A short exact sequence of complexes is a chain of complex homomorphisms

such that $0\xrightarrow{} A_k\xrightarrow{f_k}B_k\xrightarrow{g_k}C_k\xrightarrow{} 0$ is a short exact sequence for each $k$

[Cohomological version]

A short exact sequence of complexes is a chain of complex homomorphisms

such that $0\xrightarrow{} A^k\xrightarrow{f^k}B^k\xrightarrow{g^k}C^k\xrightarrow{} 0$ is a short exact sequence for each $k$