  This paper is dedicated to new methods of constructing weight structures and
weight-exact localizations; our arguments generalize their bounded versions
considered in previous papers of the authors. We start from a class of objects
$P$ of triangulated category $C$ that satisfies a certain negativity condition
(there are no $C$-extensions of positive degrees between elements of $P$; we
actually need a somewhat stronger condition of this sort) to obtain a weight
structure both "halves" of which are closed either with respect to
$C$-coproducts of less than $\alpha$ objects (for $\alpha$ being a fixed
regular cardinal) or with respect to all coproducts (provided that $C$ is
closed with respect to coproducts of this sort). This construction gives all
"reasonable" weight structures satisfying the latter condition. In particular,
we obtain certain weight structures on spectra (in $SH$) consisting of less
than $\alpha$ cells and on certain localizations of $SH$; these results are
new.
