ScalingStacks

Proof of Lemma 3.18. Given a collar-gluing $f:M\rightarrow[-1,1]$, there are canonical morphisms in $\mathcal{V}$

$$\int_{M^{\prime}}A\bigotimes_{\displaystyle\int_{M_{0}\times\mathbb{R}}A}\int_{M^{\prime\prime}}A\longrightarrow\int_{[-1,1]}f_{\ast}A\longrightarrow\int_{M}A~~.$$

The lefthand morphism is an equivalence by Lemma 3.12, which shows that factorization homology over the closed interval is equivalent to the bar construction, and inspection of the functor $f_{\ast}A$. The righthand morphism is an equivalence by Proposition 3.23, which grants that factorization homology pushes forward along $M\xrightarrow{f}[-1,1]$.

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