category with duals (list of them)
dualizable object (what they have)
The precise definition associated with the term “tensor category” varies somewhat in the literature.
It may mean any :
any monoidal category,
an additive (symmetric) monoidal category, so that the tensor preserves finite direct sums,
an abelian (symmetric) monoidal category, in which the tensor preserves finite colimits in separate arguments,
Deligne's theorem on tensor categories (Deligne 02) establishes Tannaka duality between sufficiently well-behaved linear tensor categories in characteristic zero and supergroups, realizing these tensor categories as categories of representations of these supergroups.
Deligne's theorem on tensor categories is due to