Enough projectives

In category theory, the notion of a projective object generalizes the notion of a projective module.

An object ${\displaystyle P}$ in a category ${\displaystyle {\mathcal {C}}}$ is projective if the hom functor

preserves epimorphisms. That is, every morphism ${\displaystyle f:P\to X}$ factors through every epi ${\displaystyle Y\to X}$.

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