Sign convention for shifts of chain complexes

Remark 6.1.11. The sign \((-1)^k\) in the differential of \(C[k]\) is a deliberate choice. It is slightly unusual: many references define the shift without any sign, setting \(d^{C[k]}_n := d^C_{n-k}\). Including the sign has the advantage that the shift agrees on the nose with the suspension of the derived \(\infty \)-category (see Theorem 6.1.30) rather than merely up to a natural isomorphism, and it is the convention that fits best with the shift functors of stable homotopy theory. As the two conventions produce isomorphic chain complexes, the choice affects none of the homological invariants.