# PlanetPhysics/Projective Object

Let us consider the category of Abelian groups .
An object of an abelian category is called *projective* if
the functor **Failed to parse (syntax error): {\displaystyle Hom_A (P,âˆ’) : \mathcal{A} \to {\mathbf Ab}_G}**
is exact.

{\mathbf Remark.}

This is equivalent to the following statement:
An object is *projective* if given a short exact sequence
**Failed to parse (syntax error): {\displaystyle 0 \to Mâ€² \to M \to Mâ€²â€² \to 0}**
in an Abelian category ,
one has that:
**Failed to parse (syntax error): {\displaystyle 0 \to Hom_{\mathcal{A}}(Mâ€², P) \to Hom_{\mathcal{A}}(M, P) \to Hom_{\mathcal{A}}(Mâ€²â€², P) \to 0}**
is exact in .