ScalingStacks

We first consider the case that the boundary of $M$ is empty, so that the natural $\Sigma_{i}$-equivariant map $\conf_{i}(M)\xrightarrow{\cong}\conf_{i}(M,\partial M)$ is a homeomorphism, and the natural functor $\disk_{n/M}\xrightarrow{\simeq}\disk^{\partial}_{n/M}$ is an equivalence of $\infty$-categories. In this case we are to show that the $\Sigma_{i}$-equivariant map of $\Sigma_{i}$-spaces

$$\colim_{U\in\disk_{n/M}}\conf_{i}(U)\longrightarrow\conf_{i}(M)$$

is an equivalence. After Proposition 2.19, it is enough to show that the $\Sigma_{i}$-equivariant map of $\Sigma_{i}$-topological spaces

(10)

$$\underset{U\in{\sf Disk}_{n/M}}{\colim}\conf_{i}(U)\longrightarrow\conf_{i}(M)$$

is a weak homotopy equivalence from the homotopy colimit. Each map $\conf_{i}(U)\to\conf_{i}(M)$ comprising this homotopy colimit is an open embedding. Also, for each element $\{1,\dots,i\}\xrightarrow{c}M$ of $\conf_{i}(M)$, choosing mutually disjoint Euclidean neighborhoods about each $c(j)\in M$ demonstrates that $c$ lies in the image of at least one such open embedding. Therefore this augmented diagram is an open cover of $\conf_{i}(M)$. This open cover of $M$ has the property that each finite intersection of its terms is covered by terms contained in this finite intersection. This is to say that this open cover of $\conf_{i}(M)$ is in fact a hypercover. That the map (10) is a weak homotopy equivalence follows from Corollary 1.6 of [DI].