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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 12th 2011
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI see that [[open map]] has some new additions, presumably in aid of Urs' attempts to characterise smooth manifolds among sheaves on the site $CartSp$. The source is Joyal-Moerdijk's 'A completeness theorem for open maps', where they discuss open and etale maps in a (pre)topos. This sort of characterisation could usefully be extended to other ambient categories, since all they seem to require to define things are sums, a terminal object and a notion of epimorphism (in a topos these latter are practically given to you, but I could think of other settings where you'd have to specify regular epi or quotient map or something). Any thoughts?

   I see that open map has some new additions, presumably in aid of Urs’ attempts to characterise smooth manifolds among sheaves on the site $CartSp$. The source is Joyal-Moerdijk’s ’A completeness theorem for open maps’, where they discuss open and etale maps in a (pre)topos. This sort of characterisation could usefully be extended to other ambient categories, since all they seem to require to define things are sums, a terminal object and a notion of epimorphism (in a topos these latter are practically given to you, but I could think of other settings where you’d have to specify regular epi or quotient map or something). Any thoughts?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 12th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, I had made a quick note on this just so I won't forget. But don't have the time to follow up on this at the moment.

   Yes, I had made a quick note on this just so I won’t forget. But don’t have the time to follow up on this at the moment.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 12th 2011
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   Author: DavidRoberts
   Format: MarkdownItexI might have a go at it. Section 5 of the Joyal-Moerdijk paper seems particularly appropriate.

   I might have a go at it. Section 5 of the Joyal-Moerdijk paper seems particularly appropriate.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJan 12th 2011
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   Author: Mike Shulman
   Format: MarkdownItexI've been thinking recently about a similar generalization, to the context of a more-or-less arbitrary *site*, where the covering families would replace the sums and notion of epimorphism.

   I’ve been thinking recently about a similar generalization, to the context of a more-or-less arbitrary site, where the covering families would replace the sums and notion of epimorphism.

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 12th 2011
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexNotably, some of the results in Joral-Moerdijk are set in lextensive categories with superextensive pretopologies

   Notably, some of the results in Joral-Moerdijk are set in lextensive categories with superextensive pretopologies

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMay 12th 2017
   • (edited May 12th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexFor better readability, I have split off _[[open map]]_ (topology) from _[[open morphism]]_ (general) and added disambiguation. Added two elementary classes of examples at _[[open map]]_.

   For better readability, I have split off open map (topology) from open morphism (general) and added disambiguation. Added two elementary classes of examples at open map.