Internal Parameterization of Hyperconnected Quotients

TL;DR

This paper introduces the concept of a local state classifier to provide an internal parameterization of hyperconnected quotients in topos theory, extending the fundamental correspondence between subtoposes and Lawvere-Tierney topologies.

Contribution

It proposes a new elementary concept, the local state classifier, to internally parameterize hyperconnected quotients, addressing a longstanding open problem in topos theory.

Findings

Provides an internal parameterization of hyperconnected quotients

Solves the Boolean case of Lawvere's open problems

Introduces the concept of a local state classifier

Abstract

One of the most fundamental facts in topos theory is the internal parameterization of subtoposes: the bijective correspondence between subtoposes and Lawvere-Tierney topologies. In this paper, we introduce a new but elementary concept, "a local state classifier," and give an analogous internal parameterization of hyperconnected quotients (i.e., hyperconnected geometric morphisms from a topos). As a corollary, we obtain a solution to the Boolean case of the first problem of Lawvere's open problems.