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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 28th 2011
   • (edited Jun 28th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have split off from [[smooth infinity-groupoid -- structures]] the section on concrete objects, creating a new entry [[concrete smooth infinity-groupoid]]. Right now there is * a proof that 0-truncated concrete smooth $\infty$-groupoids are equivalent to [[diffeological spaces]]; * and an argument that 1-truncated concrete smooth $\infty$-groupoids are equivalent to "diffeological groupoids": groupoids internal to diffeological spaces. That last one may require some polishing. I am still not exactly sure where this is headed, in that: what the deep theorems about these objects should be. For the moment the statement just is: there is a way to say "diffeological groupoid" using just very ygeneral nonsense. But I am experimenting on this subject with Dave Carchedi and I'll play around in the entry to see what happens.

   I have split off from smooth infinity-groupoid – structures the section on concrete objects, creating a new entry concrete smooth infinity-groupoid.

   Right now there is

   • a proof that 0-truncated concrete smooth $\infty$-groupoids are equivalent to diffeological spaces;

   • and an argument that 1-truncated concrete smooth $\infty$-groupoids are equivalent to “diffeological groupoids”: groupoids internal to diffeological spaces.

   That last one may require some polishing.

   I am still not exactly sure where this is headed, in that: what the deep theorems about these objects should be. For the moment the statement just is: there is a way to say “diffeological groupoid” using just very ygeneral nonsense.

   But I am experimenting on this subject with Dave Carchedi and I’ll play around in the entry to see what happens.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 29th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have written: > I am still not exactly sure where this is headed This applies to the fully concrete objects. I do have an idea what the $n$-concrete objects are good for: Over at [[schreiber:infinity-Chern-Simons theory]] we have a theorem that produces the holonomy of circle n-bundles with connection by an abstract hom-operation, but as a map of _discrete $\infty$-groupoids_ : it sends the discrete $\infty$-groupoid of circle n-bundles with connection on some $\Sigma$ to the discrete group $U(1)$. It is pretty clear that in order to refine this statement to the smooth case by instaed forming internal homs -- and thus refine it to a statement about transgression of differential cocycles to mapping spaces -- one needs to apply concretification. In the entry I have now proven this statement in the simplest non-trivial case: that for holonomy of ordinary circle bundles, hence the transgression of degree 2 differential cohomology classes to loop spaces. First notes are [here](http://ncatlab.org/nlab/show/concrete+smooth+infinity-groupoid#TransgressionOfDifferentialCocycles).

   I have written:

   I am still not exactly sure where this is headed

   This applies to the fully concrete objects. I do have an idea what the $n$-concrete objects are good for:

   Over at infinity-Chern-Simons theory (schreiber) we have a theorem that produces the holonomy of circle n-bundles with connection by an abstract hom-operation, but as a map of discrete $\infty$-groupoids : it sends the discrete $\infty$-groupoid of circle n-bundles with connection on some $\Sigma$ to the discrete group $U(1)$.

   It is pretty clear that in order to refine this statement to the smooth case by instaed forming internal homs – and thus refine it to a statement about transgression of differential cocycles to mapping spaces – one needs to apply concretification.

   In the entry I have now proven this statement in the simplest non-trivial case: that for holonomy of ordinary circle bundles, hence the transgression of degree 2 differential cohomology classes to loop spaces.

   First notes are here.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 29th 2011
   • (edited Jun 29th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> In the entry I have now proven this statement in the simplest non-trivial case: that for holonomy of ordinary circle bundles, hence the transgression of degree 2 differential cohomology classes to loop spaces. In fact, I think I prove it for circle $n$-bundles for all $n$, but so far only in codimension 0, hence over $\Sigma$ with $dim \Sigma = n$. It is precisely in higher codimension that the notion of "$n$-concreteness" becomes relevant: since $\mathbf{B}^n U(1)_{conn}$ is $n$-concrete, but not $k$-concrete for any $k \lt n$.

   In the entry I have now proven this statement in the simplest non-trivial case: that for holonomy of ordinary circle bundles, hence the transgression of degree 2 differential cohomology classes to loop spaces.

   In fact, I think I prove it for circle $n$-bundles for all $n$, but so far only in codimension 0, hence over $\Sigma$ with $dim \Sigma = n$.

   It is precisely in higher codimension that the notion of “$n$-concreteness” becomes relevant: since $\mathbf{B}^n U(1)_{conn}$ is $n$-concrete, but not $k$-concrete for any $k \lt n$.