Homotopy equivalence of chain complexes

Definition

A chain map $f$ between chain complexes $A$ and $B$is termed a homotopy equivalence if there exists a chain map in the opposite direction, say $g:B\to A$ such that $f\circ g$ is chain-homotopic to the identity map on $B$ and $g\circ f$ is chain-homotopic to the identity map on $A$.

If a homotopy equivalence exists between two chain complexes, they are termed homotopy-equivalent chain complexes.