  We show that the relative Auslander-Buchweitz context on a triangulated
category $\T$ coincides with the notion of co-$t$-structure on certain
triangulated subcategory of $\T$ (see Theorem \ref{M2}). In the Krull-Schmidt
case, we stablish a bijective correspondence between co-$t$-structures and
cosuspended, precovering subcategories (see Theorem \ref{correspond}). 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 $\omega$
induces a bounded non-degenerated co-$t$-structure on the smallest thick
triangulated subcategory of $\T$ containing $\omega.$ We also give a
description of the bounded co-$t$-structures on $\T$ (see Theorem \ref{Msc}).
Finally, as an application to the particular case of the bounded derived
category $\D(\HH),$ where $\HH$ is an abelian hereditary category which is
Hom-finite, Ext-finite and has a tilting object (see \cite{HR}), we give a
bijective correspondence between finite silting generator sets
$\omega=\add\,(\omega)$ and bounded co-$t$-structures (see Theorem \ref{teoH}).
