---
tag: J2TX4TDPCX
kind: concept
name: Enough projectives
msc: [18G05, 18G10]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_object
---

## Summary

A category has enough projectives when every object is a quotient of a [[projective object|YXMXAGVC1Q]].
In an [[abelian category|Y9CZ3MTNF1]] with enough projectives, every object has a [[projective resolution|HK2AA1YSQ2]].
This is the condition that makes left [[derived functors|NPMC3K6J2F]] such as Tor well defined.

## Explanations

### L1 human

Projective objects are the easy building blocks of a category, like free modules among all modules.
Having enough projectives means that every object can be reached from such a block by a surjection.
Then one can describe any object by blocks, then the relations by blocks, and so on.
For modules over a ring $R$, every module $M$ is a quotient of the free module with one basis vector for each element of $M$.
So the category of $R$-modules has enough projectives.