  We show for any oriented surface, possibly with a boundary, how to generalize
Kramers-Wannier duality to the world of quantum groups. The generalization is
motivated by quantization of Poisson-Lie T-duality from the string theory.
Cohomologies with quantum coefficients are defined for surfaces and their
meaning is revealed. They are functorial with respect to some glueing
operations and connected with q-invariants of 3-folds.
