Flat functors in higher topos theory

Abstract

For a small $n$-category $\mathcal{C}$ and an $n$-topos $\mathcal{X}$, we study necessary and sufficient conditions for a functor $f:\mathcal{C}\to \mathcal{X}$ to determine a geometric morphism from $\mathcal{X}$ to the $n$-topos $\mathcal{P}(\mathcal{C}{)}_n$ of presheaves on $\mathcal{C}$ for any $n\ge 1$. These results generalize and unify results of Lurie for $n=\infty$ and classical characterizations of flat functors (Diaconescu's theorem) for $n=1$. Interestingly, for $n=\infty$, our analogue of Diaconescu's theorem requires hypercompleteness. As an application, we show that the $\infty$-topos associated to an $n$-site behaves as an $n$-localic $\infty$-topos with respect to hypercomplete $\infty$-topoi.

27 pages