On coherent topoi & coherent $1$-localic $\infty$-topoi

TL;DR

This paper establishes an equivalence between the $ abla$-category of coherent ordinary topoi and that of coherent 1-localic $ abla$-topoi, enriching the understanding of their relationship and providing algebraic geometry examples.

Contribution

It proves the equivalence between coherent ordinary topoi and coherent 1-localic $ abla$-topoi, filling a gap in the literature and connecting different frameworks.

Findings

Equivalence between coherent ordinary topoi and coherent 1-localic $ abla$-topoi.

Collection of examples of coherent geometric morphisms from algebraic geometry.

Clarification of the relationship between different notions of coherence in topoi.

Abstract

In this note we prove the following useful fact that seems to be missing from the literature: the $\infty$-category of coherent ordinary topoi (in the sense of SGA4) is equivalent to the $\infty$-category of coherent $1$-localic $\infty$-topoi (in Lurie's sense). We also collect a number of examples of coherent geometric morphisms between $\infty$-topoi coming from algebraic geometry.