The category of 1-motives given by Deligne has realization functors and Cartier duality. He also proved any semi-normal complex algebraic curve has a 1-motive over

I will give the category of 1-motives with torsion and its Cartier dual functor, and show Cartier duality for this category. Some people including Barbieri-Viale already considered such a question and got some result, but this category is something different. For example, it is an abelian category but my category is not, and they need one more (in fact, dual) category to define Cartier dual but my category is closed under Cartier dual.