  We generalize Franz' independence in tensor categories with inclusions from
two morphisms (which represent generalized random variables) to arbitrary
ordered families of morphisms. We will see that this only works consistently if
the unit object is an initial object, in which case the inclusions can be
defined starting from the tensor category alone. The obtained independence for
morphisms is called categorial independence. We define categorial L\'evy
processes on every tensor category with initial unit object and present a
construction generalizing the reconstruction of a L\'evy process from its
convolution semigroup via the Daniell-Kolmogorov theorem. Finally, we discuss
examples showing that many known independences from algebra as well as from
(noncommutative) probability are special cases of categorial independence.
