Bitopological Gerbes on Differentiable Stacks

We develop a framework for bitopological gerbes on differentiable stacks by combining the theory of bitopological stacks with the groupoid-theoretic theory of differentiable stacks and gerbes. The underlying geometric object is a bitopological space (X, T₁, T₂) together with a differentiable groupoid presentation carrying two compatible topological structures. A bitopological gerbe is described by a pair of gerbes G₁ and G₂, associated with the two topologies, together with an equivalence Φ over a common refinement. We introduce the corresponding notion of a bitopological S¹-gerbe, formulate its groupoid version in terms of pairs of S¹-central extensions, and study compatibility, pullbacks, Morita invariance, connections, curvings, and Dixmier–Douady classes. The resulting characteristic data naturally take values in a fiber product of degree-three cohomology groups. We prove that the diagonal case T₁ = T₂ recovers the ordinary theory of S¹-gerbes on differentiable stacks. Thus the present construction provides a genuine two-topology extension of the usual stack-theoretic gerbe theory.