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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.
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.
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 $k \lt n$.
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