Through the Reshetikhin-Turaev construction, a three-dimensional topological field theory can be built from a semisimple modular category. Thanks to a result of Bartlett, Douglas, Schommer-Pries and Vicary, a semisimple modular category is even equivalent to a once-extended (anomalous) three-dimensional topological field theory. Non-semisimple modular categories are much more difficult to handle, especially if one is interested in a construction that takes their very non-trivial homological algebra into account. In my talk, I will explain how to organize the homological algebra of a modular category through low-dimensional topology, namely by means of a differential graded modular functor. In particular, I will discuss the construction of differential graded conformal blocks, certain chain complexes carrying homotopy coherent projective mapping class group representations, but also the differential graded Verlinde formula and its connection to the Deligne Conjecture. This is joint work with Christoph Schweigert.