Remarks
Two chain homotopic maps f and g induce the same maps on homology because (f − g) sends cycles to boundaries, which are zero in homology. In particular a homotopy equivalence is a quasi-isomorphism. (The converse is false in general.) This shows that there is a canonical functor to the derived category (if A is abelian).
Read more about this topic: Homotopy Category Of Chain Complexes
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