Proposition 13.5 (05W8) · ScalingStacks

[05W8]

Proposition 13.5. Let $A$ , $B$ , and $W$ be Segal spaces. If $f,g\colon A\rightrightarrows B$ are categorically homotopic maps, then the induced maps $W^{B}\rightrightarrows W^{A}$ are categorically homotopic. If $f\colon A\rightarrow B$ is a categorical equivalence, then the induced map $W^{B}\rightarrow W^{A}$ is a categorical equivalence.

[05W9]

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. ∎