Consider a finite modular tensor category $\mathcal A$. In [DGGPR] the authors exhibit a 3-dimensional topological field theory  $Z_{\mathcal A}: \operatorname{Bord}_{\mathcal A} \to \operatorname{Vect}$, which, in the case where $\mathcal A$ is semisimple, recovers the usual Reshetikhin-Turaev TQFT. In the present work we show that this extends naturally to a TQFT $Z_{\operatorname{Ch}(\mathcal A)}$, which takes values in the symmetric tensor category $\operatorname{Ch(Vect)}$ of linear cochains. This cochain valued theory furthermore respects (certain classes of) homotopies.

[DGGPR] M. De Renzi, A. M. Gainutdinov, N. Geer, B. Patureau-Mirand, and I. Runkel. 3-dimensional TQFTs from non-semisimple modular categories. Sel. Math. New Ser., 28(2):42, 2022.