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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 17th 2012
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   Author: Todd_Trimble
   Format: MarkdownItexThis is not a well-thought-out question, but I imagine someone like Mike might have meditated on similar things. One definition of elementary topos is: regular category $C$ such that the canonical inclusion $i: C \to Rel(C)$ has a right adjoint $p$. Is there a good notion of [[2-topos]] that attempts to categorify this idea?

   This is not a well-thought-out question, but I imagine someone like Mike might have meditated on similar things.

   One definition of elementary topos is: regular category $C$ such that the canonical inclusion $i: C \to Rel(C)$ has a right adjoint $p$.

   Is there a good notion of 2-topos that attempts to categorify this idea?

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 17th 2012
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   Author: Todd_Trimble
   Format: MarkdownItexNever mind... I think this one needs to bake longer...

   Never mind… I think this one needs to bake longer…

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeAug 17th 2012
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   Author: Mike Shulman
   Format: MarkdownItexWell, there are size issues, of course. But a sufficiently regular 2-category gives rise to a proarrow equipment whose proarrows are 2-sided fibrations, and then asking the inclusion of arrows into proarrows to have a sort of right adjoint would be related to a [[michaelshulman:classifying discrete opfibration]].

   Well, there are size issues, of course. But a sufficiently regular 2-category gives rise to a proarrow equipment whose proarrows are 2-sided fibrations, and then asking the inclusion of arrows into proarrows to have a sort of right adjoint would be related to a classifying discrete opfibration (michaelshulman).

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 19th 2012
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   Author: Todd_Trimble
   Format: MarkdownItexThanks for the response! That's the kind of thing I had in mind.

   Thanks for the response! That’s the kind of thing I had in mind.