Given a finite modular tensor category, we associate with each compact surface with boundary a cochain complex in such a way that the mapping class group of the surface acts projectively on its cohomology groups. In degree zero, this action coincides with the known projective action of the mapping class group on the space of chiral conformal blocks. In the case that the surface is a torus and the category is the representation category of a factorizable ribbon Hopf algebra, we recover our previous result on the projective action of the modular group on the Hochschild cohomology groups of the Hopf algebra.