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].

中文翻译：

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Topoi 的 Gabriel-Ulmer 二元性及其与现场演示的关系

让 $\kappa$ 是一个普通的基数。我们研究了 Gabriel-Ulmer 对偶性，当将具有 $\kappa$-可访问右伴随的局部 $\kappa$-可呈现类别的 2-类别限制到其 $\kappa$-可呈现格洛腾迪克拓扑的局部完整子 2-类别时几何 $\kappa$ 可访问态射。特别是，我们提供了对 $\kappa$-small cocomplete 类别的 2-category 的局部完整子 2-category 的完整理解，其中 $\kappa$-colimit 保留函子作为相应的 2-category 呈现通过限制。