A completeness theorem for open maps (Q1338199)

In many Grothendieck toposes, there are naturally-arising classes of ``open'' or ``étale'' morphisms whose properties play an important role in understanding the geometric structure of those categories. Attempts to axiomatize the notion of a ``class of open'' (or ``étale'') maps have been made by J. Penon and E. J. Dubuc, as well as the first author of this paper. In the present paper, the authors extend previous axiomatizations by adding a ``collection axiom'', reminiscent of the set- theoretic axiom of that name: they show that the Sierpiński topos \({\mathcal T}^ 2\) over an arbitrary topos \({\mathcal T}\) contains a ``canonical'' class of open maps satisfying the collection axiom, and their ``completeness theorem'' is the assertion that any class of open maps satisfying the collection axiom in a topos \({\mathcal E}\) is obtained from this canonical one via a geometric morphism \({\mathcal T}^ 2 \to {\mathcal E}\). (However, since the choice of \({\mathcal T}\) depends on both \({\mathcal E}\) and the class of open maps therein, one wonders whether the name ``completeness theorem'' is really justified.) Although the statement of the theorem is essentially geometric, the proof uses categorical logic: it consists mainly of a detailed calculation of the classifying topos for the theory of ``anodyne extensions'' of models of the theory classified by \({\mathcal E}\). Analogous results are proved for classes of étale maps, and for classes of open maps in pretoposes and (suitable) sites; and the authors also derive completeness results in terms of functors into the category of locales (with ``open map'' meaning what it usually does in this category), and in terms of morphisms having a ``path-lifting'' property with respect to a given (topos-indexed) family of paths in \({\mathcal E}\).