We associate an A-infinity category with a dimer model, and show that it is
derived-equivalent to the category of representations of the quiver with
relations associated with the dimer model if the dimer model is consistent. We
also associate an exact Lefschetz fibration with a pair of a dimer model and an
internal perfect matching on it, and use it to prove a version of homological
mirror symmetry for two-dimensional toric Fano stacks.