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

TL;DR

This paper introduces a new categorical normalization operator inspired by topos theory, generalizing the concept of subgroup normalization and applying it to algebraic language theory.

Contribution

It defines a normalization operator in any category with certain colimits, linking it to topos theory and algebraic language concepts, and provides explicit descriptions for local state classifiers.

Findings

Normalization operator coincides with subgroup normalizer in group actions

Provides a topos-theoretic description of local state classifiers

Prepares for studying regular languages and syntactic monoids categorically

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 $\Xi \to \Xi$ in any category that admits a colimit of all monomorphisms $\Xi$, 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.