  We extend to the context of Abelian categories the bijection, classically
given in categories of modules, between equivalence classes of tilting (resp.,
finendo quasi-tilting) modules and special (resp., semi-special) preenveloping
torsion classes. We show that, when $\mathcal{A}$ has an epi-generator, the
(semi-)special preenveloping torsion classes $\mathcal{T}$ given by
(quasi-)tilting objects are exactly those for which any object
$T\in\mathcal{T}$ is the epimorphic image of some object in
${}^{\perp_1}\mathcal{T}:=\text{Ker}
(\text{Ext}_{\mathcal{A}}^1(-,\mathcal{T}))$. The results are immediately
dualizable, thus yielding corresponding bijections involving (quasi-)cotilting
objects. Restricting to the particular situation when
$\mathcal{A}=\text{mod-}R$ is the category of finitely presented modules over a
right coherent ring $R$, gives new results and raises new questions even at the
level of classical tilting theory in categories of modules.
