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| In [[category theory]], the notion of a '''projective object''' generalizes the notion of a [[projective module]].
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| An object ''P'' in a category '''C''' is '''projective''' if the [[hom functor]]
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| :<math> \operatorname{Hom}(P,-)\colon\mathcal{C}\to\mathbf{Set}</math>
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| preserves [[epimorphism]]s. That is, every morphism ''f:P→X'' factors through every epi ''Y→X''.
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| Let <math>\mathcal{C}</math> be an [[abelian category]]. In this context, an object <math>P\in\mathcal{C}</math> is called a ''projective object'' if
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| :<math> \operatorname{Hom}(P,-)\colon\mathcal{C}\to\mathbf{Ab}</math>
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| is an [[exact functor]], where <math>\mathbf{Ab}</math> is the [[Category (mathematics)|category]] of [[abelian group]]s.
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| The dual notion of a projective object is that of an '''[[injective object]]''': An object <math>Q</math> in an abelian category <math>\mathcal{C}</math> is ''injective'' if the <math>\operatorname{Hom}(-,Q)</math> [[Hom functor|functor]] from <math>\mathcal{C}</math> to <math>\mathbf{Ab}</math> is exact.
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| ==Enough projectives==
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| Let <math>\mathcal{A}</math> be an [[abelian category]]. <math>\mathcal{A}</math> is said to have '''enough projectives''' if, for every object <math>A</math> of <math>\mathcal{A}</math>, there is a projective object <math>P</math> of <math>\mathcal{A}</math> and an [[exact sequence]]
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| :<math>P \longrightarrow A \longrightarrow 0.</math> | |
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| In other words, the [[function (mathematics)|map]] <math>p\colon P \to A</math> is "epi", or an [[epimorphism]].
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| ==Examples.==
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| Let <math>R</math> be a [[ring (mathematics)|ring]] with 1. Consider the category of left <math>R</math>-modules <math>\mathcal{M}_R.</math> <math>\mathcal{M}_R</math> is an abelian category. The projective objects in <math>\mathcal{M}_R</math> are precisely the [[projective module|projective left R-modules]]. So <math>R</math> is itself a projective object in <math>\mathcal{M}_R.</math> Dually, the injective objects in <math>\mathcal{M}_R</math> are exactly the [[injective module|injective left R-modules]].
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| The [[Category (mathematics)|category]] of left (right) <math>R</math>-modules also has enough projectives. This is true since, for every left (right) <math>R</math>-module <math>M</math>, we can take <math>F</math> to be the free (and hence projective) <math>R</math>-module generated by a generating set <math>X</math> for <math>M</math> (we can in fact take <math>X</math> to be <math>M</math>). Then the [[canonical projection]] <math>\pi\colon F\to M</math> is the required [[surjection]].
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| ==References==
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| *{{Mitchell TOC}}
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| {{PlanetMath attribution|id=6437|title=Projective object}}
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| {{PlanetMath attribution|id=6506|title=Enough projectives}}
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| [[Category:Homological algebra]]
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| [[Category:Objects (category theory)]]
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