Chain duality for categories over complexes

We show that the additive category of chain complexes parametrized by a finite simplicial complex $K$ forms a category with chain duality. This fact, never fully proven in the original reference, is fundamental for Ranicki's algebraic formulation of the surgery exact sequence of Sullivan and Wall, and his interpretation of the surgery obstruction map as the passage from local Poincar\'e duality to global Poincar\'e duality. Our paper also gives a new, conceptual, and geometric treatment of chain duality on $K$-based chain complexes.