The Localization Theorem for the Motivic Homotopy Theory of Complex Analytic Stacks and other Geometric Settings

We prove the analog of the Morel-Voevodsky localization theorem over complex analytic stacks, which is used in arXiv:2511.09371 to establish a 6-functor formalism of complex analytic motivic homotopy theory and produce an analytification map that is compatible with the six operations. Along the way, we establish general techniques for proving this theorem over other geometric settings, which also apply, for example, to the settings of algebraic stacks and differentiable stacks.