Projective object

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

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

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

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

Let ${\displaystyle {\mathcal {C}}}$ be an abelian category. In this context, an object ${\displaystyle P\in {\mathcal {C}}}$ is called a projective object if

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

is an exact functor, where ${\displaystyle \mathbf {Ab} }$ is the category of abelian groups.

The dual notion of a projective object is that of an injective object: An object ${\displaystyle Q}$ in an abelian category ${\displaystyle {\mathcal {C}}}$ is injective if the ${\displaystyle \operatorname {Hom} (-,Q)}$ functor from ${\displaystyle {\mathcal {C}}}$ to ${\displaystyle \mathbf {Ab} }$ is exact.

Enough projectives

Let ${\displaystyle {\mathcal {A}}}$ be an abelian category. ${\displaystyle {\mathcal {A}}}$ is said to have enough projectives if, for every object ${\displaystyle A}$ of ${\displaystyle {\mathcal {A}}}$, there is a projective object ${\displaystyle P}$ of ${\displaystyle {\mathcal {A}}}$ and an exact sequence

${\displaystyle P\longrightarrow A\longrightarrow 0.}$

In other words, the map ${\displaystyle p\colon P\to A}$ is "epic", or an epimorphism.

Examples

Let ${\displaystyle R}$ be a ring with 1. Consider the category of left ${\displaystyle R}$-modules ${\displaystyle {\mathcal {M}}_{R}.}$ ${\displaystyle {\mathcal {M}}_{R}}$ is an abelian category. The projective objects in ${\displaystyle {\mathcal {M}}_{R}}$ are precisely the projective left R-modules. So ${\displaystyle R}$ is itself a projective object in ${\displaystyle {\mathcal {M}}_{R}.}$ Dually, the injective objects in ${\displaystyle {\mathcal {M}}_{R}}$ are exactly the injective left R-modules.

The category of left (right) ${\displaystyle R}$-modules also has enough projectives. This is true since, for every left (right) ${\displaystyle R}$-module ${\displaystyle M}$, we can take ${\displaystyle F}$ to be the free (and hence projective) ${\displaystyle R}$-module generated by a generating set ${\displaystyle X}$ for ${\displaystyle M}$ (we can in fact take ${\displaystyle X}$ to be ${\displaystyle M}$). Then the canonical projection ${\displaystyle \pi \colon F\to M}$ is the required surjection.