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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2511.05012 |
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| _version_ | 1866912693310455808 |
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| author | Hora, Ryuya |
| author_facet | Hora, Ryuya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_05012 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalization of a subgroup, in a topos, and of a word-congruence Hora, Ryuya Category Theory 18B25 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. |
| title | Normalization of a subgroup, in a topos, and of a word-congruence |
| topic | Category Theory 18B25 |
| url | https://arxiv.org/abs/2511.05012 |