Towards higher topology

Formal model theory and higher topology

“…In [14] and [13], the author has discussed the relationship between the 2‐categories presented in the diagram below. There, we have focused on the geometrical and categorical aspects of this relationships, while this paper will focus on the more logical aspects of the story.…”

“…We should say what we mean by $$\mathsf {LGSketches}$$ and $$\mathbb {Mod}(-)$$. The adjunction $$\mathbb {O} \dashv \mathbb {pt}$$ was introduced and studied in [14] and it relates topoi to bounded ionads; we refer to [14, § 3] for the construction.…”

“…This will amount to a complete description of the diagram below. From a technical point of view we build on two previous paper of ours [13, 14], where we develop the relationship between topoi, bounded ionads, and accessible categories with directed colimits from a more geometric perspective. Indeed, in [14] we have categorified the Scott topology on a poset with directed joins and the Isbell duality between locales and topological spaces.…”

“… From a technical point of view we build on two previous paper of ours [13, 14], where we develop the relationship between topoi, bounded ionads, and accessible categories with directed colimits from a more geometric perspective. Indeed, in [14] we have categorified the Scott topology on a poset with directed joins and the Isbell duality between locales and topological spaces. We will briefly recall the results of these papers in the first section, but we must redirect the reader to the papers for a proper introduction to the topic.…”

“…The exposition is organized as follows. In § 2, after a very brief recollection of the results from [14] and [13], we put Scott adjunction in the context of functorial semantics. We trace the Scott topos back to the seminal works of Linton and Lawvere on algebraic theories and algebraic varieties.…”