Let A be an additive category.  The homotopy category K(A) is based on the following definition: if we have complexes A, B and maps f, g from A to B, a chain homotopy from f to g is a collection of maps hn:An→Bn−1{\displaystyle h^{n}\colon A^{n}\to B^{n-1}} (not a map of complexes) such that

fn−gn=dBn−1hn+hn+1dAn,{\displaystyle f^{n}-g^{n}=d_{B}^{n-1}h^{n}+h^{n+1}d_{A}^{n},} or simply f−g=dBh+hdA.{\displaystyle f-g=d_{B}h+hd_{A}.}
This can be depicted as shown in the diagram.
Image:Chain homotopy.jpg
User:Ryan Reich, User:Stannered