  We introduce the notion of a relative spherical category. We prove that such
a category gives rise to the generalized Kashaev and Turaev-Viro-type
3-manifold invariants defined in arXiv:1008.3103 and arXiv:0910.1624,
respectively. In this case we show that these invariants are equal and extend
to what we call a relative Homotopy Quantum Field Theory which is a branch of
the Topological Quantum Field Theory founded by E. Witten and M. Atiyah. Our
main examples of relative spherical categories are the categories of finite
dimensional weight modules over non-restricted quantum groups considered by C.
De Concini, V. Kac, C. Procesi, N. Reshetikhin and M. Rosso. These categories
are not semi-simple and have an infinite number of non-isomorphic irreducible
modules all having vanishing quantum dimensions. We also show that these
categories have associated ribbon categories which gives rise to re-normalized
link invariants. In the case of sl(2) these link invariants are the
Alexander-type multivariable invariants defined by Y. Akutsu, T. Deguchi, and
T. Ohtsuki.
