Projective object

In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object.

If the category C is an abelian category such as, for example, the category of abelian groups, then P is projective if and only if

The purpose of this definition is to ensure that any object A admits a projective resolution, i.e., a (long) exact sequence

Semadeni (1963) discusses the notion of projective (and dually injective) objects relative to a so-called bicategory, which consists of a pair of subcategories of "injections" and "surjections" in the given category C. These subcategories are subject to certain formal properties including the requirement that any surjection is an epimorphism. A projective object (relative to the fixed class of surjections) is then an object P so that Hom(P, −) turns the fixed class of surjections (as opposed to all epimorphisms) into surjections of sets (in the usual sense).

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

The projective objects in the category of abelian groups are the free abelian groups.

The projective objects in the category of compact Hausdorff spaces are precisely the extremally disconnected spaces. This result is due to Gleason (1958), with a simplified proof given by Rainwater (1959).