Internal higher topos theory

Abstract

We develop the theory of topoi internal to an arbitrary $\infty$-topos $\mathcal B$. We provide several characterisations of these, including an internal analogue of Lurie's characterisation of $\infty$-topoi, but also a description in terms of the underlying sheaves of $\infty$-categories, and we prove a number of structural results about these objects. Furthermore, we show that the $\infty$-category of topoi internal to $\mathcal B$ is equivalent to the $\infty$-category of $\infty$-topoi over $\mathcal B$, and use this result to derive a formula for the pullback of $\infty$-topoi. Lastly, we use our theory to relate smooth geometric morphisms of $\infty$-topoi to internal locally contractible topoi.