For a weight structure w on a triangulated category C we prove that the corresponding
weight complex functor and some other (weight-exact) functors are ‘conservative up to weight-degenerate
objects’; this improves earlier conservativity formulations. In the case w = w sph (the spherical weight
structure on SH), we deduce the following converse to the stable Hurewicz theorem: H sing
i
(M) = {0}
for all i < 0 if and only if M ∈ SH is an extension of a connective spectrum by an acyclic one. We also
prove an equivariant version of this statement.
The main idea is to study M that has no weights m,...,n (‘in the middle’). For w =w sph , this is the case
if there exists a distinguished triangle LM → M → RM, where RM is an n-connected spectrum and LM
is an m−1-skeleton (of M) in the sense of Margolis’s definition; this happens whenever H sing
i
(M) = {0}
for m ≤ i ≤ n and H sing
m−1 (M) is a free abelian group. We also consider morphisms that kill weights
m,...,n; those ‘send n-w-skeleta into m−1-w-skeleta’.