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  "body": "\n## Summary\n\nA Grothendieck category is an [[abelian category|Y9CZ3MTNF1]] with all small colimits, in which filtered colimits are exact, and which has a generator.\nGrothendieck introduced these conditions in his 1957 Tôhoku paper and proved that such categories have enough injective objects.\nThis made it possible to define sheaf cohomology as a derived functor.\nModule categories and categories of sheaves of abelian groups are Grothendieck categories.\nBy the [[Gabriel–Popescu theorem|3MP41FQHZ5]], every Grothendieck category is a localization of a module category.\n\n## Explanations\n\n### L1 human\n\nAbelian categories are a good setting for homological algebra, but some need extra properties to construct resolutions.\nA Grothendieck category asks for three things that modules have.\nOne can glue infinitely many objects together, and gluing along a directed system does not destroy exactness.\nThere is also one object, like the ring itself, from which all others can be probed.\nWith these, every object embeds in an injective object, so derived functors such as sheaf cohomology can be defined.\n",
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