ScalingStacks

Proof. After Lemma 2.5, it will suffice to prove the result for the case $B=\BTop(n)$, and so we omit $B$ from the notation and discussion. The functor ${\sf ev}_{0}$ is a Cartesian fibration of $\oo$-categories. Thus, to check finality, by Lemma 4.1.3.2 of [Lu1], it suffices to show that, for each $(V\hookrightarrow M)\in\disk_{n/M}$, the fiber $\infty$-category ${\sf ev}_{0}^{-1}V$ has contractible classifying space. That is, we show that the $\oo$-category $(\disk_{k/N})^{V/}$, of $k$-disks $U$ in $N$ equipped with an embedding $V\hookrightarrow f^{-1}U$, has a contractible classifying space.

There is an identification of spaces

$$\mathsf{B}\bigl(\disk_{k/N}\bigr)^{U/}~{}~\simeq~{}~\underset{(V\hookrightarrow N)\in\disk_{k/N}}{\colim}~\Map_{\mfld_{n/M}}(U,f^{-1}V)~.$$

Formally, the sequence of maps

$$\Map_{\mfld_{n/M}}(U,f^{-1}V)~\longrightarrow~\Map_{\mfld_{n}}(U,f^{-1}V)~\longrightarrow~\Map_{\mfld_{n}}(U,M)$$

is a fiber sequence (here the fiber is taken over any implicit morphism $U\hookrightarrow M$, thereby giving meaning to the lefthand space). So we seek to show the map from the colimit

$$\underset{(V\hookrightarrow N)\in\disk_{k/N}}{\colim}\Map_{\mfld_{n}}(U,f^{-1}V)~{}~\longrightarrow~{}~\Map_{\mfld_{n}}(U,M)$$

is an equivalence of spaces. We recognize this map of spaces as the map of fibers over $U\in\disk_{n}$ of the map of right fibrations over $\disk_{n}$:

$$\underset{(V\hookrightarrow N)\in\disk_{k/N}}{\colim}\disk_{n/f^{-1}V}\longrightarrow\disk_{n/M}~.$$

Being right fibrations, it is enough to show that this functor is an equivalence on maximal $\infty$-subgroupoids. Using Lemma 2.12 which identifies these maximal $\infty$-subgroupoids, this is the problem of showing, for each finite set $J$, that the map of spaces

$$\underset{(V\hookrightarrow N)\in\disk_{k/N}}{\colim}\conf_{J}\bigl(f^{-1}V\bigr)_{\Sigma_{J}}~\longrightarrow~\conf_{J}(M)_{\Sigma_{J}}$$

is an equivalence.

Lemma 2.19 implies the functor $\ddisk_{k/N}\to\disk_{k/N}$ is final, and so the forgetful map

$$\underset{(V\hookrightarrow N)\in\ddisk_{k/N}}{\colim}\conf_{J}\bigl(f^{-1}V\bigr)_{\Sigma_{J}}~\xrightarrow{~\simeq~}~\underset{(V\hookrightarrow N)\in\disk_{k/N}}{\colim}\conf_{J}\bigl(f^{-1}V\bigr)_{\Sigma_{J}}$$

is an equivalence of spaces. Now notice that, for each $(V\hookrightarrow N)\in\ddisk_{k/N}$, the map $\conf_{J}(f^{-1}V)\to\conf_{J}(M)$ an open embedding. Also, for each point $c\colon J\hookrightarrow M$ the image $f\bigl(c(J)\bigr)\subset N$ has cardinality at most $J$. So there is an object $(V\hookrightarrow N)$ of $\ddisk_{k/N}$ whose image contains the subset $f\bigl(c(J)\bigr)$. We see then that the collection of open embeddings

$$\Bigl\{\conf_{J}(f^{-1}V)_{\Sigma_{J}}\hookrightarrow\conf_{J}(M)_{\Sigma_{I}}\mid(V\hookrightarrow N)\in\ddisk_{k/N}\Bigr\}$$

forms an open cover.

Because $N$ is a manifold, the collection of open embeddings from Euclidean spaces into $N$ form a basis for the topology of $N$. It follows that the collection of (at most) $|J|$-tuples of disjoint open disks in $N$ forms an open cover of $N$ in such a way that any finite intersection of such is again covered by such. This is to say that this collection of open embeddings forms a hypercover of $\conf_{J}(M)_{\Sigma_{J}}$. Corollary 1.6 of [DI] gives that the map

$$\underset{(V\hookrightarrow N)\in\ddisk_{k/N}}{\colim}\conf_{J}\bigl(f^{-1}V\bigr)_{\Sigma_{J}}~\xrightarrow{~\simeq~}~\conf_{J}(M)_{\Sigma_{J}}$$

is an equivalence of spaces, which completes the proof.

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