# Gabriel-Ulmer duality for topoi and its relation with site presentations

## TL;DR

This paper explores the Gabriel-Ulmer duality within the context of topoi, analyzing how certain categories of Grothendieck topoi relate to site presentations and their categorical properties.

## Contribution

It provides a comprehensive understanding of the sub-2-category of locally presentable Grothendieck topoi and their relation to site presentations, extending Gabriel-Ulmer duality.

## Key findings

- Characterization of the sub-2-category of Grothendieck topoi
- Relation between topoi presentations and site presentations
- Identification of a reflective sub-bicategory of sites

## Abstract

Let $\kappa$ be a regular cardinal. We study Gabriel-Ulmer duality when one restricts the 2-category of locally $\kappa$-presentable categories with $\kappa$-accessible right adjoints to its locally full sub-2-category of $\kappa$-presentable Grothendieck topoi with geometric $\kappa$-accessible morphisms. In particular, we provide a full understanding of the locally full sub-2-category of the 2-category of $\kappa$-small cocomplete categories with $\kappa$-colimit preserving functors arising as the corresponding 2-category of presentations via the restriction. We analyse the relation of these presentations of Grothendieck topoi with site presentations and we show that the 2-category of locally $\kappa$-presentable Grothendieck topoi with geometric $\kappa$-accessible morphisms is a reflective sub-bicategory of the full sub-2-category of the 2-category of sites with morphisms of sites genearated by the weakly $\kappa$-ary sites in the sense of Shulman [37].