ScalingStacks

Proof. We first note that since $\ho(U\times E)=\ho U\times\ho E=\ho U\times I[1]$, we see that categorically homotopic maps of Segal spaces induce naturally isomorphic functors between their homotopy categories, and thus a categorical equivalence induces an equivalence between homotopy categories.

If $f,h\colon V\rightarrow U$ are maps together with categorical homotopies $H\colon gf\sim 1_{V}$ and $K\colon hg\sim 1_{U}$, then (13.9) applied to the diagrams

will show that $gf$ and $hg$ are Dwyer-Kan equivalences, and hence $g$ is a Dwyer-Kan equivalence using (7.5). ∎

Proof. By (7.5) it suffices to show that the map $j\colon W\rightarrow W^{E}$ induced by $E\rightarrow F(0)$ is a Dwyer-Kan equivalence. We have already noted in the first part of the proof of (13.8) that categorically equivalent Segal spaces have equivalent homotopy categories, whence $\ho(W^{E})\rightarrow\ho W$ is an equivalence of categories. Thus it suffices to show that the induced map $\map_{W}(x,y)\rightarrow\map_{W^{E}}(j(x),j(y))$ is a weak equivalence for each $x,y\in{\operatorname{ob}}W$.

We consider the diagram

where the horizontal arrows are induced by maps $F(1)\xrightarrow{i}E\xrightarrow{j}F(0)$; note that $ji=s_{0}$. We will show that the two horizontal maps marked $i^{*}$ are homotopy monomorphisms (12.2), and that the large rectangle is a homotopy pullback; this will imply that for each pair $x,y\in{\operatorname{ob}}W$ the maps of fibers

are such that $i^{*}j^{*}$ is a weak equivalence and $i^{*}$ is a homotopy monomorphism, whence $j^{*}$ is a weak equivalence, as desired.

That the map $W^{i}\colon W^{E}\rightarrow W^{F(1)}$ is a homotopy monomorphism of spaces in each simplicial degree (and thus also the $i^{*}$’s) follows since $(W^{E})_{n}=\Map_{s{\operatorname{\mathcal{S}}}}(E,W^{F(n)})$, $(W^{F(1)})_{n}=\Map_{s{\operatorname{\mathcal{S}}}}(F(1),W^{F(n)})=(W^{F(n)})_{1}$, and since $W^{F(n)}$ is a Segal space, using (6.2).

The large rectangle is isomorphic to the square

(where $W_{2}\times_{W_{1}}W_{2}$ denotes the limit of the diagram $W_{2}\xrightarrow{d_{1}}W_{1}\xleftarrow{d_{1}}W_{2}$) which is shown to be a homotopy pull-back by a straightforward computation using (12.4). ∎