  We introduce systems of objects and operators in linear monoidal categories
called $\hat \Psi$-systems. A $\hat \Psi$-system satisfying several additional
assumptions gives rise to a topological invariant of triples (a closed oriented
3-manifold $M$, a principal bundle over $M$, a link in $M$). This construction
generalizes the quantum dilogarithmic invariant of links appearing in the
original formulation of the volume conjecture. We conjecture that all quantum
groups at odd roots of unity give rise to $\hat \Psi$-systems and we verify
this conjecture in the case of the Borel subalgebra of quantum $sl_2$.
