  Modular categories are a well-known source of quantum 3-manifold invariants.
In this paper we study structures on modular categories which allow to define
refinements of quantum 3-manifold invariants involving cohomology classes or
generalized spin and complex spin structures. A crucial role in our
construction is played by objects which are invertible under tensor product.
All known examples of cohomological or spin type refinements of the
Witten-Reshetikhin-Turaev 3-manifold invariants are special cases of our
construction. In addition, we establish a splitting formula for the refined
invariants, generalizing the well-known product decomposition of quantum
invariants into projective ones and those determined by the linking matrix.
