# 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.