Cartier duality corresponds to taking the dual of the Hopf algebra, exchanging the multiplication and comultiplication.
Common cases include fppf sheaves of commutative groups over S, and complexes thereof.
These more general geometric objects can be useful when one wants to work with categories that have good limit behavior.
For loop groups of tori, Cartier duality defines the tame symbol in local geometric class field theory.
Gérard Laumon introduced a sheaf-theoretic Fourier transform for quasi-coherent modules over 1-motives that specializes to many of these equivalences.