ScalingStacks

Proof. Through the defining adjunctions for over $\infty$-categories, the first assertion follows because, for each $\infty$-category $\mathcal{K}$, the canonical diagram among $\infty$-categories

is a pushout; here, for $\mathcal{K}$ and $\mathcal{J}$ $\infty$-categories,

$$\mathcal{K}\star\mathcal{I}~:=~\mathcal{K}\underset{\mathcal{K}\times\{0\}\times\mathcal{I}}{\coprod}\mathcal{K}\times\{0<1\}\times\mathcal{I}\underset{\mathcal{K}\times\{1\}\times\mathcal{I}}{\coprod}\mathcal{I}$$

denotes the join of $\infty$-categories. The second assertion follows directly from the universal property of coproducts. The final assertion follows from Proposition 3.2.2.1 of [Lu2], which in particular gives that, for each Cartesian closed presentable $\infty$-category $\mathcal{C}$, the forgetful functor from commutative algebras $\Alg_{\sf Com}(\mathcal{C}^{\times})\to\mathcal{C}$ preserves and creates limits. Apply this result to the case $\mathcal{C}=\Cat$.

∎

Proof. By definition, the space $\Emb^{B}(\mathbb{R}^{n},\mathbb{R}^{n})$ sits in a homotopy pullback square:

There are evident equivalences of spaces $\Map_{/\BTop(n)}(\mathbb{R}^{n},\mathbb{R}^{n})\simeq\Omega\BTop(n)\simeq\Top(n)$ and likewise $\Map_{/\negthinspace B}(\mathbb{R}^{n},\mathbb{R}^{n})\simeq\Omega_{g}B$. It is standard that the the composite map of spaces

$$\Top(n)\longrightarrow\Emb(\mathbb{R}^{n},\mathbb{R}^{n})\longrightarrow\Map_{/\BTop(n)}(\mathbb{R}^{n},\mathbb{R}^{n})\simeq\Omega\BTop(n)$$

is an equivalence. By Kister–Mazur, Theorem 2.3, the first map including $\Top(n)$ into $\Emb(\mathbb{R}^{n},\mathbb{R}^{n})$ is a homotopy equivalence. It follows that the middle map in the above display is also an equivalence of spaces. We conclude that the bottom horizontal map in the above homotopy pullback square is an equivalence of spaces. This implies the top horizontal map too is an equivalence of spaces.

∎

Proof. Lemma 2.5 gives an equivalence $\disk_{n/M}^{B}\simeq\disk_{n/M}$. So it suffices to assume the case of an equality $B=\BTop(n)$.

The maximal $\infty$-subgroupoid of $\disk_{n}$ necessarily lies over the maximal $\infty$-subgroupoid of $\fin$, which is $\underset{i\geq 0}{\coprod}\mathsf{B}\Sigma_{i}$. The first assertion will be implied upon verifying, for each $i\geq 0$, that the map

$${\Sigma_{i}\wr{\sf Top}(n)}\longrightarrow\Emb(\underset{i}{\sqcup}\mathbb{R}^{n},\underset{i}{\sqcup}\mathbb{R}^{n})$$

is weakly homotopy equivalent to an inclusion of connected components. This is an immediate consequence of the Kister–Mazur Theorem 2.3. Via translation, the inclusion of the subgroup of origin preserving homeomorphisms $\Top_{0}(n)\xrightarrow{\simeq}\Top(n)$ is a homotopy equivalence, and so we recognize further the identification

$$\underset{i\geq 0}{\coprod}\mathsf{B}(\Sigma_{i}\wr\Top_{0}(n))\xrightarrow{~\simeq~}(\disk_{n})^{\sim}~.$$

Because the projection $\disk_{n/M}\to\disk_{n}$ is a right fibration, we recognize the maximal $\infty$-subgroupoid of $\disk_{n/M}$ as

$$\underset{i\geq 0}{\coprod}\Emb\bigl(\underset{i}{\sqcup}\mathbb{R}^{n},M\bigr)_{\Sigma_{i}\wr\Top_{0}(n)}\xrightarrow{~\simeq~}(\disk_{n/M})^{\sim}~.$$

