Projective object

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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

An object P in a category C is projective if the hom functor

 \operatorname{Hom}(P,-)\colon\mathcal{C}\to\mathbf{Set}

preserves epimorphisms. That is, every morphism f:P→X factors through every epi Y→X.

Let \mathcal{C} be an abelian category. In this context, an object P\in\mathcal{C} is called a projective object if

 \operatorname{Hom}(P,-)\colon\mathcal{C}\to\mathbf{Ab}

is an exact functor, where \mathbf{Ab} is the category of abelian groups.

The dual notion of a projective object is that of an injective object: An object Q in an abelian category \mathcal{C} is injective if the \operatorname{Hom}(-,Q) functor from \mathcal{C} to \mathbf{Ab} is exact.

Enough projectives[edit]

Let \mathcal{A} be an abelian category. \mathcal{A} is said to have enough projectives if, for every object A of \mathcal{A}, there is a projective object P of \mathcal{A} and an exact sequence

P \longrightarrow A \longrightarrow 0.

In other words, the map p\colon P \to A is "epi", or an epimorphism.

Examples.[edit]

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

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

References[edit]

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.