Supergroupoids, double structures, and equivariant cohomology

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.

UC Berkeley Ph.D. thesis; 111 pages

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• Mackenzie theory and Q-manifolds
• Differential graded contact geometry and Jacobi structures
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• Formality and Kontsevich--Duflo type theorems for Lie pairs
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• The geometry of graded cotangent bundles
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• Modular Classes of Lie Groupoid Representations up to Homotopy
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