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Let $E^\bullet$ and $F^\bullet$ be complexes on an abelian category; what does it mean to say that $E^\bullet$ and $F^\bullet$ are quasi-isomorphic?

Does it only mean that there is a map of complexes $f:E^\bullet \to F^\bullet$ that induces isomoprhisms between the cohomology objects?

Or does it also guarantee the existence of a map of complexes $g:F^\bullet \to E^\bullet$ inducing the inverses of $H^pf:H^p(E^\bullet)\to H^p(F^\bullet)$?

Put in another way: is quasi-isomorphism an equivalence relation?

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    I tried to edit, but apparently I have to change 6 characters. The map should say $g:F^\bullet \to E^\bullet$.2011-12-21
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    You could be interested in http://en.wikipedia.org/wiki/Derived_category2011-12-21
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    Does the equivalence relation "two complexes E and F are equivalent if there are quasi-isomorphisms f:E-->F and g:F-->E" have a particular name in the literature?2011-12-22

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