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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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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