Yet another way to say this is that
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.”
There is also a stronger notion of projective object, where preserves coequalizers. See more at tiny object, which can be defined as projective connected objects with “projective” used in this stronger sense.
A category has enough projectives if for every object there is an epimorphism where is projective.
Equivalently: if every object admits a projective presentation.
Projective objects and injective objects in abelian categories are of central interest in homological algebra. Here they appear as parts of cofibrant resolutions and fibrant resolutions, respectively, in the category of chain complexes , with respect to one of the two standard model structures on chain complexes.
The following are equivalent
For every object , the hom-functor is a left exact functor. So the second statement is equivalently that it is also right exact precisely if is projective.
of prop. 1
Let be an abelian category.
(with regarded as a complex concentrated in degree 0) such that
This means precisely that is an cofibrant resolution with respect to the standard model structure on chain complexes (see here) for which the fibrations are the positive-degreewise epimorphisms. Notice that in this model structure every object is fibrant, so that cofibrant resolutions are the only resolutions that need to be considered.
If has enough projectives in the sense of def. 3, then every object has a projective resolution.
For a commutative ring, an object in Mod, an -module, is projective (a projective module, see there for more details) precisely if it is a direct summand of a free module. See at projective module for more on this.
We list examples of classes of categories that have enough projective, according to def. 3.
See at projective module for more.
More explicitly: if and is the free module on , then a module homomorphism is specified equivalently by a function from to the underlying set of , which can be thought of as specifying the images of the unit elements in of the copies of .
By adjunction these are equivalently lifts of module homomorphisms
of prop. 3
For and its underlying set, consider the -linear map
out of the direct sum of copies of , which sends the unit element of the -labeled copy of to the corresponding element of (and is thus fixed on all other elements by -linearity).
A slightly subtle point is that there is no guarantee that the free module is actually projective, unless one assumes some form of the axiom of choice. Since the axiom of choice is not available in all toposes, one cannot use this procedure in general to construct, say, projective resolutions of abelian sheaves, hence in the abelian category of abelian group objects in a general Grothendieck topos (even though one can construct free resolutions), such as needed in general in abelian sheaf cohomology.
The idea of the proof is that under COSHEP, the underlying object of an abelian group in admits an epimorphism from a projective object in . Then the corresponding is an epimorphism out of a projective in , for this map is a composite of epimorphisms
(the first is epic because left adjoints preserve epis, whereas the second map, the component of the counit at , is epic because is faithful).
flat object, flat resolution
For instance section 4.3 of