Q-groupoids and Their Cohomology

We approach Mackenzie’s LA-groupoids from a supergeometric point of view by introducing Q-groupoids. A Q-groupoid is a groupoid object in the category of Q-manifolds, and there is a faithful functor from the category of LA-groupoids to the category of Q-groupoids. Using this approach, we associate to every LA-groupoid a double complex whose cohomology simultaneously generalizes Lie groupoid cohomology and Lie algebroid cohomology. As examples, we obtain simplicial-type models for equivariant Lie algebroid cohomology and orbifold Lie algebroid cohomology, and we obtain double complexes associated to Poisson groupoids and groupoid-algebroid “matched pairs”.