ScalingStacks

Proof. The proof is by induction on $k$. The case $k=2$ is immediate from (11.6).

Now suppose the lemma is proved for the map $\Map_{s{\operatorname{\mathcal{S}}}}(F(k-1),W)_{\hoequiv}\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(H(k-1),W)_{\hoequiv}$. From (11.9) we get a commutative square

This square would be a pullback square if we left off the “$\hoequiv$” decorations. Even with these decorations the square is a pullback (and hence a homotopy pullback), as can be seen by recalling that $H(k)_{1}=C(k)_{1}\cup d^{1}F(k-1)_{1}$.

Thus by induction we see that the map

$$a\colon\Map_{s{\operatorname{\mathcal{S}}}}(H(k),W)_{\hoequiv}\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(C(k),W)_{\hoequiv}$$

is a weak equivalence. The proof now follows from (11.11) and the fact that the map

$$W_{k}\approx W_{k-1}\times_{W_{0}}W_{1}\xrightarrow{a\times_{W_{0}}1}\Map_{s{\operatorname{\mathcal{S}}}}(C(k),W)\approx\Map_{s{\operatorname{\mathcal{S}}}}(d^{1}H(k-1),W)\times_{W_{0}}W_{1}$$

is a weak equivalence after restricting to the “$\hoequiv$” components. ∎