Lemma 5.8 (05UC) · ScalingStacks

[05UC]

Lemma 5.8. If $g\in W_{1}$ is a vertex which can be connected by a path in $W_{1}$ to a homotopy equivalence $g^{\prime}\in W_{1}$ , then $g$ is itself a homotopy equivalence.

[05UD]

Proof. Let $G\colon\Delta[1]\rightarrow W_{1}$ denote the path connecting $g$ and $g^{\prime}$ . Then it suffices to note that a dotted arrow exists in

where $H$ is a lift of $(s_{0}d_{1}g^{\prime},g^{\prime},s_{0}d_{0}g^{\prime})=(id_{x^{\prime}},g^{\prime},id_{y^{\prime}})$ to $W_{3}$ , since the right-hand vertical map is a fibration. ∎