Open strings, Lagrangian conductors and Floer functor

We introduce a contravariant functor, called Floer functor, from the category of Lagrangian conductors of a symplectic manifold to the homotopy category of bounded chain complexes of open strings in this manifold. The latter two categories are defined for all symplectic manifolds, whereas Floer functor is defined for semipositive manifolds which are either closed or convex at infinity. We then prove that when the first Chern class of the symplectic manifold vanishes, Lagrangian spheres define Lagrangian conductors so that in particular their integral Floer cohomology is well defined. This requires the introduction of singular almost-complex structures given by symplectic field theory.