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 for any epimorphism $e:E\twoheadrightarrow X$ and morphism $f:P\to X$, there is a morphism ${\overline {f}}:P\to E$ such that $e\circ {\overline {f}}=f$, i.e. the following diagram commutes:

That is, every morphism $P\to X$ factors through every epimorphism $E\twoheadrightarrow X$.

In a locally small category ${\mathcal {C}}$, the following statement is equivalent: $P$ is projective if the hom functor

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

preserves epimorphisms.

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 a locally small abelian category ${\mathcal {C}}$ is injective if the $\operatorname {Hom} (-,Q)$ functor from ${\mathcal {C}}$ to $\mathbf {Ab}$ is exact.

Properties

• The coproduct of two projective objects is projective.
• The retract of a projective object is projective.

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

The statement that all sets are projective is equivalent to the axiom of choice.

Let $R$ be a ring with 1. Consider the (abelian) category of left $R$-modules ${\mathcal {M}}_{R}$. The projective objects in ${\mathcal {M}}_{R}$ are precisely the projective left R-modules. Consequently, $R$ 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. Vol. 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.