  We consider the linear lambda-calculus extended with the sup type
constructor, which provides an additive conjunction along with a
non-deterministic destructor. The sup type constructor has been introduced in
the context of quantum computing. In this paper, we study this type constructor
within a simple linear logic categorical model, employing the category of
semimodules over a commutative semiring. We demonstrate that the
non-deterministic destructor, either weighted or not, finds a suitable model in
a weighted codiagonal map. Our approach offers a valid and insightful
alternative to interpreting non-determinism and probability calculi, in
instances where the conventional Moggi's Powerset Monad interpretation does not
align with the category's structure, as is the case with the category of
semimodules. The validity of this alternative relies on the presence of
biproducts within the category.
