In the case of 1-categories, we know there is a functor category $PSh(C):=[C^{op},Set]$, where $C$ is a small category, and this functor category is a topos. I am hoping this will extend to the case of 2-categories and especially the 2-category of groupoids.
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9$\begingroup$ Sorry, but what does "admits a topos of presheaves" mean? $\endgroup$– Todd TrimbleCommented Jul 26, 2019 at 19:02
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2$\begingroup$ the category of 2-functors and natural transformations Gpd^op --> C, for C any 1-category is equivalent to the category Gpd^op/~ --> C of functors and natural transformations, where Gpd/~ has the same objects as Gpd but isomorphism classes of functors as arrows. The category [Gpd^op/~,Set] is a presheaf topos... $\endgroup$– David Roberts ♦Commented Jul 26, 2019 at 22:41
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4$\begingroup$ You've just edited the question to say "Normally this would mean" something that doesn't mention groupoids at all. Earlier, you wrote that the comment by @DavidRoberts "has been suggested to help with an answer." Either this comment is an answer, or you really need to clarify what the question is intended to be. $\endgroup$– Andreas BlassCommented Jul 27, 2019 at 13:28
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1$\begingroup$ @Ben what else could it be, given the universal property I state? Given a 2-category D and a 1-category C, the category of 2-functors and natural transformations 2Cat(D,C) and the category of 1-functors and natural transformations Cat(D/~,C) composition with D -> D/~ gives an equivalence Cat(D/~,C) -> 2Cat(D,C). (Here D/~ has as hom-sets the set of iso classes of the hom-categories of D) $\endgroup$– David Roberts ♦Commented Jul 27, 2019 at 22:36
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3$\begingroup$ It remains to be seen why you want to do this, and what good you might do. Personally, I think any construction that passes to D/~, for D a 2-category, is the wrong thing. $\endgroup$– David Roberts ♦Commented Jul 28, 2019 at 0:37
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Gpd is the 2-category of groupoidd.
The category of 2-functors and natural transformations $Gpd^{op} \rightarrow C$, for $C$ any 1-category is equivalent to the category $Gpd'^{op} \rightarrow C$ of functors and natural transformations, where $Gpd'$ has the same objects as $Gpd$ but isomorphism classes of functors as arrows. The category $[Gpd'^{op},Set]$ is a presheaf topos.