ScalingStacks

Proof. After Proposition 3.9 it is enough to consider the case where $B$ is equivalent to $\BTop(n)$, and so we omit it from the notation and discussion. Let $\mathcal{F}:\mfld_{n}^{\partial}\rightarrow\mathcal{V}$ be a symmetric monoidal functor satisfying the $\otimes$-excision condition, and let $A$ be the restriction of $\mathcal{F}$ to $\disk_{n}^{\partial}$. For a manifold with boundary $\overline{M}$, we prove that the canonical morphism $\int_{\overline{M}}A\rightarrow\mathcal{F}(\overline{M})$ is an equivalence. By $\otimes$-excision applied to the collar-gluing $\partial\overline{M}\times[0,1)\underset{\partial\overline{M}\times\mathbb{R}}{\bigcup}M\cong\overline{M}$, where $M$ is the interior of $\overline{M}$, we obtain a diagram in $\mathcal{V}$

in which the vertical morphisms are equivalences. It now suffices to show that the top horizontal morphism is an equivalence. The equivalences of $\int_{M}A\rightarrow\mathcal{F}(M)$ and $\int_{\partial\overline{M}\times\mathbb{R}}A\rightarrow\mathcal{F}(\partial\overline{M}\times\mathbb{R})$ are given by Theorem 3.24; the last equivalence $\int_{\partial\overline{M}\times[0,1)}A\rightarrow\mathcal{F}({\partial\overline{M}\times[0,1)})$ by follows by Theorem 3.24 and Proposition 2.16. ∎