Non-compact 3D TQFT and non-semisimplicity

We define a once extended non-compact 3-dimensional TQFT $\mathcal{Z}$ from the data of a (potentially) non-semisimple modular tensor category. This is in the framework of generators and relations of [Bartlett et al., arxiv:1509.06811 (2015)], having disallowed generating 2-morphisms whose source is the empty. Moreover, we show that the projective mapping class group representations this TQFT gives rise to, are dual to those of [Lyubashenko, arXiv:hep-th/9405167 (1994)] and [De Renzi et al., arXiv:2010.14852 (2020)]. We develop a method to decompose a closed 3-manifold in terms of 2-morphism generators. We use this to compute the value of $\mathcal{Z}$ on 3-manifolds, explaining why it should recover Lyubashenko's 3-manifold invariants [Lyubashenko, arXiv:hep-th/9405167 (1994)]. Finally, we explain that the value of the non-compact TQFT on the solid torus recovers the data of a modified trace [Geer et al., arXiv:0711.4229 (2007)].