Abstract
We give a direct, explicit and self-contained construction of a local Lie groupoid integrating a given Lie algebroid which only depends on the choice of a spray vector field lifting the underlying anchor map. This construction leads to a complete account of local Lie theory and, in particular, to a finite-dimensional proof of the fact that the category of germs of local Lie groupoids is equivalent to that of Lie algebroids.
Funding statement: Ioan Mărcuţ was supported by the NWO Veni grant 613.009.031 and the NSF grant DMS 14-05671. María Amelia Salazar was a Post-Doctorate at IMPA during part of this project, funded by CAPES-Brazil. Alejandro Cabrera would like to thank CNPq and FAPERJ for financial support.
A Differentiating integrated cochains
In this appendix, we consider the differentiation of the local Lie groupoid cochains obtained by integration through the map Ψ defined in Section 2.5. The main result is Lemma A.1 which is used in the proof of item (ii) of Proposition 2.10.
Following the notation of Section 2.5, let
Let us fix composable arrows
By using the relations
where
Applying (A.2), we can compute
where
thus, we have
Before stating the general inductive result, let us compute one more derivative fixing elements
where we have relabeled the integration variables. Using (A.2) we compute
The key step for recognizing an inductive structure in our computation is to note that (recall that M is identified with the unit section)
for any small composable string
for any
we have
from which
Continuing by induction, we obtain the following:
Lemma A.1.
For
where
Proof.
We will consider an induction over
To use (A.6) on the right-hand side above, let us introduce the following notation:
and
By using (A.1), it follows that
and that
Using these to compute (A.7) gives
Applying (A.5), we get
The lemma thus follows
by noticing that
Acknowledgements
The authors would like to thank Marius Crainic, Pedro Frejlich, Rui Loja Fernandes, Marco Gualtieri, Eckhard Meinrenken and Daniele Sepe for useful discussions, and the anonymous referee for their careful reading and suggestions.
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