nLab separated geometric morphism

Definition

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.

Examples

Proposition

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

Related concepts

• Hausdorff topological space
• quasi-separated scheme
• separated scheme, separated morphism of schemes

References

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

• Peter Johnstone, Sketches of an Elephant