# Projective object

In category theory, the notion of a **projective object** generalizes the notion of a projective module.

An object in a category is **projective** if the hom functor

preserves epimorphisms. That is, every morphism factors through every epi .

Let be an abelian category. In this context, an object is called a *projective object* if

is an exact functor, where is the category of abelian groups.

The dual notion of a projective object is that of an **injective object**: An object in an abelian category is *injective* if the functor from to is exact.

## Enough projectives

Let be an abelian category. is said to have **enough projectives** if, for every object of , there is a projective object of and an exact sequence

In other words, the map is "epic", or an epimorphism.

## Examples

Let be a ring with 1. Consider the category of left -modules is an abelian category. The projective objects in are precisely the projective left R-modules. So is itself a projective object in Dually, the injective objects in are exactly the injective left R-modules.

The category of left (right) -modules also has enough projectives. This is true since, for every left (right) -module , we can take to be the free (and hence projective) -module generated by a generating set for (we can in fact take to be ). Then the canonical projection is the required surjection.

## References

- Mitchell, Barry (1965).
*Theory of categories*. Pure and applied mathematics.**17**. Academic Press. ISBN 978-0-124-99250-4. MR 0202787.

*This article incorporates material from Projective object on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*

*This article incorporates material from Enough projectives on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*