Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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
   • PermaLink
   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…