2007/01/08 by Luca Stefanini, Stefanini, Luca
Mathematics · #18D05 #22A22 (Secondary) #58H05 (Primary) 17B66 #Category Theory (math.CT) #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.CT #math.DG #math.SG #msc:17B66 #msc:18D05 #msc:22A22 #msc:58H05
paper · pdf · doi:10.48550/arxiv.math/0701231
22 Page, Published in Travaux Mathematiques
arxiv created 2009/02/12 · arxiv updated 2009/12/01
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.