Exponentiable Higher Toposes

We characterise the class of exponentiable $\infty$-toposes: $\mathcal X$ is exponentiable if and only if $\mathcal S\mathrm{h}(\mathcal X)$ is a continuous $\infty$-category. The heart of the proof is the description of the $\infty$-category of $\mathcal C$-valued sheaves on $\mathcal X$ as an $\infty$-category of functors that satisfy finite limits conditions as well as filtered colimits conditions (instead of limits conditions purely); we call such functors $\omega$-continuous sheaves. As an application, we show that when $\mathcal X$ is exponentiable, its $\infty$-category of stable sheaves $\mathcal S\mathrm{h}(\mathcal X, \mathrm{Sp})$ is a dualisable object in the $\infty$-category of presentable stable $\infty$-categories.