Let and be chain complexes and , be chain maps. A chain homotopy between and is a sequence of homomorphisms so that for each . Thus, we have the following diagram:
If there exists a chain homotopy between and , then and are said to be chain homotopic.
![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & A_{n+1} \ar[r]^{d_{n+1}} \ar[d... ... \ar[r]_{d_{n+1}^{'}} & A_{n}^{'} \ar[r]_{d_{n}^{'}} & A_{n-1}^{'} } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/1572/l2h/img11.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & A_{n+1} \ar[r]^{d_{n+1}} \ar[d... ... \ar[r]_{d_{n+1}^{'}} & A_{n}^{'} \ar[r]_{d_{n}^{'}} & A_{n-1}^{'} } } \end{xy}$")
