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Categorical aspects of congruence distributivity

Michael Hoefnagel, Diana Rodelo

TL;DR

The paper develops a categorical analogue of Jónsson's characterization of congruence distributivity by introducing Jónsson categories of order $n$ in regular categories, a framework that does not require the existence of suprema of equivalence relations. It connects this condition to a categorical trapezoid lemma, $n$-permutability, and the classical notion of equivalence distributivity, showing that a regular $n$-permutable category is equivalence distributive if and only if it is a Jónsson category of order $n-1$. The work provides a broad spectrum of examples (including duals of Top, Pos, and toposes) and establishes a proof strategy that translates varietal Jónsson-term characterizations into categorical inequalities on internal relations. The results advance understanding of distributivity phenomena beyond varieties, with implications for related categorical notions such as majority and Gumm categories and for further exploration of relational conditions in categorical algebra.

Abstract

We study a categorical condition on relations, which is a categorical formulation of Jónsson's characterisation of congruence distributive varieties. Categories satisfying these conditions need not be varieties; for instance, the dual of the categories of topological spaces, ordered sets, $G$-sets, and the dual of any (pre)topos all provide us with examples.

Categorical aspects of congruence distributivity

TL;DR

The paper develops a categorical analogue of Jónsson's characterization of congruence distributivity by introducing Jónsson categories of order in regular categories, a framework that does not require the existence of suprema of equivalence relations. It connects this condition to a categorical trapezoid lemma, -permutability, and the classical notion of equivalence distributivity, showing that a regular -permutable category is equivalence distributive if and only if it is a Jónsson category of order . The work provides a broad spectrum of examples (including duals of Top, Pos, and toposes) and establishes a proof strategy that translates varietal Jónsson-term characterizations into categorical inequalities on internal relations. The results advance understanding of distributivity phenomena beyond varieties, with implications for related categorical notions such as majority and Gumm categories and for further exploration of relational conditions in categorical algebra.

Abstract

We study a categorical condition on relations, which is a categorical formulation of Jónsson's characterisation of congruence distributive varieties. Categories satisfying these conditions need not be varieties; for instance, the dual of the categories of topological spaces, ordered sets, -sets, and the dual of any (pre)topos all provide us with examples.
Paper Structure (10 sections, 11 theorems, 36 equations)

This paper contains 10 sections, 11 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Subobjects and relations in categories
  4. Notation and conventions
  5. Regular categories
  6. $n$-permutability
  7. Main results
  8. Jónsson categories
  9. Examples of Jónsson categories
  10. Proof of Theorem \ref{['theorem: main theorem']}

Key Result

Proposition 2.5

CarboniKellyPedicchio1993 Let $R$ be a relation from $X$ to $Y$ and $S$ a relation from $Y$ to $Z$. Given any morphisms $x:S \to X$ and $z:S\to Z$ then $x(R \circ S)z$ if and only if there exists a regular epimorphism $e: E\twoheadrightarrow S$ and a morphism $y:E \rightarrow Y$ such that $xe R y$ a

Theorems & Definitions (31)

  • Definition 2.2
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.7
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 21 more