I just want to know if I'm understanding this right. I haven't really seen homology/cohomology outside of Z-related coefficients before, so this still seems kind of weird. I also haven't actually learned sheaf theory, so this might just be totally wrong. So if I have a top space and a sheaf mapping from it to something, say free modules: First if this is a cellular space, then the sheaf maps each cell to a free module. So is the cochain essentially a direct sum of free modules, one assigned to each cell? And for singular cohomology, the cochains would map the singular simplices to the free modules which are mapped to by the sheaf from the interiors of the geometric realizations of the simplices? I'm kind of just guessing this, not sure what else it would be. Also could someone give me an elementary example of where taking cohomology coefficients in a sheaf would be useful? preferably one not involving too much category or manifold theory. Thanks a lot for the help.