Covers and Envelopes in Grothendieck Categories: Flat Covers of Complexes with Applications

In the general setting of Grothendieck categories with enough projectives, we prove theorems that make possible to restrict the study of the problem of the existence of -covers and envelopes to the study of some properties of the class . We then prove the existence of flat covers and cotorsion envelopes of complexes, giving some examples. This generalizes the earlier work (J. Algebra201 (1998), 86–102) and finishes the problem of the existence of flat covers of complexes.