PlanetPhysics/Projective Object

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

{\mathbf Remark.}

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