and
nonabelian homological algebra
An object of a category is projective (with respect to epimorphisms) if for any morphism and any epimorphism , factors through by some morphism .
Another way to say this is that the hom-functor preserves epimorphisms.
A category has enough projectives if for every object there is an epimorphism where 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.”
If has pullbacks and epimorphisms are preserved by pullback, as is the case in a pretopos, then is projective iff any epimorphism is split.
The axiom of choice can be phrased as “all objects of Set are projective.” See also internally projective object and COSHEP.