ScalingStacks

Proof. The maps $W^{i_{0}},W^{i_{1}}\colon W^{E}\rightarrow W$ are Reedy trivial fibrations if $W$ is a complete Segal space. This is because of parts (2) and (3) of (6.4), together with the observation that

since $W^{F(n)}$ is a complete Segal space by (7.3). Thus, categorically homotopic maps coincide in the Reedy homotopy category, and hence are simplicially homotopic since $W$ is Reedy fibrant. ∎

Proof. The first “if and only if” is immediate from (13.2), while the second follows from the fact that complete Segal spaces are cofibrant and fibrant in the Reedy simplicial model category. ∎

Proof. If a categorical homotopy between $f$ and $g$ is given by $H\colon A\times E\rightarrow B$, then $W^{H}\colon W^{B}\rightarrow W^{A\times E}\approx(W^{A})^{E}$ is a categorical homotopy of $W^{f}$ and $W^{g}$. The statement about categorical equivalences follows. ∎

Proof. Recall from (7.2) that $f$ is a weak equivalence in the complete Segal space model category if and only if $\Map_{s{\operatorname{\mathcal{S}}}}(f,W)$ is a weak equivalence of spaces for each complete Segal space $W$. This is equivalent to supposing that $W^{f}\colon W^{V}\rightarrow W^{U}$ is a Reedy weak equivalence for each complete Segal space $W$, since $(W^{f})_{n}\approx\Map_{s{\operatorname{\mathcal{S}}}}(f,W^{F(n)})$ and since $W^{F(n)}$ is a complete Segal space by (7.3). The result now follows by noting that $W^{f}$ is a categorical equivalence between complete Segal spaces by (13.5) and (7.3), and thus is a Reedy weak equivalence by (13.4). ∎

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). ∎