ScalingStacks

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).

∎