The talk formulates a notion of independence of subobjects of an object in a general (i.e. not necessarily concrete) category. The content of subobject independence is morphism co-possibility: two subobjects of an object will be defined to be independent if any two morphisms on the two subobjects of an object are jointly implementable by a single morphism on the larger object. Subobject independence is the categorial generalization of what is known as subsystem independence in the context of local quantum physics. Standard concepts of subsystem independence in algebraic relativistic quantum field theory can be recovered as special cases of subobject independence by choosing special subclasses of the non-selective operations (completely positive unit preserving linear maps on C* algebras) as morphisms. It will be shown how categorial subsystem independence can be utilized in the framework of the categorial approach to quantum field theory.