ABSTRACT. First I will introduce a generalization of the notion of (right)-exact
functor between abelian categories to the case of non-additive functors. The
main result of this section is an extension theorem: any functor defined on a
suitable subcategory can be extended uniquely to a right exact functor defined
on the whole category.
Next I use those results to define various functors of generalized tensor
induction, associated to finite bisets, between categories attached to finite
groups. This includes a definition of tensor induction for Mackey functors, for
cohomological Mackey functors, for p-permutation modules and algebras. This
also gives a single formalism of bisets for restriction, inflation, and ordinary
tensor induction for modules.
Keywords: additive, exact, tensor induction, Burnside, Mackey functor, Green