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Let $F:\mathcal{C}\rightarrow \mathcal{C}'$ be a functor between "nice" (e.g. abelian with enough injectives) categories. If F is not exact we can form the derived functors $F',F'',...$ Is it possible to reverse this process ("integrate" $F$) or stated reversely: Are there nice conditions on a functor $F$ to be a derived functor?

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  • $\begingroup$ Yoneda observed that $L_p(F)(A) = \text{Nat}(\text{Ext}^p(\text{hom}_A), F)$, this might be helpful. $\endgroup$ – Ivan Di Liberti Jul 16 '18 at 15:34
  • $\begingroup$ Could you explain what $L_p$ is? And do you mean $\text{Ext}^p(\text{hom}_A)$ to be $Ext^p(\underline{ },A)$? $\endgroup$ – Takirion Jul 17 '18 at 8:34
  • $\begingroup$ @Takirion, I guess in this context $L_p$ is the left derivation of functors (see Wikipedia). So $L_p(F)$ is the derived functor, which you can apply to an object A. $\endgroup$ – Babelfish Jul 23 '18 at 16:40
  • $\begingroup$ Ah, yes, this seems plausible. $\endgroup$ – Takirion Jul 23 '18 at 17:17

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