Translation Groupoids and Orbifold Bredon Cohomology

We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps. We use this result to define an orbifold version of Bredon cohomology.

💡 Research Summary

The paper establishes a precise equivalence between two bicategorical frameworks for orbifolds. On one side, the authors consider the bicategory whose objects are representable orbifolds (i.e., stacks locally presented as quotients of smooth manifolds by Lie group actions) and whose 1‑morphisms are “good maps” – smooth maps that respect the local groupoid charts and are fully faithful on isotropy groups. On the other side, they work with the bicategory of translation groupoids (groupoids of the form G⋉X, where a Lie group G acts properly and étale‑ly on a manifold X) together with generalized equivariant maps, modeled as Hilsum–Skandalis spans (bibundles) between such groupoids.

The central theorem proves a biequivalence between these bicategories. The construction proceeds by assigning to each representable orbifold