Graded spherical skein 2+1-G-HQFT and modified Turaev-Viro invariants

For G a group, we present a G-graded version of chromatic maps and skein modules and use them to define a 2+1-G-HQFT out of a G-chromatic category. The construction applies to the representations of unrestricted quantum groups at root of unity and recovers the modified Turaev-Viro 3-dimensional invariants.