PlanetPhysics/Projective Object

Let us consider the category of Abelian groups ${\mathbf {A} b}_{G}$. An object $P$ of an abelian category ${\mathcal {A}}$ is called projective if the functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle Hom_A (P,−) : \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 Failed to parse (syntax error): {\displaystyle 0 \to M′ \to M \to M′′ \to 0} in an Abelian category ${\mathcal {A}}$, one has that: Failed to parse (syntax error): {\displaystyle 0 \to Hom_{\mathcal{A}}(M′, P) \to Hom_{\mathcal{A}}(M, P) \to Hom_{\mathcal{A}}(M′′, P) \to 0} is exact in ${\mathbf {A} b}_{G}$.