Proposition 6.4 (05UH) · ScalingStacks

Proposition 6.4. Let $W$ be a Segal space. The following are equivalent.

(1)

$W$ is a complete Segal space.
(2)

The map $W_{0}\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(E,W)$ induced by $E\rightarrow F(0)$ is a weak equivalence.
(3)

Either of the maps $\Map_{s{\operatorname{\mathcal{S}}}}(E,W)\rightarrow W_{0}$ induced by a map $F(0)\rightarrow E$ is a weak equivalence.
(4)

For each pair $x,y\in{\operatorname{ob}}W$ , the space $\hoequiv(x,y)$ is naturally weakly equivalent to the space of paths in $W_{0}$ from $x$ to $y$ .

Proof.

$(1)\Rightarrow(2)$ :: This follows from (6.2).
$(2)\Leftrightarrow(3)$ :: Straightforward.
$(2)\Rightarrow(4)$ :: Part (2) and (6.2) imply that $W_{0}\rightarrow W_{\hoequiv}$ is a weak equivalence, whence the result follows from the fact that the space of paths in $W_{0}$ with endpoints $x$ and $y$ is equivalent to the homotopy fiber of the map $\Delta$ in (6.3).
$(4)\Rightarrow(1)$ :: Immediate from the diagram (6.3).

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