Abstract

The concept of a “differentiable stack” is a higher geometric notion similar to algebraic or topological stacks, aiming to transfer the study of manifolds to a more general class of “differential geometric spaces”. Examples include orbifolds, quotients of Lie group actions and many moduli spaces. While understanding differentiable stacks by unraveling their definition as categories is possible, there is also another way to approach their study: It can be seen that the 2-category of differentiable stacks is equivalent to a 2-category of Lie groupoids. Using this relation, it is possible to expand the mathematical toolbox applicable to either concept, for example, when computing the cohomology of a differentiable stack. It is possible to express it in terms of the associated Lie groupoid, which, in particular, frames differentiable stack cohomology as a generalization of equivariant cohomology. The existence of established models for equivariant cohomology has motivated research into finding generalised models for differentiable stack cohomology as well. In this talk, we will explore the problem and its key notions as well as its challenges. We will cover some approaches of generalizing preexisting models for equivariant cohomology including a recent advancement in the case of proper and regular Lie groupoids.