Determinants over graded-commutative algebras, a categorical viewpoint

We generalize linear superalgebra to higher gradings and commutation factors, given by arbitrary abelian groups and bicharacters. Our central tool is an extension, to monoidal categories of modules, of the Nekludova-Scheunert faithful functor between the categories of graded-commutative and supercommutative algebras. As a result we generalize (super-)trace, determinant and Berezinian to graded matrices over graded-commutative algebras. For instance, on homogeneous quaternionic matrices, we obtain a lift of the Dieudonn\'e determinant to the skew-field of quaternions.