ScalingStacks

Proof. First observe that a split-exact sequence ${\mathcal{A}}\overset{f}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}{\mathcal{B}}\overset{g}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}{\mathcal{C}}$ is completely determined by the projection $g:{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ together with its section $j:{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$. This is because $f:{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$ is the fiber of $g$, which we may identify with the full subcategory of ${\mathcal{B}}$ spanned by the $b\in{\mathcal{B}}$ such that $g(b)\simeq 0$, and, since $f$ is fully faithful, $i:{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}$ is determined by the composite $f\circ i:{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}$, the fiber

$$f\circ i\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\id_{{\mathcal{B}}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}j\circ g$$

of the unit map of the adjunction $(g,j)$. Hence $\mathrm{Split}(\Cat_{\infty}^{\perf})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Loc}(\Cat_{\infty}^{\perf})$ has contractible (homotopy) fibers and is therefore and equivalence. ∎