Lemma 11.6 (05VU) · ScalingStacks

[05VU]

Lemma 11.6. Given a Segal space $W$ and $f\in\map(x,y)$ and $g\in\map(y,z)$ such that $f$ is a homotopy equivalence, the induced map

\map(x,y,z)_{f,g}\xrightarrow{(d_{1},d_{2})}\map(x,z)_{g\circ f}\times\map(x,y)_{f}

is a weak equivalence.

[05VV]

Proof. This follows from the diagram

Here the vertical column is a weak equivalence since $f$ is a homotopy equivalence (restricting to the fiber over $f\in\map(x,y)_{f}$ of the projections to $\map(x,y)_{f}$ gives exactly the zig-zag which defines $f^{*}\colon\map(y,z)_{g}\rightarrow\map(x,z)_{g\circ f}$ ). Since $(d_{0},d_{2})$ is a weak equivalence, the lemma follows. ∎