On morphisms killing weights and Hurewicz-type theorems (1904.12853v4)

Abstract: We study "canonical weight decompositions" slightly generalizing that defined by J. Wildeshaus. For an triangulated category $C$, any integer $n$, and a weight structure $w$ on $C$ a triangle $LM\to M\to RM\to LM[1]$, where $LM$ is of weights at most $m-1$ and $RM$ is of weights at least $n+1$ for some $m\le n$, is determined by $M$ if exists. This happens if and only if the weight complex $t(M)\in Obj K(Hw)$ ($Hw$ is the heart of $w$) is homotopy equivalent to a complex with zero terms in degrees $-n,\dots, -m$; hence this condition can be "detected" via pure functors. One can also take $m=-\infty$ or $n=+\infty$ to obtain that the weight complex functor is "conservative and detects weights up to objects of infinitely small and infinitely large weights"; this is a significant improvement over previously known bounded conservativity results. Applying this statement we "calculate intersections of purely generated subcategories" and prove that certain weight-exact functors are conservative up to weight-degenerate objects. The main tool is the new interesting notion of morphisms killing weights $m,\dots, n$ that we study in detail as well. We apply general results to equivariant stable homotopy categories and spherical weight structures for them (as introduced in the previous paper) and obtain a certain converse to the (equivariant) stable Hurewicz theorem. In particular, the singular homology of a spectrum $E$ vanishes in negative degrees if and only if $E$ is an extension of a connective spectrum by an acyclic one.