The q-model category of multipointed d-spaces is not left proper

Abstract

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 $\Delta$-Hausdorff $\Delta$-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.