ScalingStacks

Proof of (11.1). It is clear that for $k\geq 3$ every map

$$\Map_{s{\operatorname{\mathcal{S}}}}(E^{(k)},W)\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(F(1),W)\approx W_{1}$$

induced by an inclusion $F(1)\rightarrow E^{(k)}$ must factor through $W_{\hoequiv}\subseteq W_{1}$, since each point of the mapping space maps to a homotopy equivalence in the sense of (5.5). Let $r_{k}$ denote the map $\Map_{s{\operatorname{\mathcal{S}}}}(E^{(k)},W)\rightarrow W_{1}$ associated to the inclusion $F(1)\rightarrow E^{(k)}$ classifying the point $xy\in E^{(k)}_{1}$. We have that $\Map_{s{\operatorname{\mathcal{S}}}}(E^{(k)},W)=\Map_{s{\operatorname{\mathcal{S}}}}(E^{(k)},W)_{\hoequiv}$ for $k\geq 3$, and even when $k=2$ we have that

$$\Map_{s{\operatorname{\mathcal{S}}}}(E^{(2)},W)_{\hoequiv}\approx\Map_{s{\operatorname{\mathcal{S}}}}(E^{(2)},W)\times_{W_{1}}W_{\hoequiv}.$$

Then we must show that for each $k\geq 2$ the fiber of $r_{k}$ over any point in the subspace $W_{\hoequiv}\subset W_{1}$ is contractible. The result now follows from (11.10) applied to the pushout diagrams (11.8). ∎