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.
Definition An object P {\displaystyle P} in a category C {\displaystyle {\mathcal {C}}} is projective if for any epimorphism e : E ↠ X {\displaystyle e:E\twoheadrightarrow X} and morphism f : P → X {\displaystyle f:P\to X} , there is a morphism f ¯ : P → E {\displaystyle {\overline {f}}:P\to E} such that e ∘ f ¯ = f {\displaystyle e\circ {\overline {f}}=f} , i.e. the following diagram commutes:
That is, every morphism P → X {\displaystyle P\to X} factors through every epimorphism E ↠ X {\displaystyle E\twoheadrightarrow X} . If C is locally small, i.e., in particular Hom C ( P , X ) {\displaystyle \operatorname {Hom} _{C}(P,X)} 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)
Hom ( P , − ) : C → S e t {\displaystyle \operatorname {Hom} (P,-)\colon {\mathcal {C}}\to \mathbf {Set} }
preserves epimorphisms.
Projective objects in abelian categories 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
Hom ( P , − ) : C → A b {\displaystyle \operatorname {Hom} (P,-)\colon {\mathcal {C}}\to \mathbf {Ab} }
is an exact functor, where Ab is the category of abelian groups. An abelian category A {\displaystyle {\mathcal {A}}} is said to have enough projectives if, for every object A {\displaystyle A} of A {\displaystyle {\mathcal {A}}} , there is a projective object P {\displaystyle P} of A {\displaystyle {\mathcal {A}}} and an epimorphism from P to A or, equivalently, a short exact sequence
0 → K → P → A → 0. {\displaystyle 0\to K\to P\to A\to 0.}
The purpose of this definition is to ensure that any object A admits a projective resolution, i.e., a (long) exact sequence
… P 2 → P 1 → P 0 → A → 0 {\displaystyle \dots P_{2}\to P_{1}\to P_{0}\to A\to 0}
where the objects P 0 , P 1 , … {\displaystyle P_{0},P_{1},\dots } are projective.
Projectivity with respect to restricted classes 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).
Properties The coproduct of two projective objects is projective. A retract of a projective object is projective.
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