Short exact sequence of chain complexes

Definition
A short exact sequence of chain complexes is a short exact sequence in the category of chain complexes with chain maps, viewed in an obvious way as an abelian category. More explicitly it is a collection of data of the form:


 * chain complexes $$A,B,C$$.
 * chain maps $$i:A \to B$$ and $$p:B \to C$$

satisfying the following: For every $$n$$, the induced sequence of maps:

$$0 \to A_n \stackrel{i_n}{\to} B_n \stackrel{p_n}{\to} C_n \to 0$$

is a short exact sequence. (If we are working over the category of abelian groups, then this must be a short exact sequence of abelian groups; if we are working over the category of modules over a commutative unital ring, then this must be a short exact sequence of modules. Note that exactness depends only on the underlying abelian group structure so we can view everything as abelian groups).

Facts

 * Short exact sequence of chain complexes gives long exact sequence of homology