Gerbes on Bitopological Orbifolds: Groupoid Presentations, Dixmier–Douady Classes, and Morita Invariance

We develop a groupoid-based framework for gerbes on orbifolds equipped with two compatible topologies. The smooth topology determines the underlying orbifold, whereas a second topology records additional topological structure on the levels of its presenting groupoid. We define paired banded gerbes, formulate their Dixmier–Douady invariants, and establish classification and Morita-invariance results under explicit descent and coefficient hypotheses. In particular, we prove that decorated pairs of circle-banded gerbes are classified by a product of two groupoid-cohomology groups. We then formulate a relative classification for compatible pairs using a homotopy fibre product of gerbe-classification groupoids. We prove that both constructions are invariant under Morita equivalences that preserve the two topological sites and their coefficient systems. A finite-quotient example demonstrates why orbifold isotropy cannot in general be discarded in favor of the coarse orbit space. The example also provides a test that any proposed bitopological Dixmier–Douady theory must satisfy. The standard orbifold-gerbe theory is distinguished throughout from the additional paired structure introduced here. The classification statements are conditional on explicitly stated descent hypotheses; no automatic equivalence between arbitrary pairs of topologies is assumed.