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added to Haefliger groupoid some of the pertinent facts proven in Carchedi 12.
I am wondering if the following has been considered in the literature, and under which name.
A useful way to think of the Haefliger groupoid of a manifold is the following. Imagine the “germ bundle” over the manifold, whose fiber over a point is the germ of the manifold at that point. Then the Haefliger groupoid is the “Atiyah groupoid” of this bundle, whose objects are the points of the manifold, and whose morphisms between two points are the isomorphisms between the fibers of the bundle over these points.
Viewed this way, there is an evident generalization given by replacing germes by $n$th order infinitesimal neighbourhoods, for $1 \leq n \leq \infty$ (with $n = \infty$ corresponding to formal neighbourhoods). This gives smooth groupoids with the same objects, and morphsims between objects being the isomorphisms between the $n$th order formal neighbourhoods over these points.
This must have been considered under some name somewhere, I suppose. Would anyone know a reference that considers this?
Are these not just jet groupoids? These are discussed in a bunch of places, but in particular also in the thesis of Arne Lorenz, which is already cited at the bottom of the same n-Lab page.
Ah, right, thanks!
I have split off a subsection Haefliger groupoid – Variants and a stub entry jet groupoid, in order not to forget it again.
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