A Completeness Theorem for Open Maps

This paper provides a partial solution to the completeness problem for Joyal’s axiomatization of open and etale maps, under the additional assumption that a collection axiom (related to the set-theoretical axiom with the same name) holds. In many categories of geometric objects, there are natural classes of open maps, of proper maps and of etale maps. Some of the forma1 properties of these classes in the category of schemes were isolated by Grothendieck [S]. In the late 70’s, an attempt was made by the first author to give a full axiomatization of classes of open (and of etale) maps in topoi: along with a list of axioms, he suggested some concrete candidates for “universal” classes of open (etale) maps in path-topoi such as T2 and Y’ which could be used to test the completeness of the axioms. In 1988 E. Dubuc proved a completeness theorem for these axioms for etale maps relative to the Sierpinski topos Y2, under some special assumptions on the site; see [2]. In this paper we prove a completeness theorem for any class R of open maps (in any topos b) which satisfies an additional axiom called the Collection Axiom. More precisely, for any topos Y-, the class of morphisms in Y2 which are quasi-pullback squares (in T) satisfies the axioms for open maps as well as this Collection Axiom. Given d and R, we construct a suitable topos LT and a geometric morphism cp : F2 + 8, with the property that an arrowfin I belongs to the class R iff its inverse image q*(f) is a quasi-pullback. We also prove a completeness theorem of the following form: for a topos d and a class R of open maps satisfying the Collection Axiom, there exists a suitable geometric realization functor from d into a category of locales, with the property that a mapfin d belongs to R iff its geometric realization is an open (in the usual sense) map of locales. Analogous results hold for pretopoi and exact categories. Furthermore, we show that these completeness theorems yield similar theorems for classes of etale maps, thereby improving Dubuc’s result.