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.