On purely generated $\alpha$-smashing weight structures and weight-exact localizations

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.