ScalingStacks

Proof. It is clear that a Reedy weak equivalence between any two Segal spaces is a Dwyer-Kan equivalence.

Conversely, suppose $f\colon U\rightarrow V$ is a Dwyer-Kan equivalence between complete Segal spaces. Then $\pi_{0}U_{0}\approx{\operatorname{ob}}U/{\sim}$ and $\pi_{0}V_{0}\approx{\operatorname{ob}}V/{\sim}$ by (6.5), so that $\pi_{0}U_{0}\rightarrow\pi_{0}V_{0}$ is a bijection. In the commutative diagram

the right-hand square is a homotopy pullback (since the induced maps of fibers are of the form $\map_{U}(x,y)\rightarrow\map_{V}(fx,fy)$, which is assumed to be a weak equivalence), and the large rectangle is a homotopy pullback (since by (6.4) the induced maps of fibers are of the form $\hoequiv_{U}(x,y)\rightarrow\hoequiv_{V}(fx,fy)$, which is also a weak equivalence). We conclude that $U_{0}\rightarrow V_{0}$ is a weak equivalence, and therefore that $U_{1}\rightarrow V_{1}$ is a weak equivalence. Since both $U$ and $V$ are Segal spaces, it follows that the map $f\colon U\rightarrow V$ is a Reedy weak equivalence as desired. ∎