# On local integration of Lie brackets

## TL;DR

This paper presents an explicit method to construct local Lie groupoids from Lie algebroids using spray vector fields, providing a comprehensive local Lie theory and a finite-dimensional proof of their categorical equivalence.

## Contribution

It introduces a direct, explicit construction of local Lie groupoids from Lie algebroids relying solely on spray vector fields, simplifying the integration process.

## Key findings

- Provides a self-contained construction method for local Lie groupoids.
- Establishes the equivalence between germs of local Lie groupoids and Lie algebroids.
- Offers a finite-dimensional proof of the local Lie theory correspondence.

## 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.