Integration of Lie Algebroid Comorphisms

TL;DR

This paper establishes an equivalence between integrable Lie algebroids with complete comorphisms and source 1-connected Lie groupoids, enabling new constructions in Poisson geometry and highlighting conditions for integrability.

Contribution

It proves the path construction integration is an equivalence for integrable Lie algebroids and comorphisms, and introduces a symplectization functor in Poisson geometry.

Findings

Path construction integration is an equivalence under completeness.

Constructs a symplectization functor in Poisson geometry.

Integrability of comorphisms may fail without completeness.

Abstract

We show that the path construction integration of Lie algebroids by Lie groupoids is an actual equivalence from the category of integrable Lie algebroids and complete Lie algebroid comorphisms to the category of source 1-connected Lie groupoids and Lie groupoid comorphisms. This allows us to construct an actual symplectization functor in Poisson geometry. We include examples to show that the integrability of comorphisms and Poisson maps may not hold in the absence of a completeness assumption.