Chain homotopy

Summary

A chain homotopy between two chain maps $f, g \colon C_\bullet \to D_\bullet$ is a family of maps $h_n \colon C_n \to D_{n+1}$ with $f - g = dh + hd$. Chain homotopic maps induce the same maps on homology. The notion is the algebraic shadow of a homotopy between continuous maps, and it is used to define the homotopy category of chain complexes.