\begin{definition}[Definition:Projective Object]
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''' {{iff}}:
: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 $e$'''.
\end{definition}