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Equ-saturating categories

Dominique Bourn

TL;DR

The paper introduces Equ-saturating categories, a categorical generalization of syntactic equivalence relations with universal saturation properties. It develops both foundational notation and concrete instances across monoids, groups, semirings, rings, and left skew braces, then extends to generalized relations $\forall_US$ and their largest saturations. The core results show that every variety and many fiber categories (such as $\mathsf{Cat}_X\mathbb{E}$ for Set or Mal'tsev and Gumm bases) are Equ-saturating, and that in protomodular contexts each internal relation admits a centralizer, with pointed and additive cases offering further simplifications. The framework links internal category theory, centralizers of equivalence relations, and fiberwise saturation, yielding structural insights for normalizers, centralizers, and decomposition in a range of algebraic and categorical settings.

Abstract

Starting from the varietal notion of syntactic equivalence relation, we generalized it to a categorical concept; namely Equ-saturating category. We produce various examples and focuse our attention on the protomodular context in which any equivalence relation is then shown to have a centralizer.

Equ-saturating categories

TL;DR

The paper introduces Equ-saturating categories, a categorical generalization of syntactic equivalence relations with universal saturation properties. It develops both foundational notation and concrete instances across monoids, groups, semirings, rings, and left skew braces, then extends to generalized relations and their largest saturations. The core results show that every variety and many fiber categories (such as for Set or Mal'tsev and Gumm bases) are Equ-saturating, and that in protomodular contexts each internal relation admits a centralizer, with pointed and additive cases offering further simplifications. The framework links internal category theory, centralizers of equivalence relations, and fiberwise saturation, yielding structural insights for normalizers, centralizers, and decomposition in a range of algebraic and categorical settings.

Abstract

Starting from the varietal notion of syntactic equivalence relation, we generalized it to a categorical concept; namely Equ-saturating category. We produce various examples and focuse our attention on the protomodular context in which any equivalence relation is then shown to have a centralizer.

Paper Structure

This paper contains 22 sections, 46 theorems, 6 equations.

Table of Contents

  1. Notations
  2. Syntactic equivalence relations
  3. Monoids
  4. Groups
  5. The varietal case
  6. Semirings
  7. Rings
  8. Left skew braces
  9. Generalized syntactic equivalence relations
  10. Monoids
  11. Groups
  12. Semirings, left skew braces
  13. The case of protomodular categories
  14. $\mathrm{Equ}$-saturating protomodular categories
  15. $\mathrm{Equ}$-saturating exact protomodular categories
  16. ...and 7 more sections

Key Result

Lemma 1.1

Given any category $\mathbb{E}$ and any monormorphism $u: U \rightarrowtail X$, 1) if $u$ is saturated w.r. to $R$ and $S\subset R$, then $u$ is saturated w.r. to $S$; 2) if $u$ is normal to $R$, then $u$ is saturated w.r. to any equivalence relation $S\subset R$; 3) when the following square is a

Theorems & Definitions (55)

  • Definition 1.1
  • Lemma 1.1
  • Definition 2.1
  • Proposition 2.1
  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.2
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.2
  • ...and 45 more