ScalingStacks

Proof. Let $S\subset\disk^{\partial,{\sf or}}_{1/[-1,1]}$ be the full $\infty$-subcategory of embeddings from oriented intervals that surject onto the endpoints. That is, $S$ consists of oriented embeddings among 1-manifolds with boundary of the form $[-1,0)\sqcup\mathbb{R}^{\sqcup\ell}\sqcup(0,1]\hookrightarrow[-1,1]$, for $\ell\geq 0$. To show the inclusion $S\subset\disk^{\partial,{\sf or}}_{1/[-1,1]}$ is final, by Quillen’s Theorem A, we can show the under $\infty$-category $S^{U/}$ has a contractible classifying space for every finite disjoint union of subintervals $U\subset[-1,1]$. This is immediate, because $S^{U/}$ has an initial object, which is a disjoint union of $U$ with connected open neighborhoods of the endpoints not contained in $U$.

Lastly, the result follows because there is an equivalence $S\to\bdelta^{\op}$. On objects this is given by assigning to $(U\hookrightarrow[-1,1])$ the set connected components of the complement $\bigl[[-1,1]\smallsetminus U\bigr]$ together with the linear order inherited from that of $[-1,1]$. That this assignment defines a functor is routine. That this functor is an equivalence of $\infty$-categories follows because each comopnent of the space of morphisms of $\disk^{\partial,\sf or}_{1/[-1,1]}$ is contractible.

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