Let \(\mathcal {Q}\) be a class of objects in an abelian category \(\mathcal {A}\) . In this paper we investigate when homological and homotopical properties of \(\mathcal {Q}_0=\mathcal {Q}\) are inherited by \(\mathcal {Q}_n\) , the class of objects with \(\mathcal {Q}\) -(co)resolution dimension at most an integer \(n\ge 0\) . It is shown that if \(\mathcal {Q}\) is the first class of a Hovey triple \(\mathcal {M}=(\mathcal {Q},\mathcal {W},\mathcal {R})\) in \(\mathcal {A}\) satisfying some mild assumptions, then \(\mathcal {Q}_n\) is the first class of a hereditary Hovey triple \(\mathcal {M}_n=(\mathcal {Q}_n,\mathcal {W}_{\mathcal {Q},n},\mathcal {R}_{\mathcal {Q},n})\) with \(\mathcal {W}_{\mathcal {Q},n}\) and \(\mathcal {R}_{\mathcal {Q},n}\) explicitly described. Consequently, \(\mathcal {Q}_n\) is the left-hand side of a complete hereditary cotorsion pair, and hence a special precovering class. The dual problem is also addressed, and applications in the setting of Gorenstein homological algebra are given.
