PlanetPhysics/Projective Object

Let us consider the [[../Cod/|category]] of [[../TrivialGroupoid/|Abelian groups]] ${𝐀b}_G$. An [[../TrivialGroupoid/|object]] $P$ of an [[../AbelianCategory2/|abelian category]] $𝒜$ is called projective if the [[../TrivialGroupoid/|functor]] Failed to parse (syntax error): {\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 $𝒜$, 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 ${𝐀b}_G$.