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  "body": "\n## Summary\n\nThe derived functors of a left exact functor $F$ are functors $R^n F$, for $n \\ge 0$, with $R^0 F = F$.\nThey are computed by applying $F$ to an [[injective resolution|ZW3W6FX4PW]] and taking cohomology.\nThey turn a short exact sequence into a long exact sequence, and so measure how far $F$ is from being exact.\n[[Ext|4P7GBWJ4RR]], [[Tor|DDQRERAQCV]], [[group cohomology|YXJXA0Q52R]] and sheaf cohomology are all derived functors.\n\n## Explanations\n\n### L1 human\n\nMany natural functors keep injective maps injective but can destroy surjectivity, or the other way round.\nThe derived functors are correction terms that record exactly what is lost.\nOne computes them by replacing an object with a resolution by simple objects, applying the functor, and taking cohomology.\nFor abelian groups, the functor \"elements killed by $2$\" sends the surjection $\\Z \\to \\Z/2\\Z$ to the non-surjective map $0 \\to \\Z/2\\Z$.\nIts first derived functor is $\\operatorname{Ext}^1(\\Z/2\\Z, -)$, which repairs this failure.\n",
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