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Jul 20, 2016 at 6:15 comment added mabli In fact, this has annoyed me for quite a bit longer than a few weeks already. Seeing the answer of @Jacob Lurie it seems you are right in your assessment that this is an artefact of the shortcomings of triangulated categories which are resolved in a better context...
Jul 19, 2016 at 19:13 history edited Jeremy Rickard CC BY-SA 3.0
added comments on revised question
Jul 19, 2016 at 10:01 comment added mabli Yes indeed, very good point. It reminds me that I forgot to add an assumption on $F$, namely that $F$ is left exact in the sense that $F(D^{\geq 0}(\mathcal{A})) \subseteq D^{\geq 0}(\mathcal{B}). I edited the question accordingly...
Jul 18, 2016 at 15:33 history answered Jeremy Rickard CC BY-SA 3.0