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We show that the relative Auslander–Buchweitz context on a triangulated category coincides with the notion of co-t-structure on certain triangulated subcategory of (see Theorem 3.8). In the Krull–Schmidt case, we establish a bijective correspondence between co-t-structures and cosuspended, precovering subcategories (see Theorem 3.11). We also give a characterization of bounded co-t-structures in terms of relative homological algebra. The relationship between silting classes and co-t-structures is also studied. We prove that a silting class ω induces a bounded non-degenerated co-t-structure on the smallest thick triangulated subcategory of containing ω. We also give a description of the bounded co-t-structures on (see Theorem 5.10). Finally, as an application to the particular case of the bounded derived category where is an abelian hereditary category which is Hom-finite, Ext …