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  "body": "\n## Summary\n\nA category has enough projectives when every object is a quotient of a [[projective object|YXMXAGVC1Q]].\nIn an [[abelian category|Y9CZ3MTNF1]] with enough projectives, every object has a [[projective resolution|HK2AA1YSQ2]].\nThis is the condition that makes left [[derived functors|NPMC3K6J2F]] such as Tor well defined.\n\n## Explanations\n\n### L1 human\n\nProjective objects are the easy building blocks of a category, like free modules among all modules.\nHaving enough projectives means that every object can be reached from such a block by a surjection.\nThen one can describe any object by blocks, then the relations by blocks, and so on.\nFor modules over a ring $R$, every module $M$ is a quotient of the free module with one basis vector for each element of $M$.\nSo the category of $R$-modules has enough projectives.\n",
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