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

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].