A projective presentation of an object is a realization of that object as a suitable quotient of a projective object.
In homological algebra projective presentations can sometimes be used in place of genuine projective resolutions in the computation of derived functors. See for instance at Ext-functor for examples.
The dual notion is that of injective presentation.
Let $\mathcal{A}$ be an abelian category. For $X \in \mathcal{A}$ any object, a projective presentation of $X$ is a short exact sequence of the form
hence exhibiting $X$ as the cokernel
such that $P$ is a projective object.
projective object, projective presentation, projective cover, projective resolution
injective object, injective presentation, injective envelope, injective resolution
flat object, flat resolution