Categorical traces and a relative Lefschetz–Verdier formula

Abstract We prove a relative Lefschetz–Verdier theorem for locally acyclic objects over Noetherian base scheme. This is done by studying duals and traces in the symmetric monoidal $2$ -category of cohomological correspondences. show that local acyclicity equivalent to dualisability deduce duality preserves acyclicity. As another application category correspondences, we nearby cycle functor Henselian valuation ring duals, generalising Gabber.