Quantum field theory is outside the real realm of pure mathematics, makes contact with the real world and features chain complexes and cohomology.
The current paradigm for gauge theories such as the standard model is based on Yang-Mills theories coupled to matter. The quantisation of nonabelian (and and, depending on your choice of gauge fixing function, also abelian) Yang-Mills theories features a cohomology theory known by the moniker of BRST, after the inventors: Becchi, Rouet, Stora and, independently, Tyutin. The cleanest proofs of the renormalizability of Yang-Mills theories are cohomological in nature.
