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Internal Parameterization of Hyperconnected Quotients

Ryuya Hora

TL;DR

The paper introduces the local state classifier $\Xi$ to achieve an internal parameterization of hyperconnected quotients of a topos $\mathcal{E}$, proving a natural bijection between hyperconnected quotients, internal filters of $\Xi$, and internal semilattice homomorphisms $\Xi\to\Omega$. It develops the existence and internal semilattice structure of $\Xi$, analyzes a broader correspondence with coherent subobject families, and establishes the main theorem in this internal-structure framework. The results yield a partial solution to Lawvere's open problem, notably for Boolean toposes where all quotients become hyperconnected, and provide concrete classifications in examples such as directed graphs and group-action toposes. The work opens avenues for applying internal parameterization to other quotient classes and exploring the interaction of hyperconnected quotients with other internal topos structures.

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.

Internal Parameterization of Hyperconnected Quotients

TL;DR

The paper introduces the local state classifier to achieve an internal parameterization of hyperconnected quotients of a topos , proving a natural bijection between hyperconnected quotients, internal filters of , and internal semilattice homomorphisms . It develops the existence and internal semilattice structure of , analyzes a broader correspondence with coherent subobject families, and establishes the main theorem in this internal-structure framework. The results yield a partial solution to Lawvere's open problem, notably for Boolean toposes where all quotients become hyperconnected, and provide concrete classifications in examples such as directed graphs and group-action toposes. The work opens avenues for applying internal parameterization to other quotient classes and exploring the interaction of hyperconnected quotients with other internal topos structures.

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.
Paper Structure (21 sections, 24 theorems, 34 equations, 3 figures)

This paper contains 21 sections, 24 theorems, 34 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Terminology and preliminaries
  3. Hyperconnected geometric morphism
  4. Local state classifier
  5. Necessity and inevitability
  6. Definition and intuition
  7. Examples
  8. Local state classifier of a Grothendieck topos
  9. Internal semilattice structure
  10. Internal parameterization of hyperconnected quotients
  11. Broader correspondence
  12. Main theorem
  13. Examples and Applications
  14. Number of hyperconnected quotients
  15. Toy example: the topos of directed graphs
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\mathcal{E}$ be a topos with a local state classifier $\{\xi_{X}\colon X\to\Xi\}_{X\in \mathrm{ob}(\mathcal{E})}$ (for example, an arbitrary Grothendieck topos). Then the following three concepts correspond bijectively.

Figures (3)

  • Figure 1: Similarity with the internal parameterization of embeddings
  • Figure 2: Factorization systems and Internal parameterizations
  • Figure 3: Locally determined cocones

Theorems & Definitions (85)

  • Theorem : \ref{['mainTheorem']}
  • Definition 2.1: Hyperconnected geometric morphism
  • Example 2.2: Full and bijective on objects functor
  • Example 2.3: Topological monoid action topos
  • Example 2.4: Relativized two-valuedness
  • Example 2.5: Localic topos
  • Example 2.6: Atomic quotients and well-founded part
  • Lemma 3.1: Hyperconnected quotients are determined by local states.
  • proof
  • Definition 3.2: Locally determined cocone
  • ...and 75 more