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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 is a presheaf. For an object , denote by the group of isomorphisms There is a canonical action of on Is there a special name for those for which all of these actions are transitive? Has this situation been studied? Does this imply any nice categorical consequences? If is also a sheaf for a Grothendiek topology on does satisfy any nice conditions in the associated topos of sheaves?
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.
Thanks Urs. However, in my case, this is a special property for and does not hold for all presheaves on my category (and the fact that it holds even for this is non-trivial). This seems like a pretty natural condition, so I was hoping it popped up somewhere before.
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. :-)
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…
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