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  "body": "\n## Summary\n\nAn exact functor is an [[additive functor|KAN07YC9T2]] between [[abelian categories|Y9CZ3MTNF1]] that sends [[short exact sequences|KJKNJMGKZZ]] to short exact sequences.\nMany natural functors, such as $\\Hom(M, -)$ and $M \\otimes -$, preserve only part of the exactness; they are called left exact or right exact.\nMeasuring how far such a functor is from being exact is the purpose of derived functors such as [[Ext|4P7GBWJ4RR]] and [[Tor|DDQRERAQCV]].\n\n## Explanations\n\n### L1 human\n\nAn exact sequence records how an object is built from a subobject and a quotient.\nAn exact functor carries this information over without loss: the image of a subobject stays a subobject, and the image of the quotient stays the quotient.\nFor example, tensoring abelian groups with $\\Q$ is exact.\nTensoring with $\\Z/2\\Z$ is not: it turns the injective map \"multiply by $2$\" on $\\Z$ into the zero map on $\\Z/2\\Z$.\n",
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