Projective object

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.

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.

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Projective object at PlanetMath.