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

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 Sierpi\'nski 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.