Towards Higher Topology

TL;DR

This paper develops a categorification of the classical locale-space adjunction, establishing an idempotent adjunction between generalized bounded ionads and topoi, and relating it to the Scott adjunction in category theory.

Contribution

It introduces a categorified adjunction between ionads and topoi, extending classical dualities and connecting to the Scott adjunction in a novel categorical framework.

Findings

The adjunction between ionads and topoi is shown to be idempotent.

The categorified adjunction relates to the Scott adjunction.

Hints at a 0-dimensional adjunction within the categorified setting.

Abstract

We categorify the adjunction between locales and topological spaces, this amounts to an adjunction between (generalized) bounded ionads and topoi. We show that the adjunction is idempotent. We relate this adjunction to the Scott adjunction, which was discussed from a more categorical point of view in [Liba]. We hint that $0$-dimensional adjunction inhabits the categorified one.