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

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.