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An Introduction to Regular Categories

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New Perspectives in Algebra, Topology and Categories

Part of the book series: Coimbra Mathematical Texts ((CMT,volume 1))

Abstract

This paper provides a short introduction to the notion of regular category and its use in categorical algebra. We first prove some of its basic properties, and consider some fundamental algebraic examples. We then analyse the algebraic properties of the categories satisfying the additional Mal’tsev axiom, and then the weaker Goursat axiom. These latter contexts can be seen as the categorical counterparts of the properties of 2-permutability and of 3-permutability of congruences in universal algebra. Mal’tsev and Goursat categories have been intensively studied in the last years: we present here some of their basic properties, which are useful to read more advanced texts in categorical algebra.

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Acknowledgements

A part of the material presented in this survey article is based on [7, 8, 16]. The author is grateful to Tomas Everaert for an important suggestion concerning Theorem 1.16. Many thanks also to Maria Manuel Clementino, Diana Rodelo, Idriss Tchoffo Nguefeu, David Broodryk and the anonymous referee for carefully proofreading a first version of the article and suggesting some useful changes and corrections.

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  1. Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Chemin du Cyclotron 2 bte L7.01.02, 1348, Louvain-la-Neuve, Belgium

    Marino Gran

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Correspondence to Marino Gran .

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Editors and Affiliations

  1. Departamento de Matematica, Universidade de Coimbra, Coimbra, Portugal

    Maria Manuel Clementino

  2. Dipartimento di Matematica, University of Padova, Padua, Italy

    Alberto Facchini

  3. Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Louvain-la-Neuve, Belgium

    Marino Gran

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Gran, M. (2021). An Introduction to Regular Categories. In: Clementino, M.M., Facchini, A., Gran, M. (eds) New Perspectives in Algebra, Topology and Categories. Coimbra Mathematical Texts, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-030-84319-9_4

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