A 1 a function : D = dom ! A, where D I and a is in nite for each a 2 A. These enumerations carry a natural top ology, whose basic op en sets are the sets V = f j u g; 2 u here u is any function fi ;:::;i g!A de ned on a nite subset of I , and u 1 n means that i 2 dom and i =ui , for k =1;:::;n.Leaving the index set k k k I implicit, we denote this top ological space by En A; and call it the enumeration space of A. The space X ,involved in the group oid, is de ned by gluing several of these enumeration spaces together. A p ointofX is an equivalence class of pairs p; , where p is a small p ointof E and 2 En S isanenumeration of the stalk S . p p 0 0 Twosuch pairs p; andp ; are equivalent, i.e., de ne the same p ointof X , 0 0 if there exists a natural isomorphism : p ! p for which = . Note that S 0 for sucha , its comp onent is uniquely determined by and , b ecause is S surjective. In what follows, we will generally simply denote a p ointofX byp; , and we will not distinguish such pairs from their equivalence classes whenever we can do so without causing p ossible confusion. The top ology on the space X is given n by the basic op en sets U , de ned for any i ;:::;i 2 I and any B S ,as i ;:::;i ;B 1 n n 1 U = fp; j i ;:::; i 2 B g: 3 i ;:::;i ;B 1 n p n 1 0 0 Observe that this is well{de ned on equivalence classes; i.e., if p; p ; 0 0 , where we write by an isomorphism as ab ove, then i 2 B i i 2 B p p 0 i= i ;:::; i and similarly for . 1 n Next, we de ne the space G of arrows. The p oints of G are equivalence classes of quintuples p; ! q; ; where p; and q; are p oints of X as ab ove, and : p ! q is a natural isomorphism. Wedonot require that = . Twosuchp; ! q; S 0 0 0 0 0 and p ; ! q ; represent the same p ointofG whenever there are isomor- 0 0 0 0 phisms : p ! p and : q ! q such that = and = , S S 0 while in addition = . The top ology on G is given by the basic op en sets V = V de ned by i ;:::;i ;B ;j ;:::;j ;C i;B ;j;C n n 1 1 V = fp; ! q; j i 2 B ; j 2 C ; and i = j g: i;B ;j;C p q Here wehave again used the shorter notation i for i ;:::; i , etc. Note, as 1 n ab ove, that these basic op en sets are well{de ned on equivalence classes. It remains to de ne the structure maps x11 of the group oid. For an arrow g =[p; ! q; ], its source and target are de ned by sg =p; and tg = q; : 3 The maps s and t are well{de ned on equivalence classes, and are easily seen to b e continuous for the top ologies on X and G as just de ned. For two arrows g = 0 0 0 0 [p; ! q; ] and h =[q ; ! r; ] for which[q; ]= [q ; ] as p oints of X , 0 0 the comp osition h g in G is de ned as follows: since q; q ; , there is an 0 0 isomorphism : q ! q so that = . De ne h g to b e the equivalence class S of S p; ! r; : It is easy to check that his de nition do es not dep end on the choice of ,isagain well{de ned on equivalence classes, and is continuous for the given top ology on G and the bred pro duct top ology on G G.