Exponentiable Higher Toposes

We characterise the class of exponentiable $\mathrm{\infty }$-toposes: $\mathcal{X}$ is exponentiable if and only if $\mathcal{S}\mathrm{h}(\mathcal{X})$ is a continuous $\mathrm{\infty }$-category. The heart of the proof is the description of the $\mathrm{\infty }$-category of $\mathcal{C}$-valued sheaves on $\mathcal{X}$ as an $\mathrm{\infty }$-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 $\mathrm{\infty }$-category of stable sheaves $\mathcal S$\mathrm{h}$(\mathcal X, $\mathrm{S}\mathrm{p}$)$isadualisableobjectinthe$\infty$-category of presentable stable $\mathrm{\infty }$-categories.