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I asked a similar question on math.stackexchange but did not get any responses, so I thought I'd kick it up to mathoverflow.

In Crainic and Fernandes's "Integrability of Lie Brackets" (and the accompanying lecture notes), they use Lie II to prove the equivalence between $A$-paths (Lie algebroid morphisms $TI \to A$, where $TI$ is the tangent bundle over the unit interval) and $G$-paths (functors from the pair groupoid over $I$ into $G$). This is, of course, rock solid as a proof, but I find it a bit unsatisfying to have a fairly important step of a proof handled by applying a very powerful theorem to one of the simplest possible cases where it applies.

I first tried unwinding one of the original proofs of Lie II for groupoids from Moerdijk and Mrcun's book, where I assumed the source-simply-connected Lie groupoid in question is the pair groupoid over $I$. However, I once again found myself using powerful theorems about foliations while dealing with one of the simplest possible cases (the foliation of $I \times I$ by $I$). This feels like a result that should be amenable to elementary techniques and is nestled away in a masters thesis somewhere or a paper from the early days of Lie algebroids, but I can't seem to find a reference. I'm also sure that someone working in this field can probably pull the construction off the top of their head, but I'm having trouble finding the construction myself.

In summary:

  1. Does anyone know where I can find an explicit/direct proof of the bijection between functors on the pair groupoid over the unit inverval $I$ into a Lie groupoid $G$ and Lie algebroid morphisms $TI \to Lie(G)$?
  2. It seems as though the construction in (1) should be derivable from something like "the pair groupoid over a vector space $V$ integrates the trivial lie algebroid $TV \cong V \times V \xrightarrow{\pi_0} V$". This would be the ideal construction to find.

Update 1: The natural next step is to consider the path integration method on a Lie group. There is an explicit proof of the bijection in Proposition 1.13.4 of Duistermaat and Kolk's Lie Groups.

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