Q-groupoids and their cohomology

Abstract

We approach Mackenzie's LA-groupoids from a supergeometric point of view by introducing Q-groupoids, which are groupoid objects in the category of Q-manifolds. There is a faithful functor from the category of LA-groupoids to the category of Q-groupoids. We associate to every Q-groupoid a double complex that provides a model for the Q-cohomology of the classifying space. As examples, we obtain models for equivariant Q- and orbifold Q-cohomology, and for equivariant Lie algebroid and orbifold Lie algebroid cohomology. We obtain double complexes associated to Poisson groupoids and groupoid-algebroid "matched pairs".

v2 is the published version. Significant revision over previous version, particularly in section 5.2

References in corpus (3)

• Supergroupoids, double structures, and equivariant cohomology
• Mackenzie theory and Q-manifolds
• On the integration of LA-groupoids and duality for Poisson groupoids

Cited by in corpus (18)

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• LA-Courant algebroids and their applications
• Van Est isomorphism for homogeneous cochains
• On homotopy Poisson actions and reduction of symplectic Q-manifolds
• Infinitesimal Automorphisms of VB-groupoids and algebroids
• Multiplicative Dirac structures
• Atiyah classes and dg-Lie algebroids for matched pairs
• Graded Bundles in the Category of Lie Groupoids
• Atiyah and Todd classes arising from integrable distributions
• Van Est differentiation and integration
• Glanon groupoids
• Graded geometry in gauge theories and beyond
• Shifted Poisson structures on differentiable stacks
• Homological sections of Lie algebroids
• The Classification of Dirac Homogeneous Spaces
• The van Est homomorphism for strict Lie 2-groups
• Manin triples for double Lie bialgebroids
• Manin triples on multiplicative Courant algebroids