Representing Topoi by Topological Groupoids

It is shown that every topos with enough points is equivalent to the classifying topos of a topological groupoid. 1 Deenitions and statement of the result We recall some standard deenitions ((1, 5, 9]). A topos is a category E which is equivalent to the category of sheaves of sets on a (small) site. Equivalently, E is a topos ii it satisses the Giraud axioms ((1], p. 303). The category of sets S is a topos, and plays a role analogous to that of the one{point space in topology. In particular, a point of a topos E is a topos morphism p: S ! E. It is given by a functor p : E ! S which commutes with colimits and nite limits. For an object (sheaf) E of E, the set p (E) is also denoted E p , and called the stalk of E at p. The topos E is said to have enough points if these functors p , for all points p, are jointly conservative (see 9], p. 521, 1]). Almost all topoi arising in practice have enough points. This applies in particular to the presheaf topos ^ C on an arbitrary small category C , and to the etale topos associated to a scheme. In fact, any \coherent" topos has enough points (see Deligne, Appendix to Expos e VI in 1]). We describe a particular kind of topos with enough points. Recall that a groupoid is a category in which each arrow is an isomorphism. Such a groupoid is thus given by a set X of objects, and a set G of arrows, together with structure maps