Proposition 13.8 (05WC) · ScalingStacks

[05WC]

Proposition 13.8. If $g\colon U\rightarrow V$ is a categorical equivalence of Segal spaces, then it is a Dwyer-Kan equivalence.

[05WD]

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

and

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