  In the last few years, Lopez-Permouth and several collaborators have
introduced a new approach in the study of the classical projectivity,
injectivity and flatness of modules. This way, they introduced subprojectivity
domains of modules as a tool to measure, somehow, the projectivity level of
such a module (so not just to determine whether or not the module is
projective). In this paper we develop a new treatment of the subprojectivity in
any abelian category which shed more light on some of its various important
aspects. Namely, in terms of subprojectivity, some classical results are
unified and some classical rings are characterized. It is also shown that, in
some categories, the subprojectivity measures notions other than the
projectivity. Furthermore, this new approach allows, in addition to
establishing nice generalizations of known results, to construct various new
examples such as the subprojectivity domain of the class of Gorenstein
projective objects, the class of semi-projective complexes and particular types
of representations of a finite linear quiver. The paper ends with a study
showing that the fact that a subprojectivity domain of a class coincides with
its first right Ext-orthogonal class can be characterized in terms of the
existence of preenvelopes and precovers.
