Let and be chain complexes. A chain map is a sequence of homomorphisms such that for each . Diagramatically, this says that the following diagram commutes:
![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & A_{n} \ar[d]^{f_n} \ar[r]^{d_{... ... \ar[d]^{f_{n-1}} \ & A_{n}^{'} \ar[r]^{d_{n}^{'}} & A_{n-1}^{'} } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/1571/l2h/img7.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ & A_{n} \ar[d]^{f_n} \ar[r]^{d_{... ... \ar[d]^{f_{n-1}} \ & A_{n}^{'} \ar[r]^{d_{n}^{'}} & A_{n-1}^{'} } } \end{xy}$")