Exponentiable Higher Toposes

Abstract

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.