ScalingStacks

Proof. The following equivalences

$$\int_{M}\free_{n}(V)\simeq\colim_{U\in\disk^{\partial}_{n/M}}\coprod_{i\geq 0}\conf_{i}(U,\partial U)\underset{\Sigma_{i}}{\otimes}V^{\otimes i}\simeq\coprod_{i\geq 0}\colim_{U\in\disk^{\partial}_{n/M}}\conf_{i}(U,\partial U)\underset{\Sigma_{i}}{\otimes}V^{\otimes i}$$

follow from the commutativity of colimits. To conclude the result it therefore suffices to show that for each $i$ the canonical morphism

$$\colim_{U\in\disk^{\partial}_{n/M}}\conf_{i}(U,\partial U)\underset{\Sigma_{i}}{\otimes}V^{\otimes i}\longrightarrow\conf_{i}(M,\partial M)\underset{\Sigma_{i}}{\otimes}V^{\otimes i}$$

in $\mathcal{V}$ is an equivalence. By the assumed distributivity in the $\otimes$-presentability condition, this follows if the natural $\Sigma_{i}$-equivariant map of $\Sigma_{i}$-spaces

(9)

$$\colim_{U\in\disk^{\partial}_{n/M}}\conf_{i}(U,\partial U)\longrightarrow\conf_{i}(M,\partial M)$$

is an equivalence, which we now show.

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].

Now suppose $\partial M$ is not empty. Fix a collar-neighborhood $\partial M\times\mathbb{R}_{\geq 0}\hookrightarrow M$. Such a collar-neighborhood determines the top horizontal arrow in the diagram of topological spaces

which commutes up to homotopy – here, the homotopy colimit is indexed by the opposite of the poset of non-empty subsets of $\{1,\dots,i\}$. This collar-neighborhood also gives that this diagram is a weak homotopy pushout. The result for this case of non-empty boundary thus follows from the previous case of empty boundary applied to $\partial M$ and to $\mathring{M}$, using that homotopy colimits commute with one another.

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