Projective object

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

An object "P" in a category C is projective if the hom functor:$operatorname\{Hom\}(P,-)colonmathcal\{C\}\; omathbf\{Set\}$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 $Pinmathcal\{C\}$ is called a "projective object" if

:$operatorname\{Hom\}(P,-)colonmathcal\{C\}\; omathbf\{Ab\}$

is an exact functor, where $mathbf\{Ab\}$ is the category 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

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 $pcolon\; P\; o\; A$ 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 required surjection.

References

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