  Let $k$ be a field of any characteristic. In this paper, we construct a
functorial cofibrant resolution $\mathfrak{R}(A)$ for the $\mathbb{Z}_{\le
0}$-graded dg algebras $A$ over $k$, such that the functor $A\rightsquigarrow
\mathfrak{R}(A)$ is colax-monoidal with quasi-isomorphisms as the colax maps.
More precisely, there are maps of bifunctors $\mathfrak{R}(A\otimes B)\to
\mathfrak{R}(A)\otimes \mathfrak{R}(B)$, compatible with the projections to
$A\otimes B$, and obeying the colax-monoidal axiom.
  The main application of such resolutions (which we consider in our next
paper) is the existence of a colax-monoidal dg localization of pre-triangulated
dg categories, such that the localization is a genuine dg category, whose image
in the homotopy category of dg categories is isomorphic to the To\"{e}n's dg
localization.
