Concept
In , the notion of a projective object generalizes the notion of a projective module. Projective objects in are used in homological algebra. The dual notion of a projective object is that of an injective object. An in a category is projective if for any epimorphism and morphism , there is a morphism such that , i.e. the following diagram commutes: That is, every morphism factors through every epimorphism . If C is , i.e., in particular is a set for any object X in C, this definition is equivalent to the condition that the hom functor (also known as corepresentable functor) preserves epimorphisms. If the category C is an abelian category such as, for example, the , then P is projective if and only if is an exact functor, where Ab is the category of abelian groups. An abelian category is said to have enough projectives if, for every object of , there is a projective object of and an epimorphism from P to A or, equivalently, a short exact sequence The purpose of this definition is to ensure that any object A admits a projective resolution, i.e., a (long) exact sequence where the objects are projective. 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 coproduct of two projective objects is projective. The of a projective object is projective. 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. Let be a ring with identity. Consider the (abelian) category -Mod of left -modules.