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About groupoid of Lie algebra valued forms. You (Urs S.) have several versions of functorial ways to express a connection, including this stuff about adjoint triple of functors and flat part of BG which is the most beautiful. But let us go back to your 2007 and 2008 papers with Konrad. You used there the thin fundamental n-groupoid (which you call path groupoid) and proved the equivalence of the corresponding definition of connection/transport with the one via differential forms. The study of ODE’s has been used in one direction. Now using infinitesimal path groupoid like in synthetic geometry looks less cumbersome and more natural than the thin homotopies (everybody has intuition about calculus, but who has ever computed the thin homotopy classes). Thus, I’d rather look for a proof via synthetic methods. (any remarks?)
On the other hand, the classical approach to parallel transport is also a bit different, namely one does not talk about homotopy at all, neither thin nor not thin. One instead first defines a principal or associated bundle $P$ without connection and looks at the isomorphisms of all fibers as forming a groupoid whose object part projects onto the base manifold; then one looks at “parallel transport” functors from the groupoid of piecewise smooth paths downstairs such that the object part is a section of the projection and such that the whole prescription is smooth in the sense that the tangent to the parallel transport in any direction is well defined and all such “horizontal” tangents form a submanifold of the tangent bundle of the total space. It is classical that such “smooth transport functors into total space” are equivalent to the differential form approach. Now, Urs has found an approach without talking the total space and talking just the generic fiber; the notion of the smoothness there is a bit more complicated as it required the interpretation of thin homotopy groupoid as a diffeological space. I am looking at how to order in my head best the relation between the different approaches to the connections and would like the proofs to have minimal cumbersome detail.
P.S. Yet another difference to have in mind is that the differential forms in question in classical setup are just the forms on the manifold, and the open sets are open sets in the base manifold. Urs’s approach is rather taking the forms as depending on the objects in the site of Cartesian spaces.
Of course, I am not aware if anybody has generalized the notion of the horizontal distribution to the case of smooth higher principal bundles (with total spaces). It should not be difficult: the differential form version gives us hints. Of course while the differential forms can live abstractly in the Lie n-algebra like above in Urs’s approach, they have concrete meaning when the structure group is related to the morphisms between the fibers.
In the case when the structure n-group is replaced by n-groupoid, then part of the data for a principal bundle is the action which entails the momentum map (which has not been commented in the typical fiber approach).
Of course, I am not aware if anybody has generalized the notion of the horizontal distribution to the case of smooth higher principal bundles (with total spaces).
I had a think about this a few years back, when I was still working in the differential geometric setting, but I didn’t get anywhere. No one has, to my knowledge either, considered connections like this, except possibly Danny Stevenson - he gave a talk once involving a higher Atiyah sequence. If you like I can track it down.
I know of his work on Atiyah sequences (online slides from Danny’s talk in Minnesota, we should link it in nlab) and was thinking myself some on it, 3-4 years ago. It is related to Ehresmann appraoch to connections, that is via horizontal distributions, but the study of Atiyah sequence while useful is not necessary for that theory (historically also Atiyah’s work came later – in 1957).
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