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For questions about the derived categories of various abelian categories and questions regarding the derived category construction itself.
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Cone of a morphism in an abelian category when considered as a morphism in derived category....
Not a real answer, but hopefully a relevant comment:
There is s very close analogy between the homotopy category of spaces and chain complexes.
You can find such presentation in Peter May's notes on …
3
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Closed Model Category Structure on Chain Complexes Related to A Left-exact Functor
Let $F:A \to B$ be an additive left-exact functor of abelian categories (Do not assume that they have enough injectives / projectives.) Suppose we are given a class of objects $R$ adapted to $F$ (see …