Therefore, the second assertion follows upon showing that the $\Sigma_{i}$-equivariant continuous map

$${\sf ev}_{0}\colon\Emb\bigl(\underset{i}{\sqcup}\mathbb{R}^{n},M\bigr)_{\Top_{0}(n)^{i}}\longrightarrow\conf_{i}(M)$$

is a weak homotopy equivalence. This is implied upon showing the homotopy fiber of the continuous map

$${\sf ev}_{0}\colon\Emb(\underset{i}{\sqcup}\mathbb{R}^{n},M)\longrightarrow\conf_{i}(M)$$

is weakly homotopy equivalent to $\Top_{0}(n)^{i}$. In a standard manner, this map is a Serre fibration, and the fiber over $c\colon\{1,\dots,i\}\hookrightarrow M$ is the space $\Emb_{0}(\underset{i}{\sqcup}\mathbb{R}^{n},M)$ of embeddings under $c$. Fix such a based embedding $e_{0}\colon\underset{i}{\sqcup}\mathbb{R}^{n}\hookrightarrow M$. So we must show that the composite inclusion

$$\Top_{0}(n)^{i}\hookrightarrow\Emb((0\in\mathbb{R}^{n}),(0\in\mathbb{R}^{n}))^{i}\cong\Emb_{0}(\underset{i}{\sqcup}\mathbb{R}^{n},\underset{i}{\sqcup}\mathbb{R}^{n})\xrightarrow{-\circ e_{0}}\Emb_{0}(\underset{i}{\sqcup}\mathbb{R}^{n},M)$$

is a weak homotopy equivalence.

The Kister–Mazur Theorem gives that the first of these maps is a homotopy equivalence. The problem is thus reduced to showing that the second of these maps is a weak homotopy equivalence. This is the problem of showing that each solid diagram among topological spaces

admits a filler with respect to which the diagram commutes up to homotopy. Choose a continuous map $\phi\colon(0,1]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that $\phi_{t}$ is an origin preserving open embedding for each $t$, $\phi_{1}={\sf id}_{\mathbb{R}^{n}}$, the closure $\overline{\phi_{s}(\mathbb{R}^{n})}\subset\phi_{t}(\mathbb{R}^{n})$ whenever $s<t$, and the collection of images $\{\phi_{t}(\mathbb{R}^{n})\mid 0<t\leq 1\}$ is a basis for the topology about $0\in\mathbb{R}^{n}$. Choose a continuous map $\mathbb{D}^{k}\xrightarrow{\epsilon}(0,1]$ for which the restriction $\epsilon_{|S^{k-1}}\equiv 1$ is identically one, and the composition

$$\mathbb{D}^{k}\xrightarrow{~f~}\Emb_{0}(\underset{i}{\sqcup}\mathbb{R}^{n},M)\xrightarrow{~(\underset{i}{\sqcup}\phi_{\epsilon})^{\ast}~}\Emb_{0}(\underset{i}{\sqcup}\mathbb{R}^{n},M)$$

factors through $\Emb_{0}(\underset{i}{\sqcup}\mathbb{R}^{n},\underset{i}{\sqcup}\mathbb{R}^{n})$. Define $\widetilde{f}$ to be this factorization. By construction, the restriction $\widetilde{f}_{|S^{k-1}}=f_{0}$. The map $[0,1]\times\mathbb{D}^{k}\to\Emb_{0}(\underset{i}{\sqcup}\mathbb{R}^{n},M)$ given by $(t,p)\mapsto f\circ(\underset{i}{\sqcup}\phi_{t\epsilon+(1-t)})^{\ast}(p)$ demonstrates a homotopy making the lower triangle commute.

∎

Proof. Recognize $\mathcal{E}{\sf uc}_{n}\subset\disk_{n}$ as the full $\infty$-subcategory consisting of the connected $n$-manifolds. Specialize the second statement of Lemma 2.12 to $i=1$ to obtain an identification

$$M~\simeq~\mathcal{E}{\sf uc}_{n/M}~\simeq~\Emb(\mathbb{R}^{n},M)_{{\sf Top}(n)}\longrightarrow\BTop(n)$$

involving the homotopy ${\sf Top}(n)$-coinvariants. Manifestly, this map of spaces agrees with the tangent classifier in the case $M=\mathbb{R}^{n}$. By construction, this map is functorial in the argument $M\in\mfld_{n}$. The general case follows. ∎

Proof. This follows by the standard method of pushing off to infinity in the $\mathbb{R}_{\geq 0}$ direction (as in the Alexander trick or the contractibility of foliations on $\mathbb{R}^{n}$ up to integrable homotopy). That is, define a deformation retraction onto the subspace $\Emb(M,N)$ by defining for each $t\in[0,1]$ the map

$$h_{t}:\Emb\bigl(\mathbb{R}_{\geq 0}\times M,\mathbb{R}_{\geq 0}\times N\bigr)\longrightarrow\Emb\bigl(\mathbb{R}_{\geq 0}\times M,\mathbb{R}_{\geq 0}\times N\bigr)$$

by

$$h_{t}(g)(s,x)=\left\{\begin{array}[]{l l}(s,g_{0}(x))&\quad\text{for $s<(1-t)^{-1}-1$}\\ g(s+1-(1-t)^{-1},x)&\quad\text{for $s\geq(1-t)^{-1}-1$}\end{array}\right.$$

where $g_{0}:M\hookrightarrow N$ is the restriction of $g$ at the value $s=0$.

∎

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].) ∎