AI 中文总结
该研究证明了$\boldsymbol{\textit{G}}$型与$\boldsymbol{\textit{M}}$型多点d-空间的q-模型范畴不满足左正则性，构造了反例并揭示了路径分支合并的失效机制。
AI 中文摘要
我们通过构造一个弱等价来证明，当$\boldsymbol{\textit{P}}$属于$\boldsymbol{\textit{G}}$、$\boldsymbol{\textit{M}}$时，$\boldsymbol{\textit{P}}$-多点d-空间的q-模型范畴不是左正则的；该弱等价沿通过附加单个1维球状胞腔得到的q-上纤维化的推出不是弱等价。反例在$\boldsymbol{\textit{\u0394}}$-Hausdorff、$\boldsymbol{\textit{\u0394}}$-生成空间范畴中构造。当$\boldsymbol{\textit{P}}=\boldsymbol{\textit{G}}$时，所有对象自动饱和；当$\boldsymbol{\textit{P}}=\boldsymbol{\textit{M}}$时，原弱等价在饱和对象之间。失效源于一族非常数执行路径，其在环境映射空间中收敛到一个非执行路径的常映射；附加球状胞腔后，这种退化导致两个先前不同的路径分支合并。
英文摘要
We prove that the q-model category of $\mathcal P$-multipointed d-spaces for $\mathcal P\in\{\mathcal G,\mathcal M\}$ is not left proper by constructing a weak equivalence whose pushout along a q-cofibration obtained by attaching a single 1-dimensional globular cell is not a weak equivalence. The counterexample is constructed in the category of $Δ$-Hausdorff $Δ$-generated spaces. For $\mathcal P=\mathcal G$, all objects are automatically saturated, and for $\mathcal P=\mathcal M$, the original weak equivalence is between saturated objects. The failure is caused by a family of nonconstant execution paths that converges in the ambient mapping space to a constant map which is not an execution path; after the globular cell is attached, this degeneration causes two previously distinct path components to merge.
Comments8 pages; v2: for completeness, the definition of the globe added
