Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Enriched aspects of calculus of relations and $2$-permutability

Maria Manuel Clementino, Diana Rodelo

Abstract

The aim of this work is to further develop the calculus of (internal) relations for a regular Ord-category C. To capture the enriched features of a regular Ord-category and obtain a good calculus, the relations we work with are precisely the ideals in C. We then focus on an enriched version of the 1-dimensional algebraic 2-permutable (also called Mal'tsev) property and its well-known equivalent characterisations expressed through properties on ordinary relations. We introduce the notion of Ord-Mal'tsev category and show that these may be characterised through enriched versions of the above mentioned properties adapted to ideals. Any Ord-enrichment of a 1-dimensional Mal'tsev category is necessarily an Ord-Mal'tsev category. We also give some examples of categories which are not Mal'tsev categories, but are Ord-Mal'tsev categories.

Enriched aspects of calculus of relations and $2$-permutability

Abstract

The aim of this work is to further develop the calculus of (internal) relations for a regular Ord-category C. To capture the enriched features of a regular Ord-category and obtain a good calculus, the relations we work with are precisely the ideals in C. We then focus on an enriched version of the 1-dimensional algebraic 2-permutable (also called Mal'tsev) property and its well-known equivalent characterisations expressed through properties on ordinary relations. We introduce the notion of Ord-Mal'tsev category and show that these may be characterised through enriched versions of the above mentioned properties adapted to ideals. Any Ord-enrichment of a 1-dimensional Mal'tsev category is necessarily an Ord-Mal'tsev category. We also give some examples of categories which are not Mal'tsev categories, but are Ord-Mal'tsev categories.
Paper Structure (7 sections, 25 theorems, 52 equations)

This paper contains 7 sections, 25 theorems, 52 equations.

Table of Contents

  1. Ord-enriched categories
  2. Relations in the 1-dimensional regular context
  3. Relations in the Ord-enriched context
  4. Order ideals and their calculus of relations
  5. Mal'tsev property in the Ord-enriched context
  6. Examples of Ord-Mal'tsev categories
  7. An object-wise approach to Ord-Mal'tsev categories

Key Result

Lemma 1.1

Let $m\colon X\to Y$ and $n\colon Y\to Z$ be morphisms in an $\mathsf{Ord}$-category $\mathbb{C}$. Then:

Theorems & Definitions (69)

  • Lemma 1.1
  • Definition 1.2
  • Lemma 1.3
  • Definition 1.4
  • Lemma 1.5
  • Remark 1.6
  • Remark 1.7
  • Example 1.8
  • Lemma 1.9
  • proof
  • ...and 59 more