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1. • CommentRowNumber1.
   • CommentAuthordomenico_fiorenza
   • CommentTimeJun 7th 2011
   • PermaLink
   Author: domenico_fiorenza
   Format: MarkdownItexLet us work over a field $k$, and take the recursive definition of $n$-vector spaces. then a $2$-vector space is a $k$-algebra. now consider a 2-vector bundle over some space $X$. how does one see that its global sections are a 2-vector space? the answer somehow depends on the notion of section one adopt, but in any case the relation between the various definitions should be investigated. basically we have two notions (and their duals): i) natural transformations from the trivial 2-bundle to the given bundle. this is a very clearly defined object, but it is not clear (to me) that this is a 2-vector space: which is the underlying algebra? ii) the limit in 2-Vect of the functor $X\to 2Vect$ defining the 2-bundle. this is manifestly 2-vector space, but it is not clear (to me) that this limit exists. clearly the dream statement here would be that i) has a natural structure of 2-vector space, and that this 2-vector space represents the limit ii). (or the dual version of the above)

   Let us work over a field $k$, and take the recursive definition of $n$-vector spaces. then a $2$-vector space is a $k$-algebra. now consider a 2-vector bundle over some space $X$. how does one see that its global sections are a 2-vector space?

   the answer somehow depends on the notion of section one adopt, but in any case the relation between the various definitions should be investigated. basically we have two notions (and their duals):

   i) natural transformations from the trivial 2-bundle to the given bundle. this is a very clearly defined object, but it is not clear (to me) that this is a 2-vector space: which is the underlying algebra?

   ii) the limit in 2-Vect of the functor $X\to 2Vect$ defining the 2-bundle. this is manifestly 2-vector space, but it is not clear (to me) that this limit exists.

   clearly the dream statement here would be that i) has a natural structure of 2-vector space, and that this 2-vector space represents the limit ii).

   (or the dual version of the above)

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 7th 2011
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   Author: Urs
   Format: MarkdownItexGood question. For ordinary vector bundles one can use the fact that the forgetful functor $Vect \to Set$ is right adjoint to deduce that the underlying set of the vector space $$ \Gamma(V : X \to Vect) := \lim_\leftarrow (X \stackrel{V}{\to} Vect) $$ is itself the limit over $X \stackrel{V}{\to} Vect \to Set$, hence the set of plain sections. So maybe one should try to lift this by looking at a suitable forgetful functor out of $2 Vect$. But I am not sure.

   Good question.

   For ordinary vector bundles one can use the fact that the forgetful functor $Vect \to Set$ is right adjoint to deduce that the underlying set of the vector space

   $$\Gamma(V : X \to Vect) := \lim_\leftarrow (X \stackrel{V}{\to} Vect)$$

   is itself the limit over $X \stackrel{V}{\to} Vect \to Set$, hence the set of plain sections.

   So maybe one should try to lift this by looking at a suitable forgetful functor out of $2 Vect$. But I am not sure.

3. • CommentRowNumber3.
   • CommentAuthordomenico_fiorenza
   • CommentTimeJun 7th 2011
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   Author: domenico_fiorenza
   Format: MarkdownItex>Good question. thanks! I'm going to ask also on MO to see if we can get feedback there

   Good question.

   thanks! I’m going to ask also on MO to see if we can get feedback there