  We present a generalization of free fermionic topological insulators that are
composed of topological subsystems of differing dimensionality. We specifically
focus on topological subsystems of nonzero co-dimension are embedded within a
trivial insulating environment. A general procedure is described to isolate and
classify such embedded topological insulators and we present three
representative examples in varying dimensions and symmetry classes. Moreover,
we demonstrate with concrete examples that the presence of periodically
embedded topological insulators in an otherwise trivially classified system can
lead to topologically non-trivial physical phenomena on crystalline defects;
namely, novel topological surface/edge modes at stacking faults and partial
edge dislocations.
