Tetrahedral forms in monoidal categories and 3-manifold invariants

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 $sl_2$.