For $N \in \mathcal{A}$ an object, a projective cover is a morphism $\tilde N \to N$ out of a projective object which is a superfluous epimorphism.
projective object, projective presentation, projective cover, projective resolution
injective object, injective presentation, injective envelope, injective resolution
flat object, flat resolution
U. Shukla, On the projective cover of a module and related results, Pacific J. Math. Volume 12, Number 2 (1962), 709-717 (Euclid)
Wikipedia, Projective cover