Category

Projective object

Let 𝖢 be a category. An object 𝑃 ∈𝖢 is said to be projective iff it has the following (left) lifting property against epimorphisms: For any morphism 𝑓 :𝑃 →𝐵 and epimorphism 𝑞 :𝐴 ↠𝐵, there exists a factorization ¯𝑓 :𝑃 →𝐴 so that 𝑞¯𝑓 =𝑓. #m/def/cat

Equivalently, the covariant hom-functor 𝖢(𝑃, −) preserves epimorphisms.

See also

• Projective module, the original case.
• Injective object, the dual notion.

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