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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2011
    • (edited Jun 28th 2011)

    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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2011

    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 nn-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 Σ\Sigma to the discrete group U(1)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.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2011
    • (edited Jun 29th 2011)

    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 nn-bundles for all nn, but so far only in codimension 0, hence over Σ\Sigma with dimΣ=ndim \Sigma = n.

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