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Normalization of a subgroup, in a topos, and of a word-congruence

Ryuya Hora

TL;DR

The paper develops a generalized normalization operator in categories equipped with a local state classifier $\Xi$, showing that in the group-action topos this recovers the usual normalizer $\mathrm{N}_G(H)$ and that hyperconnected quotients of a topos correspond bijectively to internal filters of $\Xi$. It then proves a main theorem giving an explicit description of the local state classifier of hyperconnected quotients, with corollaries for topoi such as $\mathbf{Cont}(G)$ and $\Sigma\text{-}\mathbf{Set}_{\mathrm{o.f.}}$. This framework is connected to algebraic language theory by interpreting right and two-sided congruences (Nerode and syntactic congruences) within the local state classifier, underpinning a topos-theoretic approach to Nerode-type minimization and Myhill–Nerode theory. The motivating example on word actions shows how the Nerode congruence arises from $\xi_{(Q,\delta)}$ and how regular languages correspond to orbit-finite quotients, enabling a categorical lens on automata theory and the synthesis of syntactic monoids via the normalization operator $\xi_{\Xi}$.

Abstract

This paper provides a new categorical definition of a normalization operator motivated by topos theory and its applications to algebraic language theory. We first define a normalization operator $Ξ\to Ξ$ in any category that admits a colimit of all monomorphisms $Ξ$, which we call a local state classifier. In the category of group actions for a group $G$, this operator coincides with the usual normalization operator, which takes a subgroup $H\subset G$ and returns its normalizer subgroup $\mathrm{Nor}_G(H)\subset G$. Using this generalized normalization operator, we prove a topos-theoretic proposition that provides an explicit description of a local state classifier of a hyperconnected quotient of a given topos. We also briefly explain how these results serve as preparation for a topos-theoretic study of regular languages, congruences of words, and syntactic monoids.

Normalization of a subgroup, in a topos, and of a word-congruence

TL;DR

The paper develops a generalized normalization operator in categories equipped with a local state classifier , showing that in the group-action topos this recovers the usual normalizer and that hyperconnected quotients of a topos correspond bijectively to internal filters of . It then proves a main theorem giving an explicit description of the local state classifier of hyperconnected quotients, with corollaries for topoi such as and . This framework is connected to algebraic language theory by interpreting right and two-sided congruences (Nerode and syntactic congruences) within the local state classifier, underpinning a topos-theoretic approach to Nerode-type minimization and Myhill–Nerode theory. The motivating example on word actions shows how the Nerode congruence arises from and how regular languages correspond to orbit-finite quotients, enabling a categorical lens on automata theory and the synthesis of syntactic monoids via the normalization operator .

Abstract

This paper provides a new categorical definition of a normalization operator motivated by topos theory and its applications to algebraic language theory. We first define a normalization operator in any category that admits a colimit of all monomorphisms , which we call a local state classifier. In the category of group actions for a group , this operator coincides with the usual normalization operator, which takes a subgroup and returns its normalizer subgroup . Using this generalized normalization operator, we prove a topos-theoretic proposition that provides an explicit description of a local state classifier of a hyperconnected quotient of a given topos. We also briefly explain how these results serve as preparation for a topos-theoretic study of regular languages, congruences of words, and syntactic monoids.

Paper Structure

This paper contains 20 sections, 18 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Abstract definitions of a normalizer subgroup
  3. Local state classifier in a hyperconnected quotient topos
  4. Word combinatorics in algebraic language theory
  5. Preliminaries on Hyperconnected quotients and local state classifier
  6. Hyperconnected quotients
  7. Local state classifier
  8. Definition
  9. Inducing full subcategories
  10. The order structure
  11. The classification theorem
  12. The self-referential aspect of the local state classifier
  13. Normalization operator in a category with a local state classifier
  14. Definition and examples
  15. The normalization lemma
  16. ...and 5 more sections

Key Result

Proposition 2.3

Every Grothendieck topos $\mathcal{E}$ has a local state classifier.

Theorems & Definitions (42)

  • Definition : Paraphrase of Definition \ref{['def:NormalizationOperator']}
  • Definition 2.1: Hyperconnected geometric morphisms
  • Definition 2.2: hora2024internal
  • Proposition 2.3: hora2024internal
  • Proposition 2.4: hora2024internal
  • Theorem 2.5: hora2024internal
  • Theorem 3.1
  • Example 3.2: The topos of graphs: [Not being a loop] is a loop. (1/2)
  • Definition 4.1: Normalization operator
  • Example 4.2: The topos of group actions
  • ...and 32 more