# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1904.01877