ScalingStacks

Proof. Manifestly, the functor is essentially surjective, and it carries the $\infty$-subcategory $\mathcal{I}_{M}$ to the maximal $\infty$-subgroupoid $\bigl(\disk^{B}_{n/M}\bigr)^{\sim}$. There results a functor $\bigl(\ddisk^{B}_{n/M}\bigr)[\mathcal{I}_{M}^{-1}]\to\disk^{B}_{n/M}$ from the localization. We will argue that this functor is an equivalence by showing it is an equivalence on maximal $\infty$-subgroupoids, then that it is an equivalence on spaces of morphisms. After Lemma 2.5, it is enough to consider the case of $B=\BTop(n)$. We will adopt the following notation for this proof:

$$\mathsf{D}_{M}~:=~\ddisk_{n/M}\qquad\text{ and }\qquad\mathcal{D}_{M}~:=~\disk_{n/M}~.$$

The maximal $\infty$-subgroupoid of $\mathsf{D}_{M}$ is the classifying space $\mathsf{B}\mathcal{I}_{M}$. In light of the coproduct expression in Lemma 2.12, fix a cardinality $i\geq 0$. Consider the full subcategory $\mathcal{I}_{M}^{i}\subset\mathcal{I}_{M}$ consisting of those $(U\hookrightarrow M)$ for which the cardinality of the connected components $|[U]|=i$. We thus seek to show that the resulting functor $\mathcal{I}_{M}^{i}\to\conf_{i}(M)_{\Sigma_{i}}$ witnesses an equivalence from the classifying space. We explain the following sequence of weak homotopy equivalences

	$\displaystyle\mathsf{B}\mathcal{I}^{i}_{M}$	$\displaystyle\simeq$	$\displaystyle\underset{(U\hookrightarrow M)\in\mathcal{I}^{i}_{M}}{\colim}~(\mathbb{R}^{n})^{i}$
		$\displaystyle\xrightarrow{\simeq}$	$\displaystyle\underset{(U\hookrightarrow M)\in\mathcal{I}^{i}_{M}}{\sf p.s.colim}~(\mathbb{R}^{n})^{i}$
		$\displaystyle\xrightarrow{\cong}$	$\displaystyle\conf_{i}(M)_{\Sigma_{i}}$

where ${\sf p.s.colim}$ denotes the ordinary point-set colimit of topological spaces. The first equivalence is formal, because each term in the homotopy colimit is contractible. By inspection, the category $\mathcal{I}_{M}^{i}$ forms a basis for the standard Grothendieck topology on $\conf_{i}(M)_{\Sigma_{i}}$. The third homeomorphism follows. Because $\conf_{i}(M)_{\Sigma_{i}}$ is paracompact, Corollary 1.6 of [DI] gives that the second map is a weak homotopy equivalence. In summary, we have verified that the map of maximal $\infty$-subgroupoids

$$\bigl(\mathsf{D}_{M}[\mathcal{I}_{M}^{-1}]\bigr)^{\sim}\xrightarrow{~\simeq~}\bigl(\mathcal{D}_{M}\bigr)^{\sim}$$

is an equivalence.

We now show that the functor from the localization induces an equivalence on spaces of morphisms. Consider the diagram of spaces

where a superscript ⁽¹⁾ indicates a space of morphisms, and the upper vertical arrows are given as $(V\hookrightarrow U)\mapsto\bigl((V\hookrightarrow M)\hookrightarrow(U\hookrightarrow M)\bigr)$. Our goal is to show that the middle horizontal arrow is an equivalence. We will accomplish this by showing that the diagram is a map of homotopy fiber sequences, for we have already shown that the top and bottom horizontal maps are equivalences.

The right vertical sequence is a fiber sequence is because such evaluation maps are coCartesian fibrations, in general. Then, by inspection, the fiber over $(U\hookrightarrow M)$ is the maximal $\infty$-subgroupoid of the over $\infty$-category $(\mathcal{D}_{M})_{/(U\hookrightarrow M)}$. This over $\infty$-category is canonically identified as $\mathcal{D}_{U}$.

We now show that the left vertical sequence is a homotopy fiber sequence. The space of morphisms $\bigl(\mathsf{D}_{M}[\mathcal{I}_{M}^{-1}]\bigr)^{(1)}$ is the classifying space of the subcategory of the functor category $\Fun^{\mathcal{I}_{M}}\bigl([1],\mathsf{D}_{M}\bigr)\subset\Fun\bigl([1],\mathsf{D}_{M}\bigr)$ consisting of the same objects but only those natural transformations by $\mathcal{I}$. We claim the fiber over $(U\hookrightarrow M)$ of the evaluation map is canonically identified as in the sequence

$$\bigl(\mathsf{D}_{U}[\mathcal{I}_{U}^{-1}]\bigr)^{\sim}\xrightarrow{~(U\hookrightarrow M)~}\bigl(\mathsf{D}_{M}[\mathcal{I}_{M}^{-1}]\bigr)^{(1)}\xrightarrow{~{\sf ev}_{1}~}\bigl(\mathsf{D}_{M}[\mathcal{I}_{M}^{-1}]\bigr)^{\sim}~.$$

This claim is justified through Quillen’s Theorem B, for the named fiber is the classifying space of the over $\infty$-category $(\mathcal{I}_{M})_{/(U\hookrightarrow M)}$ which is canonically isomorphic to $\mathcal{I}_{U}$. To apply Quillen’s Theorem B we must show that each morphism $(U\hookrightarrow M)\hookrightarrow(V\hookrightarrow M)$ in $\mathcal{I}$ induces an equivalence of spaces $\mathsf{B}\bigl((\mathcal{I}_{M})_{/(U\hookrightarrow M)}\bigr)\simeq\mathsf{B}\bigl((\mathcal{I}_{M})_{/(V\hookrightarrow M)}\bigr)$. This map of spaces is canonically identified as the map $\mathsf{B}\mathcal{I}_{U}\to\mathsf{B}\mathcal{I}_{V}$ induced from the inclusion $U\hookrightarrow V$, which, by design, is a bijection on connected components. Through the previous analysis of this proof, this map is further identified as the map of spaces $U\hookrightarrow V$. The Kister–Mazur Theorem 2.3 implies this inclusion $U\hookrightarrow V$ is isotopic to an isomorphism, from which it follows that the map of spaces $\mathsf{B}\mathcal{I}_{U}\xrightarrow{\simeq}\mathsf{B}\mathcal{I}_{V}$ is an equivalence. We conclude that Quillen’s Theorem B applies. (For an $\infty$-categorical account of Quillen’s Theorem B, see for instance Theorem 5.3 of [Bar].) ∎