Projective object

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

Let \mathcal{C} be an abelian category. 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} if the \operatorname{Hom}(-,Q) functor from \mathcal{C} to \mathbf{Ab} is exact.

[edit] Enough projectives

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.

[edit] Examples.

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.

This article incorporates material from Projective object on PlanetMath, which is licensed under the GFDL. This article incorporates material from Enough projectives on PlanetMath, which is licensed under the GFDL.

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