In this paper, we define surfaces with pulleys which are unions of 1- and 2-dimensional manifolds, glued together on a finite number of Z‹3Z-labeled points of their interiors. Then, by seeing them as cobordisms, we give a refinement of Bar-Natan's geometrical construction of Khovanov homology which can be applied to different notions of refined links as links in I-bundle or braid-like links.