Abstract: In this talk, I will give more mathematical details of my talk on Nov 4. I will introduce the notion of limit categories for cotangent stacks of smooth stacks as an effective version of classical limits of the categories of D-modules on them. Using the notion of limit categories, I will propose a precise formulation of the Dolbeault geometric Langlands conjecture, proposed by Donagi-Pantev as the classical limit of the geometric Langlands correspondence. I will show the existence of a semiorthogonal decomposition of the limit category into quasi-BPS categories, which (when G=GL_r) categorify BPS invariants on a non-compact Calabi–Yau 3-fold playing an important role in Donaldson-Thomas theory. This semiorthogonal decomposition is interpreted as a Langlands dual to the semiorthogonal decomposition for moduli stacks of semistable Higgs bundles, obtained in our earlier work as a categorical analogue of PBW theorem in cohomological DT theory. It in particular yields a conjectural equivalence between quasi-BPS categories, which gives a categorical version of Hausel-Thaddeus mirror symmetry for Higgs bundles (for any reductive group G). This is a joint work with Tudor Pădurariu (arXiv:2508.19624).