Unit inclusion in a non-semisimple braided tensor category and non-compact relative TQFTs

The inclusion of the unit in a braided tensor category $\mathcal{V}$ induces a 1-morphism in the Morita 4-category of braided tensor categories $BrTens$. We give criteria for the dualizability of this morphism. When $\mathcal{V}$ is a semisimple (resp. non-semisimple) modular category, we show that the unit inclusion induces under the Cobordism Hypothesis a (resp. non-compact) relative 3-dimensional topological quantum field theory. Following Jordan-Safronov, we conjecture that these relative field theories together with their bulk theories recover Witten-Reshetikhin-Turaev (resp. De Renzi-Gainutdinov-Geer-Patureau-Mirand-Runkel) theories, in a fully extended setting. In particular, we argue that these theories can be obtained by the Cobordism Hypothesis.