Q-groupoids and their cohomology

Rajan Amit Mehta

Vol. 242 (2009), No. 2, 311–332

DOI: 10.2140/pjm.2009.242.311

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”.

Keywords

Lie algebroids, Lie groupoids, simplicial manifolds, Q-manifolds, equivariant cohomology

Mathematical Subject Classification 2000

Primary: 22A22

Secondary: 58A50, 55N91, 53D17

Milestones

Received: 22 January 2007

Revised: 23 June 2009

Accepted: 2 July 2009

Published: 1 October 2009