An `unsitely' result on atomic morphisms (Q1840717)

Although the notion of a geometric morphism between elementary toposes makes sense without the additional assumption of boundedness, it is not known whether certain constructions and results obtained in the more restricted setting admit the more general counterparts. One such construction is that of the bipullback of a pair of geometric morphisms with the same codomain, well-known to exist if at least one of these two morphisms is assumed bounded. In turn, this unsettled question derives in possible restrictions on results which involve bipullbacks, for instance the representation theorem [\textit{A. Joyal} and \textit{M. Tierney}, ``An extension of the Galois theory of Grothendieck'', Mem. Am. Math. Soc. 309 (1984; Zbl 0541.18002)] which states that every topos \(F\) bounded over a topos \(E\) via a geometric morphism \(f : F\to E\) is the classifying topos of a continuous open groupoid in \(E\). The main tool is the existence of an open surjection \(p: E\to F\) over \(E\) and the descent theorem for open surjections. Of interest in this connection is the result [Joyal and Tierney, op. cit.] that if the geometric morphism \(f:F\to E\) is (connected) atomic (and bounded), then any point \(p:E\to F\) over \(E\) is open (surjective). The author shows here that there is a site-free proof of this result and that therefore, in principle, it would seem not to be necessary to assume that \(f:F\to E\) be bounded. However, somewhat surprinsingly although easy to show, the mere existence of a point already implies boundedness. The value of the paper therefore lies not so much on the result itself as in the method of proof. For instance, the fact that for an atomic geometric morphism \(f:F\to E\), the left adjoint \(f\)! to the inverse image part \(f^*\) preserves monos is exploited in a clever way in order to establish the result without the choice of a (atomic) site. The paper is instructive for anybody working in topos theory.