Killing Weights from the Perspective of $t$-Structures

This paper is devoted to morphisms killing weights in a range (as defined by the first author) and to objects without these weights (as essentially defined by J. Wildeshaus) in a triangulated category endowed with a weight structure $w$. We describe several new criteria for morphisms and objects to be of these types. In some of them we use virtual $t$-truncations and a $t$-structure adjacent to $w$. In the case where the latter exists, we prove that a morphism kills weights $m,\dots ,n$ if and only if it factors through an object without these weights; we also construct new families of torsion theories and projective and injective classes. As a consequence, we obtain some “weakly functorial decompositions” of spectra (in the stable homotopy category $\mathrm {SH}$) and a new description of those morphisms that act trivially on the singular cohomology $H_{\mathrm{sing}}^0(-,\Gamma )$ with coefficients in an arbitrary abelian group $\Gamma $.