Let $A$ be an abelian category and $indA$ be its ind category. I want to know the relations between $D^b(A)$ and $D^b(indA)$. For example, I find in another question that if $A$ is thick in $indA$, then $D^b(A)$ is fully embedded in $D^b(indA)$. In particular, I wonder if under certain conditions about $A$, we can construct the triangulated category $D^b(indA)$ directly from $D^b(A)$ by just using its triangulated structure (and not using t-structure)?
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1$\begingroup$ I'm affraid that only the semi-simplicity of $A$ can help here. Otherwise it's quite difficult to compute the $D(indA)$-morphisms FROM an object of $A$ to a shift of an ind-object of $A$; this involves the higher projective limit functors. $\endgroup$– Mikhail BondarkoCommented Oct 13, 2014 at 15:56
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