We develop a local-to-global formalism for constructing Calabi–Yau structures for global sections of constructible sheaves or cosheaves of differential graded categories. The required data (a morphism between the sheafified Hochschild homology with the topological dualizing sheaf, satisfying a nondegeneracy condition) specializes to the classical notion of orientation when applied to the category of local systems on a manifold. We apply this construction to the cosheaves on arboreal skeleta arising in the microlocal approach to the A-model.
