Non-semisimple extended topological quantum field theories

We develop the general theory for construction of Extended Topological Quantum Field Theories (ETQFTs) associated with Costantino-Geer-Patureau quantum invariants closed 3-manifolds. In order to do so, we introduce relative modular categories, a class ribbon categories which are modeled on representations unrolled groups, and can be thought as non-semisimple analogue categories. Our approach exploits 2-categorical version universal introduced by Blanchet, Habegger, Masbaum, Vogel. The 1+1+1-EQFTs thus obtained realized symmetric monoidal 2-functors defined over non-rigid 2-categories admissible cobordisms decorated colored graphs cohomology classes, take values in complete graded linear particular, our extends family 2+1-TQFTs $\mathfrak{sl}_2$ Costantino, Geer, Patureau new ETQFTs. non-semisimplicity is witnessed presence critical 1-manifolds.