Definition:Projective Object

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Let $\mathbf C$ be a metacategory.

Let $P \in \mathbf C_0$ be an object of $\mathbf C$.

Then $P$ is said to be projective if and only if:

For all epimorphisms $e: E \twoheadrightarrow X$ and morphisms $f: P \to X$, there exists $\bar f: P \to E$ such that:
$\begin{xy} <0em,0em>*+{P} = "P", <4em,0em>*+{X} = "X", <4em,4em>*+{E} = "E", "P";"E" **@{-} ?>*@{>} ?*!/_.6em/{\bar f}, "P";"X" **@{-} ?>*@{>} ?*!/^.6em/{f}, "E";"X" **@{-} ?>*@2{>} ?<>(.7)*{\vee} ?*!/_.6em/{e}, \end{xy}$
is a commutative diagram, i.e. such that $f = e \circ \bar f$.

In this situation, $f$ is said to lift across $e$.