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1. • CommentRowNumber1.
   • CommentAuthorDavidCarchedi
   • CommentTimeMay 30th 2013
   • (edited May 30th 2013)
   • PermaLink
   Author: DavidCarchedi
   Format: MarkdownItexThis is a blatant cross-post of an MO question of mine, but I thought perhaps this would be a more appropriate forum to ask: Suppose that $F:C^{op} \to Set$ is a presheaf. For an object $C$, denote by $Aut(c)$ the group of isomorphisms $f:c\to c.$ There is a canonical action $Aut(c)$ of on $F(c).$ Is there a special name for those $F$ for which all of these actions are transitive? Has this situation been studied? Does this imply any nice categorical consequences? If $F$ is also a sheaf for a Grothendiek topology on $C,$ does $F$ satisfy any nice conditions in the associated topos of sheaves?

   This is a blatant cross-post of an MO question of mine, but I thought perhaps this would be a more appropriate forum to ask:

   Suppose that $F:C^{op} \to Set$ is a presheaf. For an object $C$, denote by $Aut(c)$ the group of isomorphisms $f:c\to c.$ There is a canonical action $Aut(c)$ of on $F(c).$ Is there a special name for those $F$ for which all of these actions are transitive? Has this situation been studied? Does this imply any nice categorical consequences? If $F$ is also a sheaf for a Grothendiek topology on $C,$ does $F$ satisfy any nice conditions in the associated topos of sheaves?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 30th 2013
   • (edited May 30th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexNot sure if I have seen what you are after. But what it reminds me of a bit is of the theory of presheaves on [[generalized Reedy categories]], where these automorphisms at least play a major role.

   Not sure if I have seen what you are after. But what it reminds me of a bit is of the theory of presheaves on generalized Reedy categories, where these automorphisms at least play a major role.

3. • CommentRowNumber3.
   • CommentAuthorDavidCarchedi
   • CommentTimeMay 30th 2013
   • PermaLink
   Author: DavidCarchedi
   Format: MarkdownItexThanks Urs. However, in my case, this is a special property for $F,$ and does not hold for all presheaves on my category $C$ (and the fact that it holds even for this $F$ is non-trivial). This seems like a pretty natural condition, so I was hoping it popped up somewhere before.

   Thanks Urs. However, in my case, this is a special property for $F,$ and does not hold for all presheaves on my category $C$ (and the fact that it holds even for this $F$ is non-trivial). This seems like a pretty natural condition, so I was hoping it popped up somewhere before.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 30th 2013
   • (edited May 30th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHi David, that doesn't contradict what I was pointing to. Let me clarify: On a generalized Reedy model category one considers _normal presheaves_ which are roughly such that that automorphism action that you are thinking of is _free_. See at _[[generalized Reedy model structure]]_ the section _[natural monomorphisms](http://ncatlab.org/nlab/show/generalized+Reedy+category#NormalMorphisms)_. Certainly this is a special property of a presheaf! (Of course the famous application of this is in the [[model structure for dendroidal sets]], where the normal dendroidal sets are the cofibrant objects.) So this is not the transitivity condition on a presheaf that you are after, but instead a free-ness condition, but in spirit it seems not too far away. That still doesn't mean that it is at all helpful for what you are after. But it is at least reminiscent. :-)

   Hi David,

   that doesn’t contradict what I was pointing to. Let me clarify:

   On a generalized Reedy model category one considers normal presheaves which are roughly such that that automorphism action that you are thinking of is free. See at generalized Reedy model structure the section natural monomorphisms. Certainly this is a special property of a presheaf!

   (Of course the famous application of this is in the model structure for dendroidal sets, where the normal dendroidal sets are the cofibrant objects.)

   So this is not the transitivity condition on a presheaf that you are after, but instead a free-ness condition, but in spirit it seems not too far away. That still doesn’t mean that it is at all helpful for what you are after. But it is at least reminiscent. :-)

5. • CommentRowNumber5.
   • CommentAuthorDavidCarchedi
   • CommentTimeMay 30th 2013
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   Author: DavidCarchedi
   Format: MarkdownItexAh, I see. Thanks for the clarification :). At any rate, my indexing category isn't anything like a Reedy category I am afraid. It's a category of manifolds and embeddings...

   Ah, I see. Thanks for the clarification :). At any rate, my indexing category isn’t anything like a Reedy category I am afraid. It’s a category of manifolds and embeddings…