For a weight structure w on a triangulated category $\underline {C}$ 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 $M\in SH$ 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 $m,\dots ,n$ (‘in the middle’). For $w=w^{sph}$, this is the case if there exists a distinguished triangle $LM\to M\to RM$, 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 $m\le i\le n$ and $H^{sing}_{m-1}(M)$ is a free abelian group. We also consider morphisms that kill weights $m,\dots ,n$; those ‘send n-w-skeleta into $m-1$-w-skeleta’.

对于三角类别$\underline {C}$上的权重结构w，我们证明相应的权重复函子和其他一些（权重精确）函子是“保守到权重退化对象的”；这改进了早期的保守性公式。在$w=w^{sph}$（$SH$上的球形权重结构）的情况下，我们推导出以下与稳定Hurewicz定理相反的内容：$H^{sing}_{i}(M)=\{ 0\}$对于所有$i<0$当且仅当$M\in SH$是非循环连接谱的扩展。我们还证明了该陈述的等变版本。

主要思想是研究没有权重$m,\dots ,n$（“在中间”）的M。对于$w=w^{sph}$ ，如果存在一个区分三角形$LM\to M\to RM$，则属于这种情况，其中$RM$是n连通谱，$LM$是$m- 1$ -Margolis 定义意义上的（M的）骨架；每当$H^{sing}_i(M)=\{0\}$对于$m\le i\le n$并且$H^{sing}_{m-1}(M)$是免费的时，就会发生这种情况阿贝尔群。我们还考虑了杀死权重$m,\dots ,n$的态射；那些“将n - w -骨架发送到$m-1$ - w -骨架”。