Supergroupoids, double structures, and equivariant cohomology

TL;DR

This paper introduces supergroupoids and superalgebroids with homological vector fields, linking them to double structures and equivariant cohomology models, including the BRST and Drinfel'd double models.

Contribution

It establishes Q-groupoids and Q-algebroids as new structures connecting Mackenzie's double structures with equivariant cohomology models.

Findings

Q-groupoids serve as intermediaries between LA-groupoids and double complexes.

A double complex for Q-algebroids generalizes the BRST model.

A supergroupoid version of the van Est map connects double complexes.

Abstract

Q-groupoids and Q-algebroids are, respectively, supergroupoids and superalgebroids that are equipped with compatible homological vector fields. These new objects are closely related to the double structures of Mackenzie; in particular, we show that Q-groupoids are intermediary objects between Mackenzie's LA-groupoids and double complexes, which include as a special case the simplicial model of equivariant cohomology. There is also a double complex associated to a Q-algebroid, which in the above special case is the BRST model of equivariant cohomology. Other special cases include models for the Drinfel'd double of a Lie bialgebra and Ginzburg's equivariant Poisson cohomology. Finally, a supergroupoid version of the van Est map is used to give a homomorphism from the double complex of a Q-groupoid to that of a Q-algebroid.