The category of chain complexes:

We have associated to each topological space $X$ a sequence $\{S_n(X)\}$ of free abelian groups and group homomorphisms $\partial _n:S_n(X)\longrightarrow S_{n-1}(X)$ satisfying (29.7). It is useful to describe these in an abstract setting where the groups in question need not be free abelian and prove some general results about them. This paragraph serves as an algebraic prerequisite for the study homology theory.