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 ,