nLab
separated geometric morphism

Context

Topos Theory

Could not include topos theory - contents

Contents

Definition

A geometric morphism f:𝒳→𝒴f : \mathcal{X} \to \mathcal{Y} of toposes is separated if the diagonal 𝒳→𝒳× 𝒴𝒳\mathcal{X} \to \mathcal{X} \times_{\mathcal{Y}} \mathcal{X} is a proper geometric morphism.

In particular if 𝒴\mathcal{Y} is the terminal object in Topos, hence the canonical base topos Set, we say that a topos 𝒳\mathcal{X} is a Hausdorff topos if 𝒳→𝒳×𝒳\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 GG a discrete group and BG=(G→→*)\mathbf{B}G = (G \stackrel{\to}{\to} *) its delooping groupoid, the presheaf topos GSet≃[BG,Set]G Set \simeq [\mathbf{B}G, Set] is Hausdorff precisely if GG is a finite group.

In (Johnstone) this is example C3.2.24

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

Revised on May 9, 2012 03:54:31 by Zoran Å koda (31.45.129.110)