  In this work we introduce notions in Auslander-Buchweitz theory and cotorsion
theory in extriangulated categories which extend the given ones for abelian
categories. Although these notions have been already developed for
extriangulated categories in remarkable works, in this paper, we tackle them in
a relative sense by considering subcategories of objects. This approach not
only covers the existing theory given on abelian, exact and triangulated
categories, but it also shows how to get similar results with an appropriate
treatment of local properties.
