nLab
projective object

Definition

An object $P$ of a category $C$ is projective (with respect to epimorphisms) if for any morphism $f:P\to B$ and any epimorphism $q:A\to B$, $f$ factors through $q$ by some morphism $P\to A$.

$$\begin{array}{ccc}& & A\\ & {}^{\exists }\nearrow & {\downarrow }^q\\ P& \stackrel{f}{\to }& B\end{array}.$$\array{ && A \\ &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{q}} \\ P &\stackrel{f}{\to}& B } \,.

Another way to say this is that the hom-functor $\mathrm{Hom}(P,-)$ preserves epimorphisms.

A category $C$ has enough projectives if for every object $X$ there is an epimorphism $P\to X$ where $P$ is projective.

Remarks

• Projective object generalizes the notion of projective module over a ring.

• The dual notion is injective object.

• There are variations of the definition where “epimorphism” is replaced by some other type of morphism, such as a regular epimorphism or strong epimorphism or the left class in some orthogonal factorization system. In this case one may speak of regular projectives and so on. In a regular category “projective” almost always means “regular projective.”

Properties

• If $C$ has pullbacks and epimorphisms are preserved by pullback, as is the case in a pretopos, then $P$ is projective iff any epimorphism $Q\to P$ is split.

• The axiom of choice can be phrased as “all objects of Set are projective.” See also internally projective object and COSHEP.

Examples

• An object in Ab, an abelian group, is projective precisely if it is a free group.