On purely generated $\alpha$smashing weight structures and weightexact localizations
Abstract
This paper is dedicated to new methods of constructing weight structures and weightexact 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 arXiv:
 arXiv:1712.00850
 Bibcode:
 2017arXiv171200850B
 Keywords:

 Mathematics  KTheory and Homology;
 Mathematics  Category Theory;
 Mathematics  Representation Theory;
 Primary: 18E30;
 18E35;
 18E40 Secondary: 16S85 18E05;
 18G35;
 16D90;
 18G25
 EPrint:
 Several minor corrections made