Proposition 13.2 (05W4) · ScalingStacks

[05W4]

Proposition 13.2. If $U$ is a Segal space and $W$ is a complete Segal space, then a pair of maps $f,g\colon U\rightrightarrows W$ are categorically homotopic if and only if they are homotopic in the usual sense; i.e., if there exists a map $K\colon U\times\Delta[1]\rightarrow W$ which restricts to $f$ and $g$ on the endpoints of $\Delta[1]$ .

[05W5]

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

(W^{E})_{n}\approx\Map_{s{\operatorname{\mathcal{S}}}}(E,W^{F(n)})\approx(W^{F(n)})_{0}

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. ∎