Grothendieck topoi with a left adjoint to a left adjoint to a left adjoint to the global sections functor

TL;DR

The paper defines complete connectedness of Grothendieck topoi via the existence of a left adjoint to a left adjoint to a left adjoint to global sections, and develops a parallel between this notion and local connectedness through a dual site characterisation. It introduces the container object as a central reflector for the subcategory of connected objects and proves that every topos can be embedded as a closed subtopos of a completely connected one. A detailed analysis of adjoint-string lengths between Set and a topos yields five classes of topoi, with substantial results on presheaf, localic, and family topoi. The work also presents extensive examples, including presheaf topoi, the Sierpiński topos, trees, augmented simplicial sets, and classifying topoi of algebraic theories, illustrating the ubiquity and utility of complete connectedness and its dualities with local topoi.

Abstract

This paper introduces the notion of complete connectedness of a Grothendieck topos, defined as the existence of a left adjoint to a left adjoint to a left adjoint to the global sections functor, and provides many examples. Typical examples include presheaf topoi over a category with an initial object, such as the topos of sets, the Sierpiński topos, the topos of trees, the object classifier, the topos of augmented simplicial sets, and the classifying topos of many algebraic theories, such as groups, rings, and vector spaces. We first develop a general theory on the length of adjunctions between a Grothendieck topos and the topos of sets. We provide a site characterisation of complete connectedness, which turns out to be dual to that of local topoi. We also prove that every Grothendieck topos is a closed subtopos of a completely connected Grothendieck topos.

Figures (5)

• Figure 1: An object of the Sierpiński topos
• Figure 2: The container object of the Sierpiński topos looks like "an empty box."
• Figure 3: The initial object of the Sierpiński topos looks like "nothing."
• Figure 4: The correspondence between rooted trees and rooted forests
• Figure 5: The underlying topological space of topos of trees, which has an open dense point.

Theorems & Definitions (54)

• Definition 2.1
• Remark 2.2: Meaning of each $\gamma_i$
• Example 2.3: The topos of idempotents
• Example 2.4: The Sierpiński topos
• Example 2.5: The sheaf topos over a circle
• Example 2.6: The sheaf topos over the Cantor space
• Theorem 2.7
• proof
• Remark 2.8: Possible lengths of maximal adjoint strings
• Remark 2.9: Relationship with quality types