- Projective object
In
category theory , the notion of a projective object generalizes the notion offree module .An object "P" in a category C is projective if the
hom functor :operatorname{Hom}(P,-)colonmathcal{C} omathbf{Set}preservesepimorphism s. 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 Pinmathcal{C} is called a "projective object" if:operatorname{Hom}(P,-)colonmathcal{C} omathbf{Ab}
is an
exact functor , where mathbf{Ab} is the category ofabelian group s.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
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 anexact sequence :P longrightarrow A longrightarrow 0.
In other words, the map pcolon P o A is "epi", or an epimorphism.
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 picolon F o M is the requiredsurjection .References
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