Projective object

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

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.

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

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Projective object at PlanetMath. Enough projectives at PlanetMath.