Proposition 13.4 (05W6) · ScalingStacks

[05W6]

Proposition 13.4. A map $g\colon U\rightarrow V$ between complete Segal spaces is a categorical equivalence if and only if it is a simplicial homotopy equivalence, if and only if it is a Reedy weak equivalence.

[05W7]

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. ∎