  This (partially expository) paper discusses Lagrangian Floer cohomology in
the context of Lefschetz fibrations, with emphasis on the algebraic structures
encountered there. In addition to the well-known directed A_infinity algebras
which appear in this situation, one has additional information encoded in a
certain bimodule homomorphism. There are two approaches to constructing this
homomorphism: in terms of the (noncompact) Lefschetz thimbles in the total
space, or else in terms of vanishing cycle in the fibre. We prove a comparison
result, which shows that (up to a certain remaining ambiguity) the two
approaches are equivalent.
