# Can we “integrate” functors?

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?

• Yoneda observed that $L_p(F)(A) = \text{Nat}(\text{Ext}^p(\text{hom}_A), F)$, this might be helpful. – Ivan Di Liberti Jul 16 '18 at 15:34
• Could you explain what $L_p$ is? And do you mean $\text{Ext}^p(\text{hom}_A)$ to be $Ext^p(\underline{ },A)$? – Takirion Jul 17 '18 at 8:34
• @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. – Babelfish Jul 23 '18 at 16:40
• Ah, yes, this seems plausible. – Takirion Jul 23 '18 at 17:17