Project Euclid

We study a particular structure on a topos $\cal E$, related to the notion of a `class of étale maps' due to Joyal and Moerdijk and to Bénabou's notion of `calibration', which corresponds to giving for each object $A$ of $\cal E$ a `natural' comparison between the slice category ${\cal E}/A$ and a smaller `petit topos' associated with $A$. We show that there are many naturally-arising examples of such structures; but rather few of them satisfy the condition that the relation between the `gros' and `petit' toposes of every object is expressed by a local geometric morphism.