On the integration of LA-groupoids and duality for Poisson groupoids

TL;DR

This paper develops a functorial framework for integrating LA-groupoids into double Lie groupoids, establishing conditions for fibred product integrability and linking LA-groupoids to symplectic double groupoids in the Poisson setting.

Contribution

It introduces a functorial approach to integrate LA-groupoids into double Lie groupoids and clarifies conditions for fibred product integrability, extending to Poisson groupoids.

Findings

Fibred product of Lie algebroids along integrable morphisms is always integrable.

LA-groupoids with integrable top structures can be associated with integrating double Lie groupoids.

Conditions are identified under which Poisson groupoids yield symplectic double groupoids.

Abstract

In this note a functorial approach to the integration problem of an LA-groupoid to a double Lie groupoid is discussed. To do that, we study the notions of fibred products in the categories of Lie groupoids and Lie algebroids, giving necessary and sufficient conditions for the existence of such. In particular, it turns out, that the fibred product of Lie algebroids along integrable morphisms is always integrable by a fibred product of Lie groupoids. We show that to every LA-groupoid with integrable top structure one can associate a differentiable graph in the category of Lie groupoids, which is an integrating double Lie groupoid, whenever some lifting conditions for suitable Lie algebroid homotopies are fulfilled; the result specializes to the case of a Poisson groupoid, yielding a symplectic double groupoid, provided our conditions on the associated LA-groupoid are satisfied.