Flat functors in higher topos theory

TL;DR

This paper generalizes classical and higher categorical results to characterize flat functors in higher topos theory, linking functor properties to geometric morphisms across different levels of n-topoi.

Contribution

It provides necessary and sufficient conditions for functors to induce geometric morphisms in n-topos theory, extending Diaconescu's theorem to higher categories and including hypercompleteness considerations.

Findings

Generalized Diaconescu's theorem for n-topoi

Characterized flat functors in higher topos theory

Showed n-site topoi behave as n-localic hypercomplete topoi

Abstract

For a small $n$-category $\mathscr{C}$ and an $n$-topos $\mathscr{X}$, we study necessary and sufficient conditions for a functor $f \colon \mathscr{C} \to \mathscr{X}$ to determine a geometric morphism from $\mathscr{X}$ to the $n$-topos $\mathcal{P}(\mathscr{C})_n$ of presheaves on $\mathscr{C}$ for any $n \geq 1$. 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.