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1. • CommentRowNumber1.
   • CommentAuthorStephan A Spahn
   • CommentTimeMay 2nd 2012
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   Author: Stephan A Spahn
   Format: MarkdownItexI added some closure properties of the class of proper morphisms of toposes and the proposition saying that a morphism of toposes is proper iff it satisfies the stable weak Beck-Chevalley condition to [[proper geometric morphism]].

   I added some closure properties of the class of proper morphisms of toposes and the proposition saying that a morphism of toposes is proper iff it satisfies the stable weak Beck-Chevalley condition to proper geometric morphism.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 2nd 2012
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   Author: Urs
   Format: MarkdownItexThanks! I have added some hyperlinks to the various notions of geometric morphisms involved. Eventually we should add comments on what kinds of pullbacks of toposes are involved here. By the way, concerning those free loop spaces: since proper geometric morphisms are stable under pullback, it follows that with $X$ a compact [[Hausdorff topos]] also $\mathcal{L}X$ is compact (being the pullback of the diagonal $X \to X \times X$, which is proper iff $X$ is Hausdorff). Not sure if that is what you need, but it might be worth mentioning.

   Thanks!

   I have added some hyperlinks to the various notions of geometric morphisms involved.

   Eventually we should add comments on what kinds of pullbacks of toposes are involved here.

   By the way, concerning those free loop spaces: since proper geometric morphisms are stable under pullback, it follows that with $X$ a compact Hausdorff topos also $\mathcal{L}X$ is compact (being the pullback of the diagonal $X \to X \times X$, which is proper iff $X$ is Hausdorff). Not sure if that is what you need, but it might be worth mentioning.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 7th 2012
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   Author: Urs
   Format: MarkdownItexI have added to the [References](http://ncatlab.org/nlab/show/proper+geometric+morphism#References) pointers to Jacob Lurie's discussion of higher compactness conditions for higher toposes.

   I have added to the References pointers to Jacob Lurie’s discussion of higher compactness conditions for higher toposes.

4. • CommentRowNumber4.
   • CommentAuthorTim Campion
   • CommentTimeSep 28th 2018
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   Author: Tim Campion
   Format: MarkdownItexI'm a little concerned about the current state of this page. The material on proper and tidy morphisms in ordinary 1-topos theory seems fine -- it's in the literature, after all. But I'm not confident in the material on $n$-proper morphisms of $\infty$-topoi (which, as far as I know, doesn't appear in the literature): 1. For one thing, Lurie's definition of proper geometric morphism (Def 7.3.1.4 in HTT) is not mentioned or referenced. 2. For another, the condition that _indexed_ filtered colimits be preserved seems to have been dropped. Currently the definition on the page just says that _ordinary_ filtered colimits (of $n$-truncated objects) should be preserved. Regarding (2), Lurie conjectures that a definition along these lines equivalent to his own can be given in HTT Rmk 7.3.1.5, but he doesn't attempt to state it specifically because the requisite indexed $\infty$-category theory has not been developed. Note also that Lurie's definition of "proper" corresponds to what the definition on this page would call "$\infty$-proper", with no truncation restrictions.

   I’m a little concerned about the current state of this page.

   The material on proper and tidy morphisms in ordinary 1-topos theory seems fine – it’s in the literature, after all.

   But I’m not confident in the material on $n$-proper morphisms of $\infty$-topoi (which, as far as I know, doesn’t appear in the literature):

   1. For one thing, Lurie’s definition of proper geometric morphism (Def 7.3.1.4 in HTT) is not mentioned or referenced.

   2. For another, the condition that indexed filtered colimits be preserved seems to have been dropped. Currently the definition on the page just says that ordinary filtered colimits (of $n$-truncated objects) should be preserved.

   Regarding (2), Lurie conjectures that a definition along these lines equivalent to his own can be given in HTT Rmk 7.3.1.5, but he doesn’t attempt to state it specifically because the requisite indexed $\infty$-category theory has not been developed.

   Note also that Lurie’s definition of “proper” corresponds to what the definition on this page would call “$\infty$-proper”, with no truncation restrictions.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 28th 2018
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   Author: David_Corfield
   Format: MarkdownItex>...the requisite indexed ∞-category theory has not been developed Has it since been sufficiently developed, as in [here](https://ncatlab.org/nlab/show/Parametrized+Higher+Category+Theory+and+Higher+Algebra#Shah18)?

   …the requisite indexed ∞-category theory has not been developed

   Has it since been sufficiently developed, as in here?

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeSep 28th 2018
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   Author: Mike Shulman
   Format: MarkdownItexThe page should of course mention and compare to Lurie's definition. However, I think the definition as it stands is probably correct, albeit imprecise. First it defines a *compact* topos, which is to say one that is proper over $\infty Gpd$, and in this case (just as for 1-toposes over $Set$) the indexedness condition is automatic. Then it "defines" a proper geometric morphism to be one that exhibits its domain as "compact over its codomain as base topos", which in general means that "everything becomes indexed". But this is certainly potentially confusing and should be spelled out.

   The page should of course mention and compare to Lurie’s definition. However, I think the definition as it stands is probably correct, albeit imprecise. First it defines a compact topos, which is to say one that is proper over $\infty Gpd$, and in this case (just as for 1-toposes over $Set$) the indexedness condition is automatic. Then it “defines” a proper geometric morphism to be one that exhibits its domain as “compact over its codomain as base topos”, which in general means that “everything becomes indexed”. But this is certainly potentially confusing and should be spelled out.