Projective object

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

An object $P$ in a category ${\mathcal {C}}$ is projective if the hom functor

\operatorname {Hom}(P,-)\colon {\mathcal {C}}\to {\mathbf {Set}}

preserves epimorphisms. That is, every morphism $f:P\to X$ factors through every epi $Y\to X$ .

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 "epic", 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

Mitchell, Barry (1965). Theory of categories. Pure and applied mathematics. 17. Academic Press. ISBN 978-0-124-99250-4. MR 0202787.

This article incorporates material from Projective object on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

This article incorporates material from Enough projectives on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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