  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.
