  The technique of generating families produces obstructions to the existence
of embedded Lagrangian cobordisms between Legendrian submanifolds in the
symplectizations of 1-jet bundles. In fact, generating families may be used to
construct a TQFT-like theory that, in addition to giving the aforementioned
obstructions, yield structural information about invariants of Legendrian
submanifolds. For example, the obstructions devised in this paper show that
there is no generating family compatible Lagrangian cobordism between the
Chekanov-Eliashberg Legendrian $m(5_2)$ knots. Further, the generating family
cohomology groups of a Legendrian submanifold restrict the topology of a
Lagrangian filling. Structurally, the generating family cohomology of a
Legendrian submanifold satisfies a type of Alexander duality that, when the
Legendrian is null-cobordant, can be seen as Poincar\'e duality of the
associated Lagrangian filling. This duality implies the Arnold Conjecture for
Legendrian submanifolds with linear-at-infinity generating families. The
results are obtained by developing a generating family version of wrapped Floer
cohomology and establishing long exact sequences that arise from viewing the
the spaces underlying these cohomology groups as mapping cones.
