# PlanetPhysics/Projective Object

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

{\mathbf Remark.}

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