separated geometric morphism

A geometric morphism $f : \mathcal{X} \to \mathcal{Y}$ of toposes is **separated** if the diagonal $\mathcal{X} \to \mathcal{X} \times_{\mathcal{Y}} \mathcal{X}$ is a proper geometric morphism.

In particular if $\mathcal{Y}$ is the terminal object in Topos, hence the canonical base topos Set, we say that a topos $\mathcal{X}$ is a **Hausdorff topos** if $\mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is a proper geometric morphism.

More generally, since there is a hierarchy of notions of *proper geometric morphism*, there is accordingly a hierarchy of separatedness conditions.

For $G$ a discrete group and $\mathbf{B}G = (G \stackrel{\to}{\to} *)$ its delooping groupoid, the presheaf topos $G Set \simeq [\mathbf{B}G, Set]$ is Hausdorff precisely if $G$ is a finite group.

In (Johnstone) this is example C3.2.24

Chapter II of

- Ieke Moerdijk, Jacob Vermeulen,
*Relative compactness conditions for toposes*(pdf) and*Proper maps of toposes*, American Mathematical Society (2000)

Around def. C3.2.12 of

